Datasets:

phanerozoic
/

Coq-OddOrder

fact stringlengths 10 44.1k	type stringclasses 15 values	library stringclasses 1 value	imports listlengths 5 17	filename stringclasses 34 values	symbolic_name stringlengths 1 42
odd_p_stable gT p (G : {group gT}) : odd #\|G\| -> p.-stable G. Proof. move: gT G. pose p_xp gT (E : {group gT}) x := p.-elt x && (x \in 'C([~: E, [set x]])). suffices IH gT (E : {group gT}) x y (G := <<[set x; y]>>) : [&& odd #\|G\|, p.-group E & G \subset 'N(E)] -> p_xp gT E x && p_xp gT E y -> p.-group (G / 'C(E)). - mo...	Theorem	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	odd_p_stable
C := 'C_G(P).	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	C
defN : 'N_G(P) = G. Proof. by rewrite (setIidPl _) ?normal_norm. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	defN
nsCG : C <\| G. Proof. by rewrite -defN subcent_normal. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	nsCG
nCG := normal_norm nsCG.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	nCG
nCX := subset_trans sXG nCG. (* This is B & G, Theorem A.5.1; it does not depend on the solG assumption. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	nCX
odd_abelian_gen_stable : X / C \subset 'O_p(G / C). Proof. case/exists_eqP: genX => gX defX. rewrite -defN sub_quotient_pre // -defX gen_subG. apply/bigcupsP=> A gX_A; have [_ pA nAP cAA] := and4P gX_A. have{gX_A} sAX: A \subset X by rewrite -defX sub_gen ?bigcup_sup. rewrite -sub_quotient_pre ?(subset_trans sAX nCX) /...	Theorem	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	odd_abelian_gen_stable
odd_abelian_gen_constrained : 'O_p^'(G) = 1 -> 'C_('O_p(G))(P) \subset P -> X \subset 'O_p(G). Proof. set Q := 'O_p(G) => p'G1 sCQ_P. have sPQ: P \subset Q by rewrite pcore_max. have defQ: 'O_{p^', p}(G) = Q by rewrite pseries_pop2. have pQ: p.-group Q by apply: pcore_pgroup. have sCQ: 'C_G(Q) \subset Q. by rewrite -{2...	Theorem	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	odd_abelian_gen_constrained
Puig_char G : 'L(G) \char G. Proof. exact: gFchar. Qed.	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_char
center_Puig_char G : 'Z('L(G)) \char G. Proof. by rewrite !gFchar_trans. Qed. (* This is B & G, Lemma B.1(a). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	center_Puig_char
Puig_succS G D E : D \subset E -> 'L_[G](E) \subset 'L_[G](D). Proof. move=> sDE; apply: Puig_max (Puig_succ_sub _ _). exact: norm_abgenS sDE (Puig_gen _ _). Qed. (* This is part of B & G, Lemma B.1(b) (see also BGsection1.Puig1). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_succS
Puig_sub_even m n G : m <= n -> 'L_{m.2}(G) \subset 'L_{n.2}(G). Proof. move/subnKC <-; move: {n}(n - m)%N => n. by elim: m => [\|m IHm] /=; rewrite ?sub1G ?Puig_succS. Qed. (* This is part of B & G, Lemma B.1(b). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_sub_even
Puig_sub_odd m n G : m <= n -> 'L_{n.2.+1}(G) \subset 'L_{m.2.+1}(G). Proof. by move=> le_mn; rewrite Puig_succS ?Puig_sub_even. Qed. (* This is part of B & G, Lemma B.1(b). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_sub_odd
Puig_sub_even_odd m n G : 'L_{m.2}(G) \subset 'L_{n.2.+1}(G). Proof. elim: n m => [\|n IHn] m; first by rewrite Puig1 Puig_at_sub. by case: m => [\|m]; rewrite ?sub1G ?Puig_succS ?IHn. Qed. (* This is B & G, Lemma B.1(c). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_sub_even_odd
Puig_limit G : exists m, forall k, m <= k -> 'L_{k.2}(G) = 'L_(G) /\ 'L_{k.2.+1}(G) = 'L(G). Proof. pose L2G m := 'L_{m.2}(G); pose n := #\|G\|. have []: #\|L2G n\| <= n /\ n <= n by rewrite subset_leq_card ?Puig_at_sub. elim: {1 2 3}n => [\| m IHm leLm1 /ltnW]; first by rewrite leqNgt cardG_gt0. have [eqLm le_mn\|] := e...	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_limit
Puig_inf_sub_Puig G : 'L_(G) \subset 'L(G). Proof. exact: Puig_sub_even_odd. Qed. ( This is B & G, Lemma B.1(e). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_inf_sub_Puig
