statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
"''Ind'" | := 'Ind[_] (only parsing) : ring_scope. | Notation | ''Ind' | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIndMorph (f : {morphism D >-> rT}) (phi : 'CF(f @* H)) :
'ker f \subset H -> H \subset G -> G \subset D ->
'Ind[G] (cfMorph phi) = cfMorph ('Ind[f @* G] phi). | Proof.
move=> sKH sHG sGD; have [sHD inD] := (subset_trans sHG sGD, subsetP sGD).
apply/cfun_inP=> /= x Gx; have [Dx sKG] := (inD x Gx, subset_trans sKH sHG).
rewrite cfMorphE ?cfIndE ?morphimS // (partition_big_imset f) -morphimEsub //=.
rewrite card_morphim (setIidPr sHD) natf_indexg // invfM invrK -mulrA.
congr (_ *... | Lemma | cfIndMorph | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Dx",
"apply",
"card_morphim",
"card_rcoset",
"cfIndE",
"cfMorph",
"cfMorphE",
"cfun0",
"cfun_inP",
"eq_big",
"eq_bigr",
"groupJ",
"groupM",
"invfM",
"invrK",
"ker",
"morphJ",
"morphimEsub",
"morphimP",
"morphimS",
"morphimSGK",
"morphim_set1",
"morphism",
"mulrA",
"m... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(isoG : isom G R g) (isoH : isom H S h) (eq_hg : {in H, h =1 g}). | Hypotheses | isoG | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"eq_hg",
"isom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
cfResIsom phi : 'Res[S] (cfIsom isoG phi) = cfIsom isoH ('Res[H] phi). | Proof.
have [[injg defR] [injh defS]] := (isomP isoG, isomP isoH).
rewrite !morphimEdom in defS defR; apply/cfun_inP=> s.
rewrite -{1}defS => /imsetP[x Hx ->] {s}; have Gx := subsetP sHG x Hx.
rewrite {1}eq_hg ?(cfResE, cfIsomE) // -defS -?eq_hg ?imset_f // -defR.
by rewrite (eq_in_imset eq_hg) imsetS.
Qed. | Lemma | cfResIsom | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfIsom",
"cfIsomE",
"cfResE",
"cfun_inP",
"defR",
"eq_hg",
"eq_in_imset",
"imsetP",
"imsetS",
"imset_f",
"isoG",
"isomP",
"morphimEdom",
"sHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIndIsom phi : 'Ind[R] (cfIsom isoH phi) = cfIsom isoG ('Ind[G] phi). | Proof.
have [[injg defR] [_ defS]] := (isomP isoG, isomP isoH).
rewrite morphimEdom (eq_in_imset eq_hg) -morphimEsub // in defS.
apply/cfun_inP=> s; rewrite -{1}defR => /morphimP[x _ Gx ->]{s}.
rewrite cfIsomE ?cfIndE // -defR -{1}defS ?morphimS ?card_injm // morphimEdom.
congr (_ * _); rewrite big_imset //=; first exa... | Lemma | cfIndIsom | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"big_imset",
"card_injm",
"cfIndE",
"cfIsom",
"cfIsomE",
"cfun0",
"cfun_inP",
"defR",
"eq_bigr",
"eq_hg",
"eq_in_imset",
"groupJ",
"injmP",
"injmSK",
"isoG",
"isomP",
"morphJ",
"morphimEdom",
"morphimEsub",
"morphimP",
"morphimS",
"morphim_set1",
"sub1set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"phi ^u" | := (cfAut u phi). | Notation | phi ^u | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAut"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutZ_nat n phi : (n%:R *: phi)^u = n%:R *: phi^u. | Proof. exact: raddfZnat. Qed. | Lemma | cfAutZ_nat | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"raddfZnat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutZ_Cnat z phi : z \in Num.nat -> (z *: phi)^u = z *: phi^u. | Proof. exact: raddfZ_nat. Qed. | Lemma | cfAutZ_Cnat | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"nat",
"raddfZ_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutZ_Cint z phi : z \in Num.int -> (z *: phi)^u = z *: phi^u. | Proof. exact: raddfZ_int. Qed. | Lemma | cfAutZ_Cint | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"int",
"raddfZ_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutK : cancel (@cfAut gT G u) (cfAut (algC_invaut u)). | Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE /= algC_autK. Qed. | Lemma | cfAutK | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC_autK",
"algC_invaut",
"apply",
"cfAut",
"cfunE",
"cfunP",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutVK : cancel (cfAut (algC_invaut u)) (@cfAut gT G u). | Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE /= algC_invautK. Qed. | Lemma | cfAutVK | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC_invaut",
"algC_invautK",
"apply",
"cfAut",
"cfunE",
"cfunP",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_inj : injective (@cfAut gT G u). | Proof. exact: can_inj cfAutK. Qed. | Lemma | cfAut_inj | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAut",
"cfAutK",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_eq1 phi : (cfAut u phi == 1) = (phi == 1). | Proof. by rewrite rmorph_eq1 //; apply: cfAut_inj. Qed. | Lemma | cfAut_eq1 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfAut",
"cfAut_inj",
"rmorph_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
support_cfAut phi : support phi^u =i support phi. | Proof. by move=> x; rewrite !inE cfunE fmorph_eq0. Qed. | Lemma | support_cfAut | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfunE",
"fmorph_eq0",
"inE",
"support"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_cfAut_free S : cfAut_closed u S -> free S -> free (map (cfAut u) S). | Proof.
