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 |
|---|---|---|---|---|---|---|---|---|---|---|
conjC_pair_orthogonal S chi :
cfConjC_closed S -> ~~ has cfReal S -> pairwise_orthogonal S -> chi \in S ->
pairwise_orthogonal (chi :: chi^*%CF). | Proof.
move=> ccS /hasPn nrS oSS Schi; apply: sub_pairwise_orthogonal oSS.
by apply/allP; rewrite /= Schi ccS.
by rewrite /= inE eq_sym nrS.
Qed. | Lemma | conjC_pair_orthogonal | 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",
"... | [
"allP",
"apply",
"cfConjC_closed",
"cfReal",
"chi",
"eq_sym",
"has",
"hasPn",
"inE",
"oSS",
"pairwise_orthogonal",
"sub_pairwise_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_real_conjC phi psi : cfReal phi -> '[phi, psi^*]_G = '[phi, psi]^*. | Proof. by rewrite -cfdot_conjC => /eqcfP->. Qed. | Lemma | cfdot_real_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",
"... | [
"cfReal",
"cfdot_conjC",
"eqcfP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extend_cfConjC_subset S X phi :
cfConjC_closed S -> ~~ has cfReal S -> phi \in S -> phi \notin X ->
cfConjC_subset X S -> cfConjC_subset [:: phi, phi^* & X]%CF S. | Proof.
move=> ccS nrS Sphi X'phi [uniqX /allP-sXS ccX].
split; last 1 [by apply/allP; rewrite /= Sphi ccS | apply/allP]; rewrite /= inE.
by rewrite negb_or X'phi eq_sym (hasPn nrS) // (contra (ccX _)) ?cfConjCK.
by rewrite cfConjCK !mem_head orbT; apply/allP=> xi Xxi; rewrite !inE ccX ?orbT.
Qed. | Lemma | extend_cfConjC_subset | 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",
"... | [
"allP",
"apply",
"cfConjCK",
"cfConjC_closed",
"cfConjC_subset",
"cfReal",
"eq_sym",
"has",
"hasPn",
"inE",
"last",
"mem_head",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_cforderP n :
reflect {in G, forall x, phi x ^+ n = 1} (#[phi]%CF %| n)%N. | Proof.
apply: (iffP (dvdn_biglcmP _ _ _)); rewrite genGid => phiG1 x Gx.
by apply/eqP; rewrite -dvdn_orderC phiG1.
by rewrite dvdn_orderC phiG1.
Qed. | Lemma | dvdn_cforderP | 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",
"dvdn_biglcmP",
"dvdn_orderC",
"genGid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_cforder n : (#[phi]%CF %| n) = (phi ^+ n == 1). | Proof.
apply/dvdn_cforderP/eqP=> phi_n_1 => [|x Gx].
by apply/cfun_inP=> x Gx; rewrite exp_cfunE // cfun1E Gx phi_n_1.
by rewrite -exp_cfunE // phi_n_1 // cfun1E Gx.
Qed. | Lemma | dvdn_cforder | 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",
"cfun1E",
"cfun_inP",
"dvdn_cforderP",
"exp_cfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp_cforder : phi ^+ #[phi]%CF = 1. | Proof. by apply/eqP; rewrite -dvdn_cforder. Qed. | Lemma | exp_cforder | 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",
"dvdn_cforder"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_rmorph phi : #[f phi]%CF %| #[phi]%CF. | Proof. by rewrite dvdn_cforder -rmorphXn exp_cforder rmorph1. Qed. | Lemma | cforder_rmorph | 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",
"... | [
"dvdn_cforder",
"exp_cforder",
"rmorph1",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_inj_rmorph phi : injective f -> #[f phi]%CF = #[phi]%CF. | Proof.
move=> inj_f; apply/eqP; rewrite eqn_dvd cforder_rmorph dvdn_cforder /=.
by rewrite -(rmorph_eq1 _ inj_f) rmorphXn exp_cforder.
Qed. | Lemma | cforder_inj_rmorph | 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",
"cforder_rmorph",
"dvdn_cforder",
"eqn_dvd",
"exp_cforder",
"inj_f",
"rmorphXn",
"rmorph_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_iso_to U1 U2 W1 W2 tau :
{subset U2 <= U1} -> {subset W1 <= W2} ->
{in U1, isometry tau, to W1} -> {in U2, isometry tau, to W2}. | Proof.
by move=> sU sW [Itau Wtau]; split=> [|u /sU/Wtau/sW //]; apply: sub_in2 Itau.