abelian_norm_Puig n G A : n > 0 -> abelian A -> A <\| G -> A \subset 'L_{n}(G). Proof. case: n => // n _ cAA /andP[sAG nAG]. rewrite PuigS sub_gen // bigcup_sup // inE sAG /norm_abelian cAA andbT. exact: subset_trans (Puig_at_sub n G) nAG. Qed. (* This is B & G, Lemma B.1(f), first inclusion. *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	abelian_norm_Puig
sub_cent_Puig_at n p G : n > 0 -> p.-group G -> 'C_G('L_{n}(G)) \subset 'L_{n}(G). Proof. move=> n_gt0 pG. have /ex_maxgroup[M /(max_SCN pG)SCN_M]: exists M, (gval M <\| G) && abelian M. by exists 1%G; rewrite normal1 abelian1. have{SCN_M} [cMM [nsMG defCM]] := (SCN_abelian SCN_M, SCN_P SCN_M). have sML: M \subset 'L_{n...	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	sub_cent_Puig_at
sub_center_cent_Puig_at n G : 'Z(G) \subset 'C_G('L_{n}(G)). Proof. by rewrite setIS ?centS ?Puig_at_sub. Qed. (* This is B & G, Lemma B.1(f), third inclusion (the fourth is trivial). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	sub_center_cent_Puig_at
sub_cent_Puig_inf p G : p.-group G -> 'C_G('L_(G)) \subset 'L_(G). Proof. by apply: sub_cent_Puig_at; rewrite double_gt0. Qed. (* This is B & G, Lemma B.1(f), fifth inclusion (the sixth is trivial). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	sub_cent_Puig_inf
sub_cent_Puig p G : p.-group G -> 'C_G('L(G)) \subset 'L(G). Proof. exact: sub_cent_Puig_at. Qed. (* This is B & G, Lemma B.1(f), final remark (we prove the contrapositive). *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	sub_cent_Puig
trivg_center_Puig_pgroup p G : p.-group G -> 'Z('L(G)) = 1 -> G :=: 1. Proof. move=> pG LG1; apply/(trivg_center_pgroup pG)/trivgP. rewrite -(trivg_center_pgroup (pgroupS (Puig_sub _) pG) LG1). by apply: subset_trans (sub_cent_Puig pG); apply: sub_center_cent_Puig_at. Qed. (* This is B & G, Lemma B.1(g), second part; t...	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	trivg_center_Puig_pgroup
Puig_inf_def G : 'L_(G) = 'L_[G]('L(G)). Proof. have [k defL] := Puig_limit G. by case: (defL k) => // _ <-; case: (defL k.+1) => [\|<- //]; apply: leqnSn. Qed. ( This is B & G, Lemma B.2. *)	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_inf_def
sub_Puig_eq G H : H \subset G -> 'L(G) \subset H -> 'L(H) = 'L(G). Proof. move=> sHG sLG_H; apply/setP/subset_eqP/andP. have sLH_G := subset_trans (Puig_succ_sub _ _) sHG. have gPuig := norm_abgenS _ (Puig_gen _ _). have [[kG defLG] [kH defLH]] := (Puig_limit G, Puig_limit H). have [/defLG[_ {1}<-] /defLH[_ <-]] := (le...	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	sub_Puig_eq
norm_abgen_pgroup p X G : p.-group G -> X --> G -> generated_by (p_norm_abelian p X) G. Proof. move=> pG /exists_eqP[gG defG]. have:= subxx G; rewrite -{1 3}defG gen_subG /= => /bigcupsP-sGG. apply/exists_eqP; exists gG; congr <<_>>; apply: eq_bigl => A. by rewrite andbA andbAC andb_idr // => /sGG/pgroupS->. Qed. Varia...	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	norm_abgen_pgroup
T := 'O_p(G).	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	T
nsTG : T <\| G := pcore_normal _ _.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	nsTG
pT : p.-group T := pcore_pgroup _ _.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	pT
pS : p.-group S := pHall_pgroup sylS.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	pS
sSG := pHall_sub sylS. (* This is B & G, Lemma B.3. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	sSG
pcore_Sylow_Puig_sub : 'L_(S) \subset 'L_(T) /\ 'L(T) \subset 'L(S). Proof. have [[kS defLS] [kT defLT]] := (Puig_limit S, Puig_limit [group of T]). have [/defLS[<- <-] /defLT[<- <-]] := (leq_maxl kS kT, leq_maxr kS kT). have sL_ := subset_trans (Puig_succ_sub _ _). elim: (maxn kS kT) => [\|k [_ sL1]]; first by rewrit...	Lemma	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	pcore_Sylow_Puig_sub