set Su := map _ S => sSuS freeS; have uniqS := free_uniq freeS.
have uniqSu: uniq Su by rewrite (map_inj_uniq cfAut_inj).
have{} sSuS: {subset Su <= S} by move=> _ /mapP[phi Sphi ->]; apply: sSuS.
have [|_ eqSuS] := uniq_min_size uniqSu sSuS; first by rewrite size_map.
by rewrite (perm_free (uniq_perm uniqSu uni... | Lemma | map_cfAut_free | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfAut",
"cfAut_closed",
"cfAut_inj",
"free",
"freeS",
"free_uniq",
"map",
"mapP",
"map_inj_uniq",
"perm_free",
"size_map",
"uniq",
"uniqS",
"uniq_min_size",
"uniq_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_on A phi : (phi^u \in 'CF(G, A)) = (phi \in 'CF(G, A)). | Proof. by rewrite !cfun_onE (eq_subset (support_cfAut phi)). Qed. | Lemma | cfAut_on | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onE",
"eq_subset",
"support_cfAut"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_aut phi : cfker phi^u = cfker phi. | Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
by apply/forallP/forallP=> Kx y;
have:= Kx y; rewrite !cfunE (inj_eq (fmorph_inj u)).
Qed. | Lemma | cfker_aut | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker",
"cfunE",
"fmorph_inj",
"forallP",
"inE",
"inj_eq",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_cfuni A : ('1_A)^u = '1_A :> 'CF(G). | Proof. by apply/cfunP=> x; rewrite !cfunElock rmorph_nat. Qed. | Lemma | cfAut_cfuni | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfunElock",
"cfunP",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_aut phi : #[phi^u]%CF = #[phi]%CF. | Proof. exact: cforder_inj_rmorph cfAut_inj. Qed. | Lemma | cforder_aut | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAut_inj",
"cforder_inj_rmorph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutRes phi : ('Res[H] phi)^u = 'Res phi^u. | Proof. by apply/cfunP=> x; rewrite !cfunElock rmorphMn. Qed. | Lemma | cfAutRes | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfunElock",
"cfunP",
"rmorphMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutMorph (psi : 'CF(f @* H)) : (cfMorph psi)^u = cfMorph psi^u. | Proof. by apply/cfun_inP=> x Hx; rewrite !cfunElock Hx. Qed. | Lemma | cfAutMorph | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfMorph",
"cfunElock",
"cfun_inP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutIsom (isoGR : isom G R f) phi :
(cfIsom isoGR phi)^u = cfIsom isoGR phi^u. | Proof.
apply/cfun_inP=> y; have [_ {1}<-] := isomP isoGR => /morphimP[x _ Gx ->{y}].
by rewrite !(cfunE, cfIsomE).