Qed. | Lemma | sub_iso_to | 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",
"isometry",
"split",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometry_of_free S f :
free S -> {in S &, isometry f} ->
{tau : {linear 'CF(L) -> 'CF(G)} |
{in S, tau =1 f} & {in <<S>>%VS &, isometry tau}}. | Proof.
move=> freeS If; have defS := free_span freeS.
have [tau /(_ freeS (size_map f S))Dtau] := linear_of_free S (map f S).
have{} Dtau: {in S, tau =1 f}.
by move=> _ /(nthP 0)[i ltiS <-]; rewrite -!(nth_map 0 0) ?Dtau.
exists tau => // _ _ /defS[a -> _] /defS[b -> _].
rewrite !{1}linear_sum !{1}cfdot_suml; apply/e... | Lemma | isometry_of_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",
"... | [
"If",
"apply",
"cfdotZl",
"cfdotZr",
"cfdot_suml",
"cfdot_sumr",
"eq_big_seq",
"free",
"freeS",
"free_span",
"isometry",
"linear",
"linearZ",
"linear_of_free",
"linear_sum",
"map",
"nthP",
"nth_map",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometry_of_cfnorm S tauS :
pairwise_orthogonal S -> pairwise_orthogonal tauS ->
map cfnorm tauS = map cfnorm S ->
{tau : {linear 'CF(L) -> 'CF(G)} | map tau S = tauS
& {in <<S>>%VS &, isometry tau}}. | Proof.
move=> oS oT eq_nST; have freeS := orthogonal_free oS.
have eq_sz: size tauS = size S by have:= congr1 size eq_nST; rewrite !size_map.
have [tau defT] := linear_of_free S tauS; rewrite -[S]/(tval (in_tuple S)).
exists tau => [|u v /coord_span-> /coord_span->]; rewrite ?raddf_sum ?defT //=.
apply: eq_bigr => i _ ... | Lemma | isometry_of_cfnorm | 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",
"cfdotZl",
"cfdotZr",
"cfdot_suml",
"cfnorm",
"coord_span",
"eqVneq",
"eq_bigr",
"freeS",
"in_tuple",
"isometry",
"linear",
"linearZ",
"linear_of_free",
"map",
"mem_nth",
"nth_map",
"nth_uniq",
"orthogonal_free",
"pairwise_orthogonal",
"pairwise_orthogonalP",
"radd... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometry_raddf_inj U (tau : {additive 'CF(L) -> 'CF(G)}) :
{in U &, isometry tau} -> {in U &, forall u v, u - v \in U} ->
{in U &, injective tau}. | Proof.
move=> Itau linU phi psi Uphi Upsi /eqP; rewrite -subr_eq0 -raddfB.
by rewrite -cfnorm_eq0 Itau ?linU // cfnorm_eq0 subr_eq0 => /eqP.
Qed. | Lemma | isometry_raddf_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",
"... | [
"additive",
"cfnorm_eq0",
"isometry",
"raddfB",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_isometry : @isometry _ _ G G -%R. | Proof. by move=> x y; rewrite cfdotNl cfdotNr opprK. Qed. | Lemma | opp_isometry | 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",
"... | [
"cfdotNl",
"cfdotNr",
"isometry",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_subproof (phi : 'CF(B)) :
is_class_fun H [ffun x => phi (if H \subset G then x else 1%g) *+ (x \in H)]. | Proof.
apply: intro_class_fun => /= [x y Hx Hy | x /negbTE/=-> //].
by rewrite Hx (groupJ Hx) //; case: subsetP => // sHG; rewrite cfunJgen ?sHG.
Qed. | Fact | cfRes_subproof | 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",
"cfunJgen",
"groupJ",
"intro_class_fun",
"is_class_fun",
"sHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes phi | := Cfun 1 (cfRes_subproof phi). | Definition | 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",
"... | [
"Cfun",
"cfRes_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfResE phi : A \subset B -> {in A, cfRes phi =1 phi}. | Proof. by move=> sAB x Ax; rewrite cfunElock mem_gen ?genS. Qed. | Lemma | cfResE | 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",
"... | [
"cfRes",
"cfunElock",
"genS",
"mem_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes1 phi : cfRes phi 1%g = phi 1%g. | Proof. by rewrite cfunElock if_same group1. Qed. | Lemma | cfRes1 | 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",
"... | [
"cfRes",
"cfunElock",
"group1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_is_linear : linear cfRes. | Proof.
by move=> a phi psi; apply/cfunP=> x; rewrite !cfunElock mulrnAr mulrnDl.
Qed. | Lemma | cfRes_is_linear | 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",
"cfRes",
"cfunElock",
"cfunP",
"linear",
"mulrnAr",
"mulrnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_cfun1 : cfRes 1 = 1. | Proof.
apply: cfun_in_genP => x Hx; rewrite cfunElock Hx !cfun1Egen Hx.
by case: subsetP => [-> // | _]; rewrite group1.
