Dataset Viewer
Auto-converted to Parquet Duplicate
statement
stringlengths
1
980
proof
stringlengths
0
46.9k
type
stringclasses
17 values
symbolic_name
stringlengths
1
42
library
stringclasses
1 value
filename
stringclasses
34 values
imports
listlengths
25
84
deps
listlengths
0
64
docstring
stringlengths
0
75
source_url
stringclasses
1 value
commit
stringclasses
1 value
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)). - move=> gT G oddG P A pP /andP[/mulGsubP[_ sPG] _] /andP[sA...
Theorem
odd_p_stable
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "defC", "der1_odd_GL2_charf", "group", "nCG", "odd", "oddG", "p'Q", "p'q", "pA", "pP", "p_pr", "solE", "stable_factor_cent" ]
Theorems A.1-A.3 are essentially inlined in this proof.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(oddG : odd #|G|) (solG : solvable G) (pP : p.-group P).
Hypotheses
oddG
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "group", "odd", "pP", "solG" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(nsPG : P <| G) (sXG : X \subset G).
Hypotheses
nsPG
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(genX : generated_by (p_norm_abelian p P) X).
Hypotheses
genX
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "generated_by", "p_norm_abelian" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
C
:= 'C_G(P).
Let
C
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
defN : 'N_G(P) = G.
Proof. by rewrite (setIidPl _) ?normal_norm. Qed.
Let
defN
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nsCG : C <| G.
Proof. by rewrite -defN subcent_normal. Qed.
Let
nsCG
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "defN" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nCG
:= normal_norm nsCG.
Let
nCG
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "nsCG" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nCX
:= subset_trans sXG nCG.
Let
nCX
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "nCG" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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) //=. rewrite odd_p_stable ?normalM ?pcore_normal //. ...
Theorem
odd_abelian_gen_stable
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "cAA", "defN", "genX", "nCX", "odd_p_stable", "pA" ]
This is B & G, Theorem A.5.1; it does not depend on the solG assumption.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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}defQ solvable_p_constrained //= defQ /pHall pQ indexgg subxx. have pC: p.-group C. apply/p...
Theorem
odd_abelian_gen_constrained
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "coprime_nil_faithful_cent_stab", "defQ", "group", "nCX", "odd_abelian_gen_stable", "p'G1", "p'q", "solvable_p_constrained" ]
This is B & G, Theorem A.5.2.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
"X --> Y"
:= (generated_by (norm_abelian X) Y) (at level 70, no associativity) : group_scope.
Notation
X --> Y
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "generated_by", "norm_abelian" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
Puig_char G : 'L(G) \char G.
Proof. exact: gFchar. Qed.
Lemma
Puig_char
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
center_Puig_char G : 'Z('L(G)) \char G.
Proof. by rewrite !gFchar_trans. Qed.
Lemma
center_Puig_char
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
Puig_succS
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_gen", "Puig_max", "Puig_succ_sub", "norm_abgenS" ]
This is B & G, Lemma B.1(a).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
Puig_sub_even
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_succS" ]
This is part of B & G, Lemma B.1(b) (see also BGsection1.Puig1).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
Puig_sub_odd
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_sub_even", "Puig_succS" ]
This is part of B & G, Lemma B.1(b).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
Puig_sub_even_odd
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig1", "Puig_at_sub", "Puig_succS" ]
This is part of B & G, Lemma B.1(b).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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|] := eqVneq (L2G m.+1) (L2G m); last first. rewrite eq_sym eqEcard Puig_sub_even ?leqnSn // -ltnNge ...
Lemma
Puig_limit
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "PuigS", "Puig_at_sub", "Puig_def", "Puig_inf", "Puig_sub_even" ]
This is B & G, Lemma B.1(c).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
Puig_inf_sub_Puig G : 'L_*(G) \subset 'L(G).
Proof. exact: Puig_sub_even_odd. Qed.
Lemma
Puig_inf_sub_Puig
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_sub_even_odd" ]
BGsection1.Puig_inf_sub.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
abelian_norm_Puig
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "PuigS", "Puig_at_sub", "cAA", "norm_abelian" ]
This is B & G, Lemma B.1(e).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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}(G) by apply: abelian_norm_Puig. by apply: subset_trans (sML); rewrite -defCM se...
Lemma
sub_cent_Puig_at
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "abelian_norm_Puig", "group" ]
This is B & G, Lemma B.1(f), first inclusion.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sub_center_cent_Puig_at n G : 'Z(G) \subset 'C_G('L_{n}(G)).
Proof. by rewrite setIS ?centS ?Puig_at_sub. Qed.
Lemma
sub_center_cent_Puig_at
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_at_sub" ]
This is B & G, Lemma B.1(f), second inclusion.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
sub_cent_Puig_inf
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "group", "sub_cent_Puig_at" ]
This is B & G, Lemma B.1(f), third inclusion (the fourth is trivial).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sub_cent_Puig p G : p.-group G -> 'C_G('L(G)) \subset 'L(G).
Proof. exact: sub_cent_Puig_at. Qed.
Lemma
sub_cent_Puig
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "group", "sub_cent_Puig_at" ]
This is B & G, Lemma B.1(f), fifth inclusion (the sixth is trivial).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
trivg_center_Puig_pgroup
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_sub", "group", "sub_cent_Puig", "sub_center_cent_Puig_at" ]
This is B & G, Lemma B.1(f), final remark (we prove the contrapositive).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
Puig_inf_def
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_limit", "defL" ]
definition of 'L(G) in terms of 'L_*(G).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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[_ <-]] := (leq_maxl kG kH, leq_maxr kG kH). split; do [elim: (maxn _ _) => [|k I...
Lemma
sub_Puig_eq
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig1", "PuigS", "Puig_def", "Puig_gen", "Puig_inf_def", "Puig_limit", "Puig_max", "Puig_sub_even_odd", "Puig_succ_sub", "norm_abgenS", "sHG" ]