Qed. | Lemma | cfAutIsom | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfIsom",
"cfIsomE",
"cfunE",
"cfun_inP",
"isoGR",
"isom",
"isomP",
"morphimP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutQuo phi : (phi / H)^u = (phi^u / H)%CF. | Proof. by apply/cfunP=> Hx; rewrite !cfunElock cfker_aut rmorphMn. Qed. | Lemma | cfAutQuo | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfker_aut",
"cfunElock",
"cfunP",
"rmorphMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutMod (psi : 'CF(G / H)) : (psi %% H)^u = (psi^u %% H)%CF. | Proof. by apply/cfunP=> x; rewrite !cfunElock rmorphMn. Qed. | Lemma | cfAutMod | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfunElock",
"cfunP",
"rmorphMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutInd (psi : 'CF(H)) : ('Ind[G] psi)^u = 'Ind psi^u. | Proof.
have [sHG | not_sHG] := boolP (H \subset G).
apply/cfunP=> x; rewrite !(cfunE, cfIndE) // rmorphM /= fmorphV rmorph_nat.
by congr (_ * _); rewrite rmorph_sum; apply: eq_bigr => y; rewrite !cfunE.
rewrite !cfIndEout // linearZ /= cfAut_cfuni rmorphM rmorph_nat /=.
rewrite -cfdot_cfAut ?rmorph1 // => _ /imageP... | Lemma | cfAutInd | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfAut_cfuni",
"cfIndE",
"cfIndEout",
"cfdot_cfAut",
"cfun1E",
"cfunE",
"cfunP",
"eq_bigr",
"fmorphV",
"imageP",
"linearZ",
"rmorph1",
"rmorphM",
"rmorph_nat",
"rmorph_sum",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutDprodl (phi : 'CF(K)) : (cfDprodl KxH phi)^u = cfDprodl KxH phi^u. | Proof.
apply/cfun_inP=> _ /(mem_dprod KxH)[x [y [Kx Hy -> _]]].
by rewrite !(cfunE, cfDprodEl).
Qed. | Lemma | cfAutDprodl | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"KxH",
"apply",
"cfDprodEl",
"cfDprodl",
"cfunE",
"cfun_inP",
"mem_dprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutDprodr (psi : 'CF(H)) : (cfDprodr KxH psi)^u = cfDprodr KxH psi^u. | Proof.
apply/cfun_inP=> _ /(mem_dprod KxH)[x [y [Kx Hy -> _]]].
by rewrite !(cfunE, cfDprodEr).
Qed. | Lemma | cfAutDprodr | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"KxH",
"apply",
"cfDprodEr",
"cfDprodr",
"cfunE",
"cfun_inP",
"mem_dprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutDprod (phi : 'CF(K)) (psi : 'CF(H)) :
(cfDprod KxH phi psi)^u = cfDprod KxH phi^u psi^u. | Proof. by rewrite rmorphM /= cfAutDprodl cfAutDprodr. Qed. | Lemma | cfAutDprod | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"KxH",
"cfAutDprodl",
"cfAutDprodr",
"cfDprod",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_cfRes | := cfAutRes conjC. | Definition | conj_cfRes | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAutRes",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_conjC | := cfker_aut conjC. | Definition | cfker_conjC | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfker_aut",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_cfQuo | := cfAutQuo conjC. | Definition | conj_cfQuo | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAutQuo",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_cfMod | := cfAutMod conjC. | Definition | conj_cfMod | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAutMod",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_cfInd | := cfAutInd conjC. | Definition | conj_cfInd | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAutInd",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfconjC_eq1 | := cfAut_eq1 conjC. | Definition | cfconjC_eq1 | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAut_eq1",
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_subproof :
is_class_fun G [ffun x => phi (if y \in 'N(G) then x ^ y^-1 else x)]. | Proof.
apply: intro_class_fun => [x z _ Gz | x notGx].
have [nGy | _] := ifP; last by rewrite cfunJgen.
by rewrite -conjgM conjgC conjgM [LHS]cfunJgen // memJ_norm ?groupV.
by rewrite cfun0gen //; case: ifP => // nGy; rewrite memJ_norm ?groupV.