Qed. | Lemma | cfRes_cfun1 | 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",
"cfRes",
"cfun1Egen",
"cfunElock",
"cfun_in_genP",
"group1",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_is_monoid_morphism : monoid_morphism cfRes. | Proof.
split=> [|phi psi]; [exact: cfRes_cfun1 | apply/cfunP=> x].
by rewrite !cfunElock mulrnAr mulrnAl -mulrnA mulnb andbb.
Qed. | Lemma | cfRes_is_monoid_morphism | 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",
"cfRes",
"cfRes_cfun1",
"cfunElock",
"cfunP",
"monoid_morphism",
"mulnb",
"mulrnA",
"mulrnAl",
"mulrnAr",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_is_multiplicative | :=
(fun g => (g.2,g.1)) cfRes_is_monoid_morphism. | Definition | cfRes_is_multiplicative | 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",
"... | [
"cfRes_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Res[' H , G ]" | := (@cfRes _ H G) (only parsing) : ring_scope. | Notation | ''Res[' H , G ] | 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",
"... | [
"cfRes"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Res[' H ]" | := 'Res[H, _] : ring_scope. | Notation | ''Res[' H ] | 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 | |
"''Res'" | := 'Res[_] (only parsing) : ring_scope. | Notation | ''Res' | 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 | |
cfResEout phi : ~~ (H \subset G) -> 'Res[H] phi = (phi 1%g)%:A. | Proof.
move/negPf=> not_sHG; apply/cfunP=> x.
by rewrite cfunE cfun1E mulr_natr cfunElock !genGid not_sHG.
Qed. | Lemma | cfResEout | 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",
"cfun1E",
"cfunE",
"cfunElock",
"cfunP",
"genGid",
"mulr_natr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfResRes A phi :
A \subset H -> H \subset G -> 'Res[A] ('Res[H] phi) = 'Res[A] phi. | Proof.
move=> sAH sHG; apply/cfunP=> x; rewrite !cfunElock !genGid !gen_subG sAH sHG.
by rewrite (subset_trans sAH) // -mulrnA mulnb -in_setI (setIidPr _) ?gen_subG.
Qed. | Lemma | cfResRes | 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",
"genGid",
"gen_subG",
"in_setI",
"mulnb",
"mulrnA",
"sHG",
"setIidPr",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_id A psi : 'Res[A] psi = psi. | Proof. by apply/cfun_in_genP=> x Ax; rewrite cfunElock Ax subxx. Qed. | Lemma | cfRes_id | 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",
"cfun_in_genP",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_cfker_Res A phi :
A \subset H -> A \subset cfker phi -> A \subset cfker ('Res[H, G] phi). | Proof.
move=> sAH kerA; apply/subsetP=> x Ax; have Hx := subsetP sAH x Ax.
rewrite inE Hx; apply/forallP=> y; rewrite !cfunElock !genGid groupMl //.
by rewrite !(fun_if phi) cfkerMl // (subsetP kerA).
Qed. | Lemma | sub_cfker_Res | 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",
"cfkerMl",
"cfunElock",
"forallP",
"genGid",
"groupMl",
"inE",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_cfker_Res phi : H \subset cfker phi -> cfker ('Res[H, G] phi) = H. | Proof. by move=> kH; apply/eqP; rewrite eqEsubset cfker_sub sub_cfker_Res. Qed. | Lemma | eq_cfker_Res | 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",
"cfker_sub",
"eqEsubset",
"sub_cfker_Res"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_sub_ker phi : H \subset cfker phi -> 'Res[H, G] phi = (phi 1%g)%:A. | Proof.
move=> kerHphi; have sHG := subset_trans kerHphi (cfker_sub phi).
apply/cfun_inP=> x Hx; have ker_x := subsetP kerHphi x Hx.
by rewrite cfResE // cfunE cfun1E Hx mulr1 cfker1.
Qed. | Lemma | cfRes_sub_ker | 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",
"cfResE",
"cfker",
"cfker1",
"cfker_sub",
"cfun1E",
"cfunE",
"cfun_inP",
"mulr1",
"sHG",
"subsetP",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_Res phi : #['Res[H] phi]%CF %| #[phi]%CF. | Proof. exact: cforder_rmorph. Qed. | Lemma | cforder_Res | 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",
"... | [
"cforder_rmorph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_subproof phi :
is_class_fun <<G>>
[ffun x => phi (if G \subset D then f x else 1%g) *+ (x \in G)]. | Proof.
rewrite genGid; apply: intro_class_fun => [x y Gx Gy | x /negPf-> //].
rewrite Gx groupJ //; case subsetP => // sGD.
by rewrite morphJ ?cfunJ ?mem_morphim ?sGD.