This is B & G, Lemma B.2.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Lemma
norm_abgen_pgroup
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "defG", "generated_by", "group", "p_norm_abelian" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(oddG : odd #|G|) (solG : solvable G) (sylS : p.-Sylow(G) S).
Hypotheses
oddG
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "odd", "solG", "sylS" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
p'G1 : 'O_p^'(G) = 1.
Hypothesis
p'G1
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
T
:= 'O_p(G).
Let
T
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nsTG : T <| G
:= pcore_normal _ _.
Let
nsTG
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
pT : p.-group T
:= pcore_pgroup _ _.
Let
pT
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "group" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
pS : p.-group S
:= pHall_pgroup sylS.
Let
pS
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "group", "sylS" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sSG
:= pHall_sub sylS.
Let
sSG
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "sylS" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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 rewrite !Puig1 pcore_sub_Hall. have{sL1} gL: 'L_{k.*2.+1}(T) --> 'L_{k.*2.+2}...
Lemma
pcore_Sylow_Puig_sub
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig1", "PuigS", "Puig_at_sub", "Puig_gen", "Puig_limit", "Puig_max", "Puig_succS", "Puig_succ_sub", "group", "norm_abgenS", "norm_abgen_pgroup", "nsTG", "odd_abelian_gen_constrained", "pS", "pT", "sSG", "sub_cent_Puig_at" ]
This is B & G, Lemma B.3.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
Y
:= 'Z('L(T)).
Let
Y
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
L
:= 'L(S).
Let
L
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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 ?Puig_sub_even_odd. apply: ...
Theorem
Puig_center_normal
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_gen", "Puig_sub", "Puig_sub_even_odd", "abelian_norm_Puig", "defC", "defG", "norm_abgenS", "norm_abgen_pgroup", "nsCG", "odd_abelian_gen_stable", "pS", "pT", "pcore_Sylow_Puig_sub", "sLG", "sSG", "sub_Puig_eq", "sub_cent_Puig_at", "sylS" ]
This is B & G, Theorem B.4(b).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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 = 'Z('L(S / D)). rewrite !quotientE -(setIid S) -(morph...
Theorem
Puig_factorization
Root
theories/BGappendixAB.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "ssralg", "zmodp", "matrix", "mxalgebra", "center", "gfunctor", "commutator", "gseries", "pgroup"...
[ "Puig_center_normal", "Puig_sub", "group", "pS", "sSG", "sSN", "sylS" ]
This is B & G, Theorem B.4(a).
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nU
:= ((p ^ q).-1 %/ p.-1)%N.
Let
nU
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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}.
Variant
finFieldImage
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "sigma" ]
External statement of the finite field assumption.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(pr_p : prime p) (pr_q : prime q) (coUp1 : coprime nU p.-1).
Hypotheses
pr_p
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "nU", "pr_q" ]
These correspond to hypothesis (A) of B & G, Appendix C, Theorem C.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(defH : P ><| U = H) (fieldH : finFieldImage).
Hypotheses
defH
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "finFieldImage" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(oP : #|P| = (p ^ q)%N) (oU : #|U| = nU).
Hypotheses
oP
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "nU", "oU" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(abelQ : q.-abelem Q) (nQP0 : P0 \subset 'N(Q)).
Hypotheses
abelQ
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
These correspond to hypothesis (B) of B & G, Appendix C, Theorem C.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nU_P0Q : exists2 y, y \in Q & P0 :^ y \subset 'N(U).
Hypothesis
nU_P0Q
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
ltqp : (q < p)%N.
Hypothesis
ltqp
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
Negation of the goal of B & G, Appendix C, Theorem C.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
F
:= (PrimeCharType charFp).
Notation
F
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
galF
:= [the splittingFieldType 'F_p of F].
Notation
galF
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
Fpq : {vspace F}
:= fullv.
Let
Fpq
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
Fp : {vspace F}
:= 1%VS.
Let
Fp
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
oF : #|F| = (p ^ q)%N.
Hypothesis
oF
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
oF_p : #|'F_p| = p.
Proof. exact: card_Fp. Qed.
Let
oF_p
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
oFp : #|Fp| = p.
Proof. by rewrite (@card_vspace1 _ _ (Falgebra.class (PrimeCharType _))). Qed.
Let
oFp
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "Fp" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
oFpq : #|Fpq| = (p ^ q)%N.
Proof. by rewrite card_vspacef. Qed.
Let
oFpq
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "Fpq" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
dimFpq : \dim Fpq = q.
Proof. by rewrite primeChar_dimf oF pfactorK. Qed.
Let
dimFpq
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "Fpq", "oF" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(inj_sigma : 'injm sigma) (inj_sigmaU : 'injm sigmaU).
Hypotheses
inj_sigma
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "sigma" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
im_sigma : sigma @* P = [set: F].
Hypothesis
im_sigma
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "sigma" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
(sP0P : P0 \subset P) (sigma_s : sigma s = 1) (defP0 : <[s]> = P0).
Hypotheses
sP0P
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "sigma" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
psi u : F
:= val (sigmaU u).
Let
psi
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
inj_psi : {in U &, injective psi}.
Proof. by move=> u v Uu Uv /val_inj/(injmP inj_sigmaU)->. Qed.
Let
inj_psi
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sigmaJ : {in P & U, forall x u, sigma (x ^ u) = sigma x * psi u}.
Hypothesis
sigmaJ
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi", "sigma" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
Ps : s \in P.
Proof. by rewrite -cycle_subG defP0. Qed.
Let
Ps
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
P0s : s \in P0.
Proof. by rewrite -defP0 cycle_id. Qed.
Let
P0s
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nz_psi u : psi u != 0.
Proof. by rewrite -unitfE (valP (sigmaU u)). Qed.
Let
nz_psi