Qed. | Fact | cfConjg_subproof | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfun0gen",
"cfunJgen",
"conjgC",
"conjgM",
"groupV",
"intro_class_fun",
"is_class_fun",
"last",
"memJ_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg | := Cfun 1 cfConjg_subproof. | Definition | cfConjg | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Cfun",
"cfConjg_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"f ^ y" | := (cfConjg y f) : cfun_scope. | Notation | f ^ y | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgE phi y x : y \in 'N(G) -> (phi ^ y)%CF x = phi (x ^ y^-1)%g. | Proof. by rewrite cfunElock genGid => ->. Qed. | Lemma | cfConjgE | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfunElock",
"genGid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgEJ phi y x : y \in 'N(G) -> (phi ^ y)%CF (x ^ y) = phi x. | Proof. by move/cfConjgE->; rewrite conjgK. Qed. | Lemma | cfConjgEJ | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgE",
"conjgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgEout phi y : y \notin 'N(G) -> (phi ^ y = phi)%CF. | Proof.
by move/negbTE=> notNy; apply/cfunP=> x; rewrite !cfunElock genGid notNy.
Qed. | Lemma | cfConjgEout | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfunElock",
"cfunP",
"genGid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgEin phi y (nGy : y \in 'N(G)) :
(phi ^ y)%CF = cfIsom (norm_conj_isom nGy) phi. | Proof.
apply/cfun_inP=> x Gx.
by rewrite cfConjgE // -{2}[x](conjgKV y) cfIsomE ?memJ_norm ?groupV.
Qed. | Lemma | cfConjgEin | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgE",
"cfIsom",
"cfIsomE",
"cfun_inP",
"conjgKV",
"groupV",
"memJ_norm",
"norm_conj_isom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgMnorm phi :
{in 'N(G) &, forall y z, phi ^ (y * z) = (phi ^ y) ^ z}%CF. | Proof.
move=> y z nGy nGz.
by apply/cfunP=> x; rewrite !cfConjgE ?groupM // invMg conjgM.
Qed. | Lemma | cfConjgMnorm | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgE",
"cfunP",
"conjgM",
"groupM",
"invMg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_id phi y : y \in G -> (phi ^ y)%CF = phi. | Proof.
move=> Gy; apply/cfunP=> x; have nGy := subsetP (normG G) y Gy.
by rewrite -(cfunJ _ _ Gy) cfConjgEJ.
Qed. | Lemma | cfConjg_id | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgEJ",
"cfunJ",
"cfunP",
"normG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgM L phi :
G <| L -> {in L &, forall y z, phi ^ (y * z) = (phi ^ y) ^ z}%CF. | Proof. by case/andP=> _ /subsetP nGL; apply: sub_in2 (cfConjgMnorm phi). Qed. | Lemma | cfConjgM | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgMnorm",
"subsetP"
] | Isaacs' 6.1.b | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfConjgJ1 phi : (phi ^ 1)%CF = phi. | Proof. by apply/cfunP=> x; rewrite cfConjgE ?group1 // invg1 conjg1. Qed. | Lemma | cfConjgJ1 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgE",
"cfunP",
"conjg1",
"group1",
"invg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgK y : cancel (cfConjg y) (cfConjg y^-1 : 'CF(G) -> 'CF(G)). | Proof.
move=> phi; apply/cfunP=> x; rewrite !cfunElock groupV /=.
by case: ifP => -> //; rewrite conjgKV.
Qed. | Lemma | cfConjgK | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjg",
"cfunElock",
"cfunP",
"conjgKV",
"groupV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgKV y : cancel (cfConjg y^-1) (cfConjg y : 'CF(G) -> 'CF(G)). | Proof. by move=> phi /=; rewrite -{1}[y]invgK cfConjgK. Qed. | Lemma | cfConjgKV | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjg",
"cfConjgK",
"invgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg1 phi y : (phi ^ y)%CF 1%g = phi 1%g. | Proof. by rewrite cfunElock conj1g if_same. Qed. | Lemma | cfConjg1 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfunElock",
"conj1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_is_linear y : linear (cfConjg y : 'CF(G) -> 'CF(G)). | Proof. by move=> a phi psi; apply/cfunP=> x; rewrite !cfunElock. Qed. | Fact | cfConjg_is_linear | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjg",
"cfunElock",
"cfunP",
"linear"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_cfuniJ A y : y \in 'N(G) -> ('1_A ^ y)%CF = '1_(A :^ y) :> 'CF(G). | Proof.
move=> nGy; apply/cfunP=> x; rewrite !cfunElock genGid nGy -sub_conjgV.
by rewrite -class_lcoset -class_rcoset norm_rlcoset ?memJ_norm ?groupV.