Qed. | Fact | cfMorph_subproof | 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",
"cfunJ",
"genGid",
"groupJ",
"intro_class_fun",
"is_class_fun",
"mem_morphim",
"morphJ",
"sGD",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph phi | := Cfun 1 (cfMorph_subproof phi). | Definition | cfMorph | 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",
"cfMorph_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorphE phi x : G \subset D -> x \in G -> cfMorph phi x = phi (f x). | Proof. by rewrite cfunElock => -> ->. Qed. | Lemma | cfMorphE | 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",
"... | [
"cfMorph",
"cfunElock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph1 phi : cfMorph phi 1%g = phi 1%g. | Proof. by rewrite cfunElock morph1 if_same group1. Qed. | Lemma | cfMorph1 | 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",
"... | [
"cfMorph",
"cfunElock",
"group1",
"morph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorphEout phi : ~~ (G \subset D) -> cfMorph phi = (phi 1%g)%:A. | Proof.
move/negPf=> not_sGD; apply/cfunP=> x; rewrite cfunE cfun1E mulr_natr.
by rewrite cfunElock not_sGD.
Qed. | Lemma | cfMorphEout | 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",
"cfun1E",
"cfunE",
"cfunElock",
"cfunP",
"mulr_natr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_cfun1 : cfMorph 1 = 1. | Proof.
apply/cfun_inP=> x Gx; rewrite cfunElock !cfun1E Gx.
by case: subsetP => [sGD | _]; rewrite ?group1 // mem_morphim ?sGD.
Qed. | Lemma | cfMorph_cfun1 | 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",
"cfun1E",
"cfunElock",
"cfun_inP",
"group1",
"mem_morphim",
"sGD",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_is_linear : linear cfMorph. | Proof.
by move=> a phi psi; apply/cfunP=> x; rewrite !cfunElock mulrnAr -mulrnDl.
Qed. | Fact | cfMorph_is_linear | 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",
"cfunP",
"linear",
"mulrnAr",
"mulrnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_is_monoid_morphism : monoid_morphism cfMorph. | Proof.
split=> [|phi psi]; [exact: cfMorph_cfun1 | apply/cfunP=> x].
by rewrite !cfunElock mulrnAr mulrnAl -mulrnA mulnb andbb.
Qed. | Fact | cfMorph_is_monoid_morphism | 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",
"cfMorph_cfun1",
"cfunElock",
"cfunP",
"monoid_morphism",
"mulnb",
"mulrnA",
"mulrnAl",
"mulrnAr",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_inj : injective cfMorph. | Proof.
move=> phi1 phi2 eq_phi; apply/cfun_inP=> _ /morphimP[x Dx Gx ->].
by rewrite -!cfMorphE // eq_phi.
Qed. | Lemma | cfMorph_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",
"... | [
"Dx",
"apply",
"cfMorph",
"cfMorphE",
"cfun_inP",
"morphimP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_eq1 phi : (cfMorph phi == 1) = (phi == 1). | Proof. exact/rmorph_eq1/cfMorph_inj. Qed. | Lemma | cfMorph_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",
"... | [
"cfMorph",
"cfMorph_inj",
"rmorph_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_morph phi : cfker (cfMorph phi) = G :&: f @*^-1 (cfker phi). | Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
have Dx := subsetP sGD x Gx; rewrite Dx mem_morphim //=.
apply/forallP/forallP=> Kx y.
have [{y} /morphimP[y Dy Gy ->] | fG'y] := boolP (y \in f @* G).
by rewrite -morphM // -!(cfMorphE phi) ?groupM.
by rewrite !cfun0 ?groupMl // mem_morphim.
have [Gy | G'y... | Lemma | cfker_morph | 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",
"cfMorph",
"cfMorphE",
"cfker",
"cfun0",
"forallP",
"groupM",
"groupMl",
"inE",
"last",
"mem_morphim",
"morphM",
"morphimP",
"sGD",
"setP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_morph_im phi : f @* cfker (cfMorph phi) = cfker phi. | Proof. by rewrite cfker_morph // morphim_setIpre (setIidPr (cfker_sub _)). Qed. | Lemma | cfker_morph_im | 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",
"... | [
"cfMorph",
"cfker",
"cfker_morph",
"cfker_sub",
"morphim_setIpre",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_cfker_morph phi (A : {set aT}) :
(A \subset cfker (cfMorph phi)) = (A \subset G) && (f @* A \subset cfker phi). | Proof.
rewrite cfker_morph // subsetI; apply: andb_id2l => sAG.
by rewrite sub_morphim_pre // (subset_trans sAG).