Y := 'Z('L(T)).	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Y
L := 'L(S). (* This is B & G, Theorem B.4(b). *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	L
Puig_center_normal : 'Z(L) <\| G. Proof. have [sLiST sLTS] := pcore_Sylow_Puig_sub. have sLiLT: 'L_(T) \subset 'L(T) by apply: Puig_sub_even_odd. have sZY: 'Z(L) \subset Y. rewrite subsetI andbC subIset ?centS ?orbT //=. suffices: 'C_S('L_(S)) \subset 'L(T). by apply: subset_trans; rewrite setISS ?Puig_sub ?centS ?Pui...	Theorem	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_center_normal
Puig_factorization : 'O_p^'(G) * 'N_G('Z('L(S))) = G. Proof. set D := 'O_p^'(G); set Z := 'Z(_); have [sSG pS _] := and3P sylS. have sSN: S \subset 'N(D) by rewrite (subset_trans sSG) ?gFnorm. have p'D: p^'.-group D := pcore_pgroup _ _. have tiSD: S :&: D = 1 := coprime_TIg (pnat_coprime pS p'D). have def_Zq: Z / D = '...	Theorem	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import fintype bigop prime finset.", "From mathcomp Require Import fingroup morphism automorphism quotient.", "From mathcomp Require Import ssralg zmodp matrix mxalgebra.", "From mathcomp Require Import ce...	theories/BGappendixAB.v	Puig_factorization
nU := ((p ^ q).-1 %/ p.-1)%N. (* External statement of the finite field assumption. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	nU
finFieldImage : Prop := FinFieldImage (F : finFieldType) (sigma : {morphism P >-> F}) of isom P [set: F] sigma & sigma @^-1 <[1%R : F]> = P0 & exists2 sigmaU : {morphism U >-> {unit F}}, 'injm sigmaU & {in P & U, morph_act 'J 'U sigma sigmaU}. ( These correspond to hypothesis (A) of B & G, Appendix C, Theorem C. *) H...	Variant	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	finFieldImage
Fpq : {vspace F} := fullv.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	Fpq
Fp : {vspace F} := 1%VS. Hypothesis oF : #\|F\| = (p ^ q)%N.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	Fp
oF_p : #\|'F_p\| = p. Proof. exact: card_Fp. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	oF_p
oFp : #\|Fp\| = p. Proof. by rewrite (@card_vspace1 _ _ (Falgebra.class (PrimeCharType _))). Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	oFp
oFpq : #\|Fpq\| = (p ^ q)%N. Proof. by rewrite card_vspacef. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	oFpq
dimFpq : \dim Fpq = q. Proof. by rewrite primeChar_dimf oF pfactorK. Qed. Variables (sigma : {morphism P >-> F}) (sigmaU : {morphism U >-> {unit F}}). Hypotheses (inj_sigma : 'injm sigma) (inj_sigmaU : 'injm sigmaU). Hypothesis im_sigma : sigma @* P = [set: F]. Variable s : gT. Hypotheses (sP0P : P0 \subset P) (sigma_s...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	dimFpq
psi u : F := val (sigmaU u).	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	psi
inj_psi : {in U &, injective psi}. Proof. by move=> u v Uu Uv /val_inj/(injmP inj_sigmaU)->. Qed. Hypothesis sigmaJ : {in P & U, forall x u, sigma (x ^ u) = sigma x * psi u}.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	inj_psi
Ps : s \in P. Proof. by rewrite -cycle_subG defP0. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	Ps
P0s : s \in P0. Proof. by rewrite -defP0 cycle_id. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	P0s
nz_psi u : psi u != 0. Proof. by rewrite -unitfE (valP (sigmaU u)). Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	nz_psi
sigma1 : sigma 1%g = 0. Proof. exact: morph1. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	sigma1
sigmaM : {in P &, {morph sigma : x1 x2 / (x1 * x2)%g >-> x1 + x2}}. Proof. exact: morphM. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	sigmaM
sigmaV : {in P, {morph sigma : x / x^-1%g >-> - x}}. Proof. exact: morphV. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	sigmaV
