Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
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
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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 |
Structured dataset from odd-order — Formal proof of the Feit-Thompson Odd Order Theorem.
6afa795b9018c64ab5c7cd2f9b3c9ab5dd45d93f| 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 |
| 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 |
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.
Puig_succS | theories/BGappendixAB.vEach 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.
@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}
}