Qed. | Lemma | cfConjg_cfuniJ | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfunElock",
"cfunP",
"class_lcoset",
"class_rcoset",
"genGid",
"groupV",
"memJ_norm",
"norm_rlcoset",
"sub_conjgV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_cfuni A y : y \in 'N(A) -> ('1_A ^ y)%CF = '1_A :> 'CF(G). | Proof.
by have [/cfConjg_cfuniJ-> /normP-> | /cfConjgEout] := boolP (y \in 'N(G)).
Qed. | Lemma | cfConjg_cfuni | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgEout",
"cfConjg_cfuniJ",
"normP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_cfun1 y : (1 ^ y)%CF = 1 :> 'CF(G). | Proof.
by rewrite -cfuniG; have [/cfConjg_cfuni|/cfConjgEout] := boolP (y \in 'N(G)).
Qed. | Lemma | cfConjg_cfun1 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgEout",
"cfConjg_cfuni",
"cfuniG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_is_monoid_morphism y : monoid_morphism (cfConjg y : _ -> 'CF(G)). | Proof.
split=> [|phi psi]; first exact: cfConjg_cfun1.
by apply/cfunP=> x; rewrite !cfunElock.
Qed. | Fact | cfConjg_is_monoid_morphism | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjg",
"cfConjg_cfun1",
"cfunElock",
"cfunP",
"monoid_morphism",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_is_multiplicative y | :=
(fun g => (g.2,g.1)) (cfConjg_is_monoid_morphism y). | Definition | cfConjg_is_multiplicative | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjg_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_eq1 phi y : ((phi ^ y)%CF == 1) = (phi == 1). | Proof. by apply: rmorph_eq1; apply: can_inj (cfConjgK y). Qed. | Lemma | cfConjg_eq1 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgK",
"rmorph_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAutConjg phi u y : cfAut u (phi ^ y) = (cfAut u phi ^ y)%CF. | Proof. by apply/cfunP=> x; rewrite !cfunElock. Qed. | Lemma | cfAutConjg | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfAut",
"cfunElock",
"cfunP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_cfConjg phi y : (phi ^ y)^*%CF = (phi^* ^ y)%CF. | Proof. exact: cfAutConjg. Qed. | Lemma | conj_cfConjg | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfAutConjg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_conjg phi y : y \in 'N(G) -> cfker (phi ^ y) = cfker phi :^ y. | Proof.
move=> nGy; rewrite cfConjgEin // cfker_isom.
by rewrite morphim_conj (setIidPr (cfker_sub _)).
Qed. | Lemma | cfker_conjg | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgEin",
"cfker",
"cfker_isom",
"cfker_sub",
"morphim_conj",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDetConjg phi y : cfDet (phi ^ y) = (cfDet phi ^ y)%CF. | Proof.
have [nGy | not_nGy] := boolP (y \in 'N(G)); last by rewrite !cfConjgEout.
by rewrite !cfConjgEin cfDetIsom.
Qed. | Lemma | cfDetConjg | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgEin",
"cfConjgEout",
"cfDet",
"cfDetIsom",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia (B : {set gT}) (phi : 'CF(B)) | :=
[set y in 'N(B) | (phi ^ y)%CF == phi]. | Definition | inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''I[' phi ]" | := (inertia phi) : group_scope. | Notation | ''I[' phi ] | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inertia"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''I_' G [ phi ]" | := (G%g :&: 'I[phi]) : group_scope. | Notation | ''I_' G [ phi ] | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_set_inertia (H : {group gT}) phi : group_set 'I[phi : 'CF(H)]. | Proof.
apply/group_setP; split; first by rewrite inE group1 /= cfConjgJ1.
move=> y z /setIdP[nHy /eqP n_phi_y] /setIdP[nHz n_phi_z].
by rewrite inE groupM //= cfConjgMnorm ?n_phi_y.