Qed. | Lemma | sub_cfker_morph | 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",
"... | [
"aT",
"apply",
"cfMorph",
"cfker",
"cfker_morph",
"sAG",
"sub_morphim_pre",
"subsetI",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_morphim_cfker phi (A : {set aT}) :
A \subset G -> (f @* A \subset cfker phi) = (A \subset cfker (cfMorph phi)). | Proof. by move=> sAG; rewrite sub_cfker_morph ?sAG. Qed. | Lemma | sub_morphim_cfker | 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",
"... | [
"aT",
"cfMorph",
"cfker",
"sAG",
"sub_cfker_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_morph phi : #[cfMorph phi]%CF = #[phi]%CF. | Proof. exact/cforder_inj_rmorph/cfMorph_inj. Qed. | Lemma | cforder_morph | 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",
"... | [
"cfMorph",
"cfMorph_inj",
"cforder_inj_rmorph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfResMorph (G H : {group aT}) (phi : 'CF(f @* G)) :
H \subset G -> G \subset D -> 'Res (cfMorph phi) = cfMorph ('Res[f @* H] phi). | Proof.
move=> sHG sGD; have sHD := subset_trans sHG sGD.
apply/cfun_inP=> x Hx; have [Gx Dx] := (subsetP sHG x Hx, subsetP sHD x Hx).
by rewrite !(cfMorphE, cfResE) ?morphimS ?mem_morphim //.
Qed. | Lemma | cfResMorph | 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",
"aT",
"apply",
"cfMorph",
"cfMorphE",
"cfResE",
"cfun_inP",
"group",
"mem_morphim",
"morphimS",
"sGD",
"sHD",
"sHG",
"subsetP",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isoGR : isom G R f. | Hypothesis | isoGR | 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",
"... | [
"isom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
defR | := isom_im isoGR. | Let | defR | 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",
"... | [
"isoGR",
"isom_im"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
G1 | := (isom_inv isoGR @* R). | Notation | G1 | 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",
"... | [
"isoGR",
"isom_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
defG : G1 = G | := isom_im (isom_sym isoGR). | Let | defG | 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",
"... | [
"G1",
"isoGR",
"isom_im",
"isom_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_key : unit. | Proof. by []. Qed. | Fact | cfIsom_key | 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",
"... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom | :=
locked_with cfIsom_key (cfMorph \o 'Res[G1] : 'CF(G) -> 'CF(R)). | Definition | cfIsom | 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",
"... | [
"G1",
"cfIsom_key",
"cfMorph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_unlockable | := [unlockable of cfIsom]. | Canonical | cfIsom_unlockable | 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",
"... | [
"cfIsom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsomE phi (x : aT : finType) : x \in G -> cfIsom phi (f x) = phi x. | Proof.
move=> Gx; rewrite unlock cfMorphE //= /restrm ?defG ?cfRes_id ?invmE //.
by rewrite -defR mem_morphim.
Qed. | Lemma | cfIsomE | 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",
"... | [
"aT",
"cfIsom",
"cfMorphE",
"cfRes_id",
"defG",
"defR",
"invmE",
"mem_morphim",
"restrm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom1 phi : cfIsom phi 1%g = phi 1%g. | Proof. by rewrite -(morph1 f) cfIsomE. Qed. | Lemma | cfIsom1 | 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",
"... | [
"cfIsom",
"cfIsomE",
"morph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_is_zmod_morphism : zmod_morphism cfIsom. | Proof. rewrite unlock; exact: raddfB. Qed. | Lemma | cfIsom_is_zmod_morphism | 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",
"... | [
"cfIsom",
"raddfB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_is_additive | := cfIsom_is_zmod_morphism. | Definition | cfIsom_is_additive | 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",
"... | [
"cfIsom_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_is_monoid_morphism : monoid_morphism cfIsom. | Proof. rewrite unlock; exact: (rmorph1 _, rmorphM _). Qed. | Lemma | cfIsom_is_monoid_morphism | 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",
"... | [
"cfIsom",
"monoid_morphism",
"rmorph1",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_is_multiplicative | :=
(fun g => (g.2,g.1)) cfIsom_is_monoid_morphism. | Definition | cfIsom_is_multiplicative | 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",
"... | [
"cfIsom_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_is_scalable : scalable cfIsom. | Proof. rewrite unlock; exact: linearZ_LR. Qed. | Lemma | cfIsom_is_scalable | 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",
"... | [
"cfIsom",
"linearZ_LR",
"scalable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_cfun1 : cfIsom 1 = 1. | Proof. exact: rmorph1. Qed. | Lemma | cfIsom_cfun1 | 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",
"... | [
"cfIsom",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_isom phi : cfker (cfIsom phi) = f @* cfker phi. | Proof.
rewrite unlock cfker_morph // defG cfRes_id morphpre_restrm morphpre_invm.
by rewrite -defR !morphimIim.