sigmaX n : {in P, {morph sigma : x / (x ^+ n)%g >-> x *+ n}}. Proof. exact: morphX. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	sigmaX
psi1 : psi 1%g = 1. Proof. by rewrite /psi morph1. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	psi1
psiM : {in U &, {morph psi : u1 u2 / (u1 * u2)%g >-> u1 * u2}}. Proof. by move=> u1 u2 Uu1 Uu2; rewrite /psi morphM. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	psiM
psiV : {in U, {morph psi : u / u^-1%g >-> u^-1}}. Proof. by move=> u Uu; rewrite /psi morphV. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	psiV
psiX n : {in U, {morph psi : u / (u ^+ n)%g >-> u ^+ n}}. Proof. by move=> u Uu; rewrite /psi morphX // val_unitX. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	psiX
sigmaE := (sigma1, sigma_s, mulr1, mul1r, (sigmaJ, sigmaX, sigmaM, sigmaV), (psi1, psiX, psiM, psiV)).	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	sigmaE
psiE u : u \in U -> psi u = sigma (s ^ u). Proof. by move=> Uu; rewrite !sigmaE. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	psiE
nPU : U \subset 'N(P). Proof. by have [] := sdprodP defH. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	nPU
memJ_P : {in P & U, forall x u, x ^ u \in P}. Proof. by move=> x u Px Uu; rewrite /= memJ_norm ?(subsetP nPU). Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	memJ_P
in_PU := (memJ_P, in_group).	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	in_PU
sigmaP0 : sigma @* P0 =i Fp. Proof. rewrite -defP0 morphim_cycle // sigma_s => x. apply/cycleP/vlineP=> [] [n ->]; first by exists n%:R; rewrite scaler_nat. by exists (val n); rewrite -{1}[n]natr_Zp -in_algE rmorph_nat zmodXgE. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	sigmaP0
nt_s : s != 1%g. Proof. by rewrite -(morph_injm_eq1 inj_sigma) // sigmaE oner_eq0. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	nt_s
p_gt0 : (0 < p)%N. Proof. exact: prime_gt0. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	p_gt0
q_gt0 : (0 < q)%N. Proof. exact: prime_gt0. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	q_gt0
p1_gt0 : (0 < p.-1)%N. Proof. by rewrite -subn1 subn_gt0 prime_gt1. Qed. (* This is B & G, Appendix C, Remark I. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	p1_gt0
not_dvd_q_p1 : ~~ (q %\| p.-1)%N. Proof. rewrite -prime_coprime // -[q]card_ord -sum1_card -coprime_modl -modn_summ. have:= coUp1; rewrite /nU predn_exp mulKn //= -coprime_modl -modn_summ. congr (coprime (_ %% _) _); apply: eq_bigr => i _. by rewrite -{1}[p](subnK p_gt0) subn1 -modnXm modnDl modnXm exp1n. Qed. (* This i...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	not_dvd_q_p1
odd_p : odd p. Proof. by apply: contraLR ltqp => /prime_oddPn-> //; rewrite -leqNgt prime_gt1. Qed. (* This is the second assertion of B & G, Appendix C, Remark V. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	odd_p
odd_q : odd q. Proof. apply: contraR not_dvd_q_p1 => /prime_oddPn-> //. by rewrite -subn1 dvdn2 oddB ?odd_p. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	odd_q
qgt2 : (2 < q)%N. Proof. by rewrite odd_prime_gt2. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	qgt2
pgt4 : (4 < p)%N. Proof. by rewrite odd_geq ?(leq_ltn_trans qgt2). Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	pgt4
qgt1 : (1 < q)%N. Proof. exact: ltnW. Qed. Local Notation Nm := (galNorm Fp Fpq). Local Notation uval := (@FinRing.uval _).	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	qgt1
cycFU (FU : {group {unit F}}) : cyclic FU. Proof. exact: field_unit_group_cyclic. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	cycFU
cUU : abelian U. Proof. by rewrite cyclic_abelian // -(injm_cyclic inj_sigmaU) ?cycFU. Qed. (* This is B & G, Appendix C, Remark VII. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	cUU