Qed. | Fact | group_set_inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgJ1",
"cfConjgMnorm",
"gT",
"group",
"group1",
"groupM",
"group_set",
"group_setP",
"inE",
"setIdP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_group H phi | := Group (@group_set_inertia H phi). | Canonical | inertia_group | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"group_set_inertia"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''I[' phi ]" | := (inertia_group phi) : Group_scope. | Notation | ''I[' phi ] | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inertia_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''I_' G [ phi ]" | := (G :&: 'I[phi])%G : Group_scope. | Notation | ''I_' G [ phi ] | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertiaJ phi y : y \in 'I[phi] -> (phi ^ y)%CF = phi. | Proof. by case/setIdP=> _ /eqP->. Qed. | Lemma | inertiaJ | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"setIdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_valJ phi x y : y \in 'I[phi] -> phi (x ^ y)%g = phi x. | Proof. by case/setIdP=> nHy /eqP {1}<-; rewrite cfConjgEJ. Qed. | Lemma | inertia_valJ | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgEJ",
"setIdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Inertia_sub phi : 'I_G[phi] \subset G. | Proof. exact: subsetIl. Qed. | Lemma | Inertia_sub | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"subsetIl"
] | lemmas concerning the localized inertia group 'I_G[phi]. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
norm_inertia phi : 'I[phi] \subset 'N(H). | Proof. by rewrite ['I[_]]setIdE subsetIl. Qed. | Lemma | norm_inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"setIdE",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_inertia phi : H \subset 'I[phi]. | Proof.
by apply/subsetP=> y Hy; rewrite inE cfConjg_id ?(subsetP (normG H)) /=.
Qed. | Lemma | sub_inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjg_id",
"inE",
"normG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normal_inertia phi : H <| 'I[phi]. | Proof. by rewrite /normal sub_inertia norm_inertia. Qed. | Lemma | normal_inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"norm_inertia",
"normal",
"sub_inertia"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_Inertia phi : H \subset G -> H \subset 'I_G[phi]. | Proof. by rewrite subsetI sub_inertia andbT. Qed. | Lemma | sub_Inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"sub_inertia",
"subsetI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_Inertia phi : 'I_G[phi] \subset 'N(H). | Proof. by rewrite setIC subIset ?norm_inertia. Qed. | Lemma | norm_Inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"norm_inertia",
"setIC",
"subIset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normal_Inertia phi : H \subset G -> H <| 'I_G[phi]. | Proof. by rewrite /normal norm_Inertia andbT; apply: sub_Inertia. Qed. | Lemma | normal_Inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"norm_Inertia",
"normal",
"sub_Inertia"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_eqE phi :
H <| G ->
{in G &, forall y z, (phi ^ y == phi ^ z)%CF = (z \in 'I_G[phi] :* y)}. | Proof.
case/andP=> _ nHG y z Gy; rewrite -{1 2}[z](mulgKV y) groupMr // mem_rcoset.
move: {z}(z * _)%g => z Gz; rewrite 2!inE Gz cfConjgMnorm ?(subsetP nHG) //=.
by rewrite eq_sym (can_eq (cfConjgK y)).
Qed. | Lemma | cfConjg_eqE | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"can_eq",
"cfConjgK",
"cfConjgMnorm",
"eq_sym",
"groupMr",
"inE",
"mem_rcoset",
"mulgKV",
"nHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_sub_inertia phi : 'C(H) \subset 'I[phi]. | Proof.
apply/subsetP=> y cHy; have nHy := subsetP (cent_sub H) y cHy.
rewrite inE nHy; apply/eqP/cfun_inP=> x Hx; rewrite cfConjgE //.
by rewrite /conjg invgK mulgA (centP cHy) ?mulgK.
Qed. | Lemma | cent_sub_inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"centP",
"cent_sub",
"cfConjgE",
"cfun_inP",
"conjg",
"inE",
"invgK",
"mulgA",
"mulgK",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_sub_Inertia phi : 'C_G(H) \subset 'I_G[phi]. | Proof. exact: setIS (cent_sub_inertia phi). Qed. | Lemma | cent_sub_Inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cent_sub_inertia",
"setIS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
center_sub_Inertia phi : H \subset G -> 'Z(G) \subset 'I_G[phi]. | Proof.
by move/centS=> sHG; rewrite setIS // (subset_trans sHG) // cent_sub_inertia.