Qed. | Lemma | cfker_isom | 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",
"... | [
"cfIsom",
"cfRes_id",
"cfker",
"cfker_morph",
"defG",
"defR",
"morphimIim",
"morphpre_invm",
"morphpre_restrm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsomK : cancel (cfIsom isoGR) (cfIsom (isom_sym isoGR)). | Proof.
move=> phi; apply/cfun_inP=> x Gx; rewrite -{1}(invmE (isom_inj isoGR) Gx).
by rewrite !cfIsomE // -(isom_im isoGR) mem_morphim.
Qed. | Lemma | cfIsomK | 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",
"cfun_inP",
"invmE",
"isoGR",
"isom_im",
"isom_inj",
"isom_sym",
"mem_morphim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsomKV : cancel (cfIsom (isom_sym isoGR)) (cfIsom isoGR). | Proof.
move=> phi; apply/cfun_inP=> y Ry; pose injGR := isom_inj isoGR.
rewrite -{1}[y](invmK injGR) ?(isom_im isoGR) //.
suffices /morphpreP[fGy Gf'y]: y \in invm injGR @*^-1 G by rewrite !cfIsomE.
by rewrite morphpre_invm (isom_im isoGR).
Qed. | Lemma | cfIsomKV | 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",
"cfun_inP",
"invm",
"invmK",
"isoGR",
"isom_im",
"isom_inj",
"isom_sym",
"morphpreP",
"morphpre_invm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_inj : injective (cfIsom isoGR). | Proof. exact: can_inj cfIsomK. Qed. | Lemma | cfIsom_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",
"... | [
"cfIsom",
"cfIsomK",
"isoGR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_eq1 phi : (cfIsom isoGR phi == 1) = (phi == 1). | Proof. exact/rmorph_eq1/cfIsom_inj. Qed. | Lemma | cfIsom_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",
"... | [
"cfIsom",
"cfIsom_inj",
"isoGR",
"rmorph_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_isom phi : #[cfIsom isoGR phi]%CF = #[phi]%CF. | Proof. exact: cforder_inj_rmorph cfIsom_inj. Qed. | Lemma | cforder_isom | 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",
"... | [
"cfIsom",
"cfIsom_inj",
"cforder_inj_rmorph",
"isoGR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
H | := <<B>>%g. | Notation | H | 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 | |
cfMod : 'CF(G / B) -> 'CF(G) | := cfMorph. | Definition | 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",
"... | [
"cfMorph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_Quo (phi : 'CF(G)) | :=
[ffun Hx : coset_of B =>
phi (if B \subset cfker phi then repr Hx else 1%g) *+ (Hx \in G / B)%g]. | Definition | ffun_Quo | 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",
"coset_of",
"repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_subproof phi : is_class_fun <<G / B>> (ffun_Quo phi). | Proof.
rewrite genGid; apply: intro_class_fun => [|Hx /negPf-> //].
move=> _ _ /morphimP[x Nx Gx ->] /morphimP[z Nz Gz ->].
rewrite -morphJ ?mem_morphim ?val_coset_prim ?groupJ //= -gen_subG.
case: subsetP => // KphiH; do 2!case: repr_rcosetP => _ /KphiH/cfkerMl->.
by rewrite cfunJ.
Qed. | Fact | cfQuo_subproof | 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",
"cfkerMl",
"cfunJ",
"ffun_Quo",
"genGid",
"gen_subG",
"groupJ",
"intro_class_fun",
"is_class_fun",
"mem_morphim",
"morphJ",
"morphimP",
"repr_rcosetP",
"subsetP",
"val_coset_prim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo phi | := Cfun 1 (cfQuo_subproof phi). | Definition | 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",
"... | [
"Cfun",
"cfQuo_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"phi / 'B'" | := (cfQuo phi)
(at level 40, left associativity) : cfun_scope. | Notation | phi / 'B' | 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",
"... | [
"cfQuo"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"phi %% 'B'" | := (cfMod phi) (at level 40) : cfun_scope. | Notation | phi %% 'B' | 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",
"... | [
"cfMod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfModE phi x : B <| G -> x \in G -> (phi %% B)%CF x = phi (coset B x). | Proof. by move/normal_norm=> nBG; apply: cfMorphE. Qed. | Lemma | cfModE | 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",
"cfMorphE",
"coset",
"normal_norm"
] | stronger results are needed. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfMod1 phi : (phi %% B)%CF 1%g = phi 1%g. | Proof. exact: cfMorph1. Qed. | Lemma | cfMod1 | 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",
"... | [
"cfMorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMod_cfun1 : (1 %% B)%CF = 1. | Proof. exact: rmorph1. Qed. | Lemma | cfMod_cfun1 | 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",
"... | [
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_mod phi : B <| G -> B \subset cfker (phi %% B). | Proof.
case/andP=> sBG nBG; rewrite cfker_morph // subsetI sBG.
apply: subset_trans _ (ker_sub_pre _ _); rewrite ker_coset_prim subsetI.
by rewrite (subset_trans sBG nBG) sub_gen.