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sigma1 : sigma 1%g = 0.
Proof. exact: morph1. Qed.
Let
sigma1
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "sigma" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sigmaM : {in P &, {morph sigma : x1 x2 / (x1 * x2)%g >-> x1 + x2}}.
Proof. exact: morphM. Qed.
Let
sigmaM
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "sigma", "x1", "x2" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sigmaV : {in P, {morph sigma : x / x^-1%g >-> - x}}.
Proof. exact: morphV. Qed.
Let
sigmaV
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "sigma" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sigmaX n : {in P, {morph sigma : x / (x ^+ n)%g >-> x *+ n}}.
Proof. exact: morphX. Qed.
Let
sigmaX
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "sigma" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
psi1 : psi 1%g = 1.
Proof. by rewrite /psi morph1. Qed.
Let
psi1
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
psiM : {in U &, {morph psi : u1 u2 / (u1 * u2)%g >-> u1 * u2}}.
Proof. by move=> u1 u2 Uu1 Uu2; rewrite /psi morphM. Qed.
Let
psiM
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
psiV : {in U, {morph psi : u / u^-1%g >-> u^-1}}.
Proof. by move=> u Uu; rewrite /psi morphV. Qed.
Let
psiV
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
psiX n : {in U, {morph psi : u / (u ^+ n)%g >-> u ^+ n}}.
Proof. by move=> u Uu; rewrite /psi morphX // val_unitX. Qed.
Let
psiX
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
sigmaE
:= (sigma1, sigma_s, mulr1, mul1r, (sigmaJ, sigmaX, sigmaM, sigmaV), (psi1, psiX, psiM, psiV)).
Let
sigmaE
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi1", "psiM", "psiV", "psiX", "sigma1", "sigmaJ", "sigmaM", "sigmaV", "sigmaX" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
psiE u : u \in U -> psi u = sigma (s ^ u).
Proof. by move=> Uu; rewrite !sigmaE. Qed.
Let
psiE
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "psi", "sigma", "sigmaE" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nPU : U \subset 'N(P).
Proof. by have [] := sdprodP defH. Qed.
Let
nPU
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "defH" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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
memJ_P
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "nPU" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
in_PU
:= (memJ_P, in_group).
Let
in_PU
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "memJ_P" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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
sigmaP0
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "Fp", "sigma" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
nt_s : s != 1%g.
Proof. by rewrite -(morph_injm_eq1 inj_sigma) // sigmaE oner_eq0. Qed.
Let
nt_s
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "inj_sigma", "sigmaE" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
p_gt0 : (0 < p)%N.
Proof. exact: prime_gt0. Qed.
Let
p_gt0
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
q_gt0 : (0 < q)%N.
Proof. exact: prime_gt0. Qed.
Let
q_gt0
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
p1_gt0 : (0 < p.-1)%N.
Proof. by rewrite -subn1 subn_gt0 prime_gt1. Qed.
Let
p1_gt0
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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.
Let
not_dvd_q_p1
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "nU", "p_gt0" ]
This is B & G, Appendix C, Remark I.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
odd_p : odd p.
Proof. by apply: contraLR ltqp => /prime_oddPn-> //; rewrite -leqNgt prime_gt1. Qed.
Let
odd_p
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "ltqp", "odd" ]
This is the first assertion of B & G, Appendix C, Remark V.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
odd_q : odd q.
Proof. apply: contraR not_dvd_q_p1 => /prime_oddPn-> //. by rewrite -subn1 dvdn2 oddB ?odd_p. Qed.
Let
odd_q
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "not_dvd_q_p1", "odd", "odd_p" ]
This is the second assertion of B & G, Appendix C, Remark V.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
qgt2 : (2 < q)%N.
Proof. by rewrite odd_prime_gt2. Qed.
Let
qgt2
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
pgt4 : (4 < p)%N.
Proof. by rewrite odd_geq ?(leq_ltn_trans qgt2). Qed.
Let
pgt4
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "qgt2" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
qgt1 : (1 < q)%N.
Proof. exact: ltnW. Qed.
Let
qgt1
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
Nm
:= (galNorm Fp Fpq).
Notation
Nm
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "Fp", "Fpq" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
uval
:= (@FinRing.uval _).
Notation
uval
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
cycFU (FU : {group {unit F}}) : cyclic FU.
Proof. exact: field_unit_group_cyclic. Qed.
Let
cycFU
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "group" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
cUU : abelian U.
Proof. by rewrite cyclic_abelian // -(injm_cyclic inj_sigmaU) ?cycFU. Qed.
Let
cUU
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "cycFU" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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 <[u0 ^+ (#[u0] %/ nU)]>%G]. rewrite -cycle_sub_g...
Let
im_psi
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "Du", "Fp", "Nm", "alpha", "cycFU", "defFU", "nU", "oF", "oF_p", "oU", "psi", "uval", "x1" ]
This is B & G, Appendix C, Remark VII.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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 rmorphM. pose galFpU := Morphism fM @* [set: {unit 'F_p}]. have ->:...
Let
defFU
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "Du", "Fp", "cycFU", "nU", "oF", "oF_p", "uval" ]
This is B & G, Appendix C, Remark VIII.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
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/subsetP=> x /setIP[Px /cent1P/comm...
Let
frobH
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "im_sigma", "in_PU", "inj_sigma", "oU", "sigmaE" ]
This is B & G, Appendix C, Remark IX.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
p'q : q != p.
Proof. by rewrite ltn_eqF. Qed.
Let
p'q
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[]
of part (B) of the assumptions of Theorem C.
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
cQQ : abelian Q.
Proof. exact: abelem_abelian abelQ. Qed.
Let
cQQ
Root
theories/BGappendixC.v
[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "prime", "finset", "binomial", "order", "fingroup", "morphism", "automorphism", "quotient", "action", "gproduct", "ssralg", "finalg", "zm...
[ "abelQ" ]
https://github.com/math-comp/odd-order
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f
End of preview. Expand in Data Studio