Qed. | Lemma | center_sub_Inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"centS",
"cent_sub_inertia",
"sHG",
"setIS",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_inertia phi y : y \in 'N(H) -> 'I[phi] :^ y = 'I[phi ^ y]. | Proof.
move=> nHy; apply/setP=> z; rewrite !['I[_]]setIdE conjIg conjGid // !in_setI.
apply/andb_id2l=> nHz; rewrite mem_conjg !inE.
by rewrite !cfConjgMnorm ?in_group ?(can2_eq (cfConjgKV y) (cfConjgK y)) ?invgK.
Qed. | Lemma | conjg_inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"can2_eq",
"cfConjgK",
"cfConjgKV",
"cfConjgMnorm",
"conjGid",
"conjIg",
"inE",
"in_group",
"in_setI",
"invgK",
"mem_conjg",
"setIdE",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia0 : 'I[0 : 'CF(H)] = 'N(H). | Proof. by apply/setP=> x; rewrite !inE linear0 eqxx andbT. Qed. | Lemma | inertia0 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqxx",
"inE",
"linear0",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_add phi psi : 'I[phi] :&: 'I[psi] \subset 'I[phi + psi]. | Proof.
rewrite !['I[_]]setIdE -setIIr setIS //.
by apply/subsetP=> x /[!(inE, linearD)]/= /andP[/eqP-> /eqP->].
Qed. | Lemma | inertia_add | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"linearD",
"setIIr",
"setIS",
"setIdE",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_sum I r (P : pred I) (Phi : I -> 'CF(H)) :
'N(H) :&: \bigcap_(i <- r | P i) 'I[Phi i]
\subset 'I[\sum_(i <- r | P i) Phi i]. | Proof.
elim/big_rec2: _ => [|i K psi Pi sK_Ipsi]; first by rewrite setIT inertia0.
by rewrite setICA; apply: subset_trans (setIS _ sK_Ipsi) (inertia_add _ _).
Qed. | Lemma | inertia_sum | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"big_rec2",
"inertia0",
"inertia_add",
"setICA",
"setIS",
"setIT",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_scale a phi : 'I[phi] \subset 'I[a *: phi]. | Proof.
apply/subsetP=> x /setIdP[nHx /eqP Iphi_x].
by rewrite inE nHx linearZ /= Iphi_x.
Qed. | Lemma | inertia_scale | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"linearZ",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_scale_nz a phi : a != 0 -> 'I[a *: phi] = 'I[phi]. | Proof.
move=> nz_a; apply/eqP.
by rewrite eqEsubset -{2}(scalerK nz_a phi) !inertia_scale.
Qed. | Lemma | inertia_scale_nz | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqEsubset",
"inertia_scale",
"scalerK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_opp phi : 'I[- phi] = 'I[phi]. | Proof. by rewrite -scaleN1r inertia_scale_nz // oppr_eq0 oner_eq0. Qed. | Lemma | inertia_opp | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inertia_scale_nz",
"oner_eq0",
"oppr_eq0",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia1 : 'I[1 : 'CF(H)] = 'N(H). | Proof. by apply/setP=> x; rewrite inE rmorph1 eqxx andbT. Qed. | Lemma | inertia1 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqxx",
"inE",
"rmorph1",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Inertia1 : H <| G -> 'I_G[1 : 'CF(H)] = G. | Proof. by rewrite inertia1 => /normal_norm/setIidPl. Qed. | Lemma | Inertia1 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inertia1",
"normal_norm",
"setIidPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_mul phi psi : 'I[phi] :&: 'I[psi] \subset 'I[phi * psi]. | Proof.
rewrite !['I[_]]setIdE -setIIr setIS //.
by apply/subsetP=> x /[!(inE, rmorphM)]/= /andP[/eqP-> /eqP->].
Qed. | Lemma | inertia_mul | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"rmorphM",
"setIIr",
"setIS",
"setIdE",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_prod I r (P : pred I) (Phi : I -> 'CF(H)) :
'N(H) :&: \bigcap_(i <- r | P i) 'I[Phi i]
\subset 'I[\prod_(i <- r | P i) Phi i]. | Proof.
elim/big_rec2: _ => [|i K psi Pi sK_psi]; first by rewrite inertia1 setIT.
by rewrite setICA; apply: subset_trans (setIS _ sK_psi) (inertia_mul _ _).