Qed. | Lemma | cfker_mod | 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",
"cfker_morph",
"ker_coset_prim",
"ker_sub_pre",
"sub_gen",
"subsetI",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuoEnorm (phi : 'CF(G)) x :
B \subset cfker phi -> x \in 'N_G(B) -> (phi / B)%CF (coset B x) = phi x. | Proof.
rewrite cfunElock -gen_subG => sHK /setIP[Gx nHx]; rewrite sHK /=.
rewrite mem_morphim // val_coset_prim //.
by case: repr_rcosetP => _ /(subsetP sHK)/cfkerMl->.
Qed. | Lemma | cfQuoEnorm | 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",
"cfkerMl",
"cfunElock",
"coset",
"gen_subG",
"mem_morphim",
"repr_rcosetP",
"sHK",
"setIP",
"subsetP",
"val_coset_prim"
] | Note that cfQuo is nondegenerate even when G does not normalize B. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfQuoE (phi : 'CF(G)) x :
B <| G -> B \subset cfker phi -> x \in G -> (phi / B)%CF (coset B x) = phi x. | Proof. by case/andP=> _ nBG sBK Gx; rewrite cfQuoEnorm // (setIidPl _). Qed. | Lemma | cfQuoE | 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",
"... | [
"cfQuoEnorm",
"cfker",
"coset",
"setIidPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo1 (phi : 'CF(G)) : (phi / B)%CF 1%g = phi 1%g. | Proof. by rewrite cfunElock repr_coset1 group1 if_same. Qed. | Lemma | cfQuo1 | 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",
"... | [
"cfunElock",
"group1",
"repr_coset1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuoEout (phi : 'CF(G)) :
~~ (B \subset cfker phi) -> (phi / B)%CF = (phi 1%g)%:A. | Proof.
move/negPf=> not_kerB; apply/cfunP=> x; rewrite cfunE cfun1E mulr_natr.
by rewrite cfunElock not_kerB.
Qed. | Lemma | cfQuoEout | 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",
"cfun1E",
"cfunE",
"cfunElock",
"cfunP",
"mulr_natr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_cfun1 : (1 / B)%CF = 1. | Proof.
apply/cfun_inP=> Hx G_Hx; rewrite cfunElock !cfun1E G_Hx cfker_cfun1 -gen_subG.
have [x nHx Gx ->] := morphimP G_Hx.
case: subsetP=> [sHG | _]; last by rewrite group1.
by rewrite val_coset_prim //; case: repr_rcosetP => y /sHG/groupM->.
Qed. | Lemma | cfQuo_cfun1 | 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_cfun1",
"cfun1E",
"cfunElock",
"cfun_inP",
"gen_subG",
"group1",
"groupM",
"last",
"morphimP",
"repr_rcosetP",
"sHG",
"subsetP",
"val_coset_prim"
] | cfQuo is only linear on the class functions that have H in their kernel. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfModK : B <| G -> cancel cfMod cfQuo. | Proof.
move=> nsBG phi; apply/cfun_inP=> _ /morphimP[x Nx Gx ->] //.
by rewrite cfQuoE ?cfker_mod ?cfModE.
Qed. | Lemma | cfModK | 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",
"cfMod",
"cfModE",
"cfQuo",
"cfQuoE",
"cfker_mod",
"cfun_inP",
"morphimP"
] | Cancellation properties | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfQuoK :
B <| G -> forall phi, B \subset cfker phi -> (phi / B %% B)%CF = phi. | Proof.
by move=> nsHG phi sHK; apply/cfun_inP=> x Gx; rewrite cfModE ?cfQuoE.