im_psi (x : F) : (x \in psi @: U) = (Nm x == 1). Proof. have /cyclicP[u0 defFU]: cyclic [set: {unit F}] by apply: cycFU. have o_u0: #[u0] = (p ^ q).-1 by rewrite orderE -defFU card_finField_unit oF. have ->: psi @: U = uval @: (sigmaU @* U) by rewrite morphimEdom -imset_comp. have /set1P[->]: (sigmaU @* U)%G \in [set <...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	im_psi
defFU : sigmaU @* U \x [set u \| uval u \in Fp] = [set: {unit F}]. Proof. have fP v: in_alg F (uval v) \is a GRing.unit by rewrite rmorph_unit ?(valP v). pose f (v : {unit 'F_p}) := FinRing.unit F (fP v). have fM: {in setT &, {morph f: v1 v2 / (v1 * v2)%g}}. by move=> v1 v2 _ _; apply: val_inj; rewrite /= -1?in_algE rmo...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	defFU
frobH : [Frobenius H = P ><\| U]. Proof. apply/Frobenius_semiregularP=> // [\|\|u /setD1P[ntu Uu]]. - by rewrite -(morphim_injm_eq1 inj_sigma) // im_sigma finRing_nontrivial. - rewrite -cardG_gt1 oU ltn_divRL ?dvdn_pred_predX // mul1n -!subn1. by rewrite ltn_sub2r ?(ltn_exp2l 0) ?(ltn_exp2l 1) ?prime_gt1. apply/trivgP/sub...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	frobH
p'q : q != p. Proof. by rewrite ltn_eqF. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	p'q
cQQ : abelian Q. Proof. exact: abelem_abelian abelQ. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	cQQ
p'Q : p^'.-group Q. Proof. exact: pi_pgroup (abelem_pgroup abelQ) _. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	p'Q
pP : p.-group P. Proof. by rewrite /pgroup oP pnatX ?pnat_id. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	pP
coQP : coprime #\|Q\| #\|P\|. Proof. exact: p'nat_coprime p'Q pP. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	coQP
sQP0Q : [~: Q, P0] \subset Q. Proof. by rewrite commg_subl. Qed. (* This is B & G, Appendix C, Remark X. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	sQP0Q
defQ : 'C_Q(P0) \x [~: Q, P0] = Q. Proof. by rewrite dprodC coprime_abelian_cent_dprod // (coprimegS sP0P). Qed. (* This is B & G, Appendix C, Remark XI. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	defQ
nU_P0QP0 : exists2 y, y \in [~: Q, P0] & P0 :^ y \subset 'N(U). Proof. have [_ /(mem_dprod defQ)[z [y [/setIP[_ cP0z] QP0y -> _]]]] := nU_P0Q. by rewrite conjsgM (normsP (cent_sub P0)) //; exists y. Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	nU_P0QP0
E := [set x : galF \| Nm x == 1 & Nm (2 - x) == 1].	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	E
E_1 : 1 \in E. Proof. by rewrite !inE -addrA subrr addr0 galNorm1 eqxx. Qed. (* This is B & G, Appendix C, Lemma C.1. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	E_1
Einv_gt1_le_pq : E = [set x^-1 \| x in E] -> (1 < #\|E\|)%N -> (p <= q)%N. Proof. rewrite (cardsD1 1) E_1 ltnS card_gt0 => Einv /set0Pn[/= a /setD1P[not_a1 Ea]]. pose tau (x : F) := (2 - x)^-1. have Etau x: x \in E -> tau x \in E. rewrite inE => Ex; rewrite Einv (imset_f (fun y => y^-1)) //. by rewrite inE andbC opprD add...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	Einv_gt1_le_pq
E_gt1 : (1 < #\|E\|)%N. Proof. have [q_gt4 \| q_le4] := ltnP 4 q. pose inK x := enum_rank_in (classes1 H) (x ^: H). have inK_E x: x \in H -> enum_val (inK x) = x ^: H. by move=> Hx; rewrite enum_rankK_in ?mem_classes. pose j := inK s; pose k := inK (s ^+ 2)%g; pose e := gring_classM_coef j j k. have cPP: abelian P by rewr...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	E_gt1
Qy : y \in Q. Proof. by rewrite (subsetP sQP0Q). Qed.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	Qy
t := s ^ y.	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	t
P1 := P0 :^ y. (* This is B & G, Appendix C, Lemma C.3, Step 1. *)	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	P1
splitH x : x \in H -> exists2 u, u \in U & exists2 v, v \in U & exists2 s1, s1 \in P0 & x = u * s1 * v. Proof. case/(mem_sdprod defH) => z [v [Pz Uv -> _]]. have [-> \| nt_z] := eqVneq z 1. by exists 1 => //; exists v => //; exists 1; rewrite ?mulg1. have nz_z: sigma z != 0 by rewrite (morph_injm_eq1 inj_sigma). have /(...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	splitH