Qed. | Lemma | inertia_prod | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"big_rec2",
"inertia1",
"inertia_mul",
"setICA",
"setIS",
"setIT",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_injective (chi : 'CF(H)) :
{in H &, injective chi} -> 'I[chi] = 'C(H). | Proof.
move=> inj_chi; apply/eqP; rewrite eqEsubset cent_sub_inertia andbT.
apply/subsetP=> y Ichi_y; have /setIdP[nHy _] := Ichi_y.
apply/centP=> x Hx; apply/esym/commgP/conjg_fixP.
by apply/inj_chi; rewrite ?memJ_norm ?(inertia_valJ _ Ichi_y).
Qed. | Lemma | inertia_injective | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"centP",
"cent_sub_inertia",
"chi",
"commgP",
"conjg_fixP",
"eqEsubset",
"inertia_valJ",
"memJ_norm",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_irr_prime p i :
#|H| = p -> prime p -> i != 0 -> 'I['chi[H]_i] = 'C(H). | Proof. by move=> <- pr_H /(irr_prime_injP pr_H); apply: inertia_injective. Qed. | Lemma | inertia_irr_prime | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"chi",
"inertia_injective",
"irr_prime_injP",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_irr0 : 'I['chi[H]_0] = 'N(H). | Proof. by rewrite irr0 inertia1. Qed. | Lemma | inertia_irr0 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"chi",
"inertia1",
"irr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_iso y : isometry (cfConjg y : 'CF(H) -> 'CF(H)). | Proof.
move=> phi psi; congr (_ * _).
have [nHy | not_nHy] := boolP (y \in 'N(H)); last by rewrite !cfConjgEout.
rewrite (reindex_astabs 'J y) ?astabsJ //=.
by apply: eq_bigr=> x _; rewrite !cfConjgEJ.
Qed. | Lemma | cfConjg_iso | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"astabsJ",
"cfConjg",
"cfConjgEJ",
"cfConjgEout",
"eq_bigr",
"isometry",
"last",
"reindex_astabs"
] | Isaacs' 6.1.c | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfdot_Res_conjg psi phi y :
y \in G -> '['Res[H, G] psi, phi ^ y] = '['Res[H] psi, phi]. | Proof.
move=> Gy; rewrite -(cfConjg_iso y _ phi); congr '[_, _]; apply/cfunP=> x.
rewrite !cfunElock !genGid; case nHy: (y \in 'N(H)) => //.
by rewrite !(fun_if psi) cfunJ ?memJ_norm ?groupV.
Qed. | Lemma | cfdot_Res_conjg | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjg_iso",
"cfunElock",
"cfunJ",
"cfunP",
"genGid",
"groupV",
"memJ_norm"
] | Isaacs' 6.1.d | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfConjg_char (chi : 'CF(H)) y :
chi \is a character -> (chi ^ y)%CF \is a character. | Proof.
have [nHy Nchi | /cfConjgEout-> //] := boolP (y \in 'N(H)).
by rewrite cfConjgEin cfIsom_char.
Qed. | Lemma | cfConjg_char | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgEin",
"cfConjgEout",
"cfIsom_char",
"character",
"chi"
] | Isaac's 6.1.e | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfConjg_lin_char (chi : 'CF(H)) y :
chi \is a linear_char -> (chi ^ y)%CF \is a linear_char. | Proof. by case/andP=> Nchi chi1; rewrite qualifE/= cfConjg1 cfConjg_char. Qed. | Lemma | cfConjg_lin_char | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjg1",
"cfConjg_char",
"chi",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjg_irr y chi : chi \in irr H -> (chi ^ y)%CF \in irr H. | Proof. by rewrite !irrEchar cfConjg_iso => /andP[/cfConjg_char->]. Qed. | Lemma | cfConjg_irr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjg_char",
"cfConjg_iso",
"chi",
"irr",
"irrEchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_Iirr i y | := cfIirr ('chi[H]_i ^ y)%CF. | Definition | conjg_Iirr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfIirr",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_IirrE i y : 'chi_(conjg_Iirr i y) = ('chi_i ^ y)%CF. | Proof. by rewrite cfIirrE ?cfConjg_irr ?mem_irr. Qed. | Lemma | conjg_IirrE | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjg_irr",
"cfIirrE",
"conjg_Iirr",
"mem_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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