Qed. | Lemma | cfQuoK | 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",
"cfModE",
"cfQuoE",
"cfker",
"cfun_inP",
"nsHG",
"sHK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMod_eq1 psi : B <| G -> (psi %% B == 1)%CF = (psi == 1). | Proof. by move/cfModK/can_eq <-; rewrite rmorph1. Qed. | Lemma | cfMod_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",
"... | [
"can_eq",
"cfModK",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_eq1 phi :
B <| G -> B \subset cfker phi -> (phi / B == 1)%CF = (phi == 1). | Proof. by move=> nsBG kerH; rewrite -cfMod_eq1 // cfQuoK. Qed. | Lemma | cfQuo_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",
"... | [
"cfMod_eq1",
"cfQuoK",
"cfker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"phi / H" | := (cfQuo H phi) : cfun_scope. | Notation | phi / H | 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",
"... | [
"cfQuo"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"phi %% H" | := (@cfMod _ _ H phi) : cfun_scope. | Notation | phi %% H | 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",
"... | [
"cfMod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfResMod H K (psi : 'CF(G / K)) :
H \subset G -> K <| G -> ('Res (psi %% K) = 'Res[H / K] psi %% K)%CF. | Proof. by move=> sHG /andP[_]; apply: cfResMorph. Qed. | Lemma | cfResMod | 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",
"cfResMorph",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_cfker_mod (A : {set gT}) K (psi : 'CF(G / K)) :
K <| G -> (cfker (psi %% K) / K)%g = cfker psi. | Proof. by case/andP=> _ /cfker_morph_im <-. Qed. | Lemma | quotient_cfker_mod | 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",
"cfker_morph_im",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_cfker_mod (A : {set gT}) K (psi : 'CF(G / K)) :
K <| G -> A \subset 'N(K) ->
(A \subset cfker (psi %% K)) = (A / K \subset cfker psi)%g. | Proof.
by move=> nsKG nKA; rewrite -(quotientSGK nKA) ?quotient_cfker_mod// cfker_mod.
Qed. | Lemma | sub_cfker_mod | 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",
"cfker_mod",
"gT",
"nsKG",
"quotientSGK",
"quotient_cfker_mod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_quo H phi :
H <| G -> H \subset cfker (phi) -> cfker (phi / H) = (cfker phi / H)%g. | Proof.
move=> nsHG /cfQuoK {2}<- //; have [sHG nHG] := andP nsHG.
by rewrite cfker_morph 1?quotientGI // cosetpreK (setIidPr _) ?cfker_sub.
Qed. | Lemma | cfker_quo | 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",
"... | [
"cfQuoK",
"cfker",
"cfker_morph",
"cfker_sub",
"cosetpreK",
"nHG",
"nsHG",
"quotientGI",
"sHG",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuoEker phi x :
x \in G -> (phi / cfker phi)%CF (coset (cfker phi) x) = phi x. | Proof. by move/cfQuoE->; rewrite ?cfker_normal. Qed. | Lemma | cfQuoEker | 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",
"... | [
"cfQuoE",
"cfker",
"cfker_normal",
"coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfaithful_quo phi : cfaithful (phi / cfker phi). | Proof. by rewrite cfaithfulE cfker_quo ?cfker_normal ?trivg_quotient. Qed. | Lemma | cfaithful_quo | 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",
"... | [
"cfaithful",
"cfaithfulE",
"cfker",
"cfker_normal",
"cfker_quo",
"trivg_quotient"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfResQuo H K phi :
K \subset cfker phi -> K \subset H -> H \subset G ->
('Res[H / K] (phi / K) = 'Res[H] phi / K)%CF. | Proof.
move=> kerK sKH sHG; apply/cfun_inP=> xb Hxb; rewrite cfResE ?quotientS //.
have{xb Hxb} [x nKx Hx ->] := morphimP Hxb.
by rewrite !cfQuoEnorm ?cfResE// 1?inE ?Hx ?(subsetP sHG)// sub_cfker_Res.
Qed. | Lemma | cfResQuo | 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",
"cfQuoEnorm",
"cfResE",
"cfker",
"cfun_inP",
"inE",
"morphimP",
"quotientS",
"sHG",
"sub_cfker_Res",
"subsetP"
] | Note that there is no requirement that K be normal in H or G. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfQuoInorm K phi :
K \subset cfker phi -> (phi / K)%CF = 'Res ('Res['N_G(K)] phi / K)%CF. | Proof.
move=> kerK; rewrite -cfResQuo ?subsetIl ?quotientInorm ?cfRes_id //.
by rewrite subsetI normG (subset_trans kerK) ?cfker_sub.
Qed. | Lemma | cfQuoInorm | 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",
"... | [
"cfResQuo",
"cfRes_id",
"cfker",
"cfker_sub",
"normG",
"quotientInorm",
"subsetI",
"subsetIl",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_mod H (psi : 'CF(G / H)) : H <| G -> #[psi %% H]%CF = #[psi]%CF. | Proof. by move/cfModK/can_inj/cforder_inj_rmorph->. Qed. | Lemma | cforder_mod | 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",
"... | [
"cfModK",
"cforder_inj_rmorph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_quo H phi :
H <| G -> H \subset cfker phi -> #[phi / H]%CF = #[phi]%CF. | Proof. by move=> nsHG kerHphi; rewrite -cforder_mod ?cfQuoK. Qed. | Lemma | cforder_quo | 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",
"... | [
"cfQuoK",
"cfker",
"cforder_mod",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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