not_splitU s1 s2 u : s1 \in P0 -> s2 \in P0 -> u \in U -> s1 * u * s2 \in U -> (s1 == 1) && (s2 == 1) \|\| (u == 1) && (s1 * s2 == 1). Proof. move=> P0s1 P0s2 Uu; have [_ _ _ tiPU] := sdprodP defH. have [Ps1 Ps2]: s1 \in P /\ s2 \in P by rewrite !(subsetP sP0P). have [-> \| nt_s1 /=] := altP (s1 =P 1). by rewrite mul1g gr...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	not_splitU
tiH_P1 t1 : t1 \in P1^# -> H :&: H :^ t1 = U. Proof. case/setD1P=>[nt_t1 P1t1]; set X := H :&: _. have [nsPH sUH _ _ tiPU] := sdprod_context defH. have sUX: U \subset X. by rewrite subsetI sUH -(normsP nUP0y t1 P1t1) conjSg. have defX: (P :&: X) * U = X. by rewrite setIC group_modr // (sdprodW defH) setIAC setIid. have...	Let	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	tiH_P1
BGappendixC3_Ediv : E = [set x^-1 \| x in E]%R. Proof. suffices sEV_E: [set x^-1 \| x in E]%R \subset E. by apply/esym/eqP; rewrite eqEcard sEV_E card_imset //=; apply: invr_inj. have /mulG_sub[/(subset_trans sP0P)/subsetP-sP0H /subsetP-sUH] := sdprodW defH. have Hs := sP0H s P0s; have P1t: t \in P1 by rewrite memJ_conjg...	Fact	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	BGappendixC3_Ediv
BGappendixC_inner_subproof : (p <= q)%N. Proof. have [y QP0y nUP0y] := nU_P0QP0. by apply: Einv_gt1_le_pq E_gt1; apply: BGappendixC3_Ediv nUP0y. Qed.	Fact	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	BGappendixC_inner_subproof
prime_dim_normed_finField : (p <= q)%N. Proof. apply: wlog_neg; rewrite -ltnNge => ltqp. have [F sigma /isomP[inj_sigma im_sigma] defP0] := fieldH. case=> sigmaU inj_sigmaU sigmaJ. have oF: #\|F\| = (p ^ q)%N by rewrite -cardsT -im_sigma card_injm. have charFp: p \in [char F] := card_finCharP oF pr_p. have sP0P: P0 \subs...	Theorem	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div.", "From mathcomp Require Import choice fintype tuple finfun bigop prime finset.", "From mathcomp Require Import binomial order.", "From mathcomp Require Import fingroup morphism automorphism quotient action.", "From mathcomp Requi...	theories/BGappendixC.v	prime_dim_normed_finField
plength_1 p (G : {set gT}) := 'O_{p^', p, p^'}(G) == G.	Definition	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.", "From mathcomp Require Import div fintype bigop prime finset binomial.", "From mathcomp Require Import fingroup morphism perm automorphism quotient.", "From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.", ...	theories/BGsection1.v	plength_1
p_elt_gen p (G : {set gT}) := <<[set x in G \| p.-elt x]>>.	Definition	theories	[ "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.", "From mathcomp Require Import div fintype bigop prime finset binomial.", "From mathcomp Require Import fingroup morphism perm automorphism quotient.", "From mathcomp Require Import action gproduct ssralg finalg zmodp matrix.", ...	theories/BGsection1.v	p_elt_gen

End of preview. Expand in Data Studio

Coq-OddOrder

Structured dataset from odd-order — Formal proof of the Feit-Thompson Odd Order Theorem.

2,201 declarations extracted from Coq source files.

Applications

Training language models on formal proofs
Fine-tuning theorem provers
Retrieval-augmented generation for proof assistants
Learning proof embeddings and representations

Source

Repository: https://github.com/math-comp/odd-order

Schema

Column	Type	Description
fact	string	Declaration body
type	string	Lemma, Definition, Theorem, etc.
library	string	Source module
imports	list	Required imports
filename	string	Source file path
symbolic_name	string	Identifier

Downloads last month: 6

Collection including phanerozoic/Coq-OddOrder

Proof Assistant Projects

Collection

Digesting proof assistant libraries for AI ingestion. • 84 items • Updated Jan 15 • 3