Coq-OddOrder

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

Source

Schema

Column Type Description
statement string Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof string Verbatim proof/body, empty if the declaration has none
type string Declaration keyword
symbolic_name string Declaration identifier
library string Sub-library
filename string Repository-relative source path
imports list[string] File-level Require/Import modules
deps list[string] Intra-corpus identifiers referenced
docstring string Preceding documentation comment, empty if absent
source_url string Upstream repository
commit string Upstream commit extracted

Statistics

  • Entries: 2,457
  • With proof: 2,323 (94.5%)
  • With docstring: 680 (27.7%)
  • Libraries: 1

By type

Type Count
Lemma 769
Let 702
Notation 332
Definition 172
Theorem 120
Hypothesis 72
Hypotheses 61
Proposition 54
Corollary 49
Remark 47
Fact 23
Canonical 21
Inductive 17
Fixpoint 7
Variant 6
Coercion 4
Record 1

Example

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.
  • type: Lemma | symbolic_name: Puig_succS | theories/BGappendixAB.v

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{coq_oddorder_dataset,
  title  = {Coq-OddOrder},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/math-comp/odd-order, commit 6afa795b9018},
  url    = {https://huggingface.co/datasets/phanerozoic/Coq-OddOrder}
}
Downloads last month
65

Collection including phanerozoic/Coq-OddOrder