fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
Ind_irr_neq0i : H \subset G -> 'Ind[G, H] 'chi_i != 0.
Proof. by move/cfInd_eq0->; rewrite ?irr_neq0 ?irr_char. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | Ind_irr_neq0 | |
Ind_Iirr(A B : {set gT}) i := cfIirr ('Ind[B, A] 'chi_i). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | Ind_Iirr | |
constt_cfRes_irri : {j | j \in irr_constt ('Res[H, G] 'chi_i)}.
Proof. apply/sigW/neq0_has_constt/Res_irr_neq0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | constt_cfRes_irr | |
constt_cfInd_irri :
H \subset G -> {j | j \in irr_constt ('Ind[G, H] 'chi_i)}.
Proof. by move=> sHG; apply/sigW/neq0_has_constt/Ind_irr_neq0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | constt_cfInd_irr | |
cfker_Resphi :
H \subset G -> phi \is a character -> cfker ('Res[H] phi) = H :&: cfker phi.
Proof.
move=> sHG Nphi; apply/setP=> x; rewrite !cfkerEchar ?cfRes_char // !inE.
by apply/andb_id2l=> Hx; rewrite (subsetP sHG) ?cfResE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfker_Res | |
cfker_Indchi :
H \subset G -> chi \is a character -> chi != 0 ->
cfker ('Ind[G, H] chi) = gcore (cfker chi) G.
Proof.
move=> sHG Nchi nzchi; rewrite !cfker_nzcharE ?cfInd_char ?cfInd_eq0 //.
apply/setP=> x; rewrite inE cfIndE // (can2_eq (mulVKf _) (mulKf _)) ?neq0CG //.
rewrite cfInd1 // mulrA -natrM Lagrange // mulr_natl -sumr_const.
apply/eqP/bigcapP=> [/normC_sum_upper ker_chiG_x y Gy | ker_chiG_x].
by rewrite mem_conjg inE ker_chiG_x ?groupV // => z _; apply: char1_ge_norm.
by apply: eq_bigr => y /groupVr/ker_chiG_x; rewrite mem_conjgV inE => /eqP.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfker_Ind | |
cfker_Ind_irri :
H \subset G -> cfker ('Ind[G, H] 'chi_i) = gcore (cfker 'chi_i) G.
Proof. by move/cfker_Ind->; rewrite ?irr_neq0 ?irr_char. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfker_Ind_irr | |
neq0CGG : (#|G|)%:R != 0 :> algC. Proof. exact: natrG_neq0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | neq0CG | |
neq0CiGG B : (#|G : B|)%:R != 0 :> algC.
Proof. exact: natr_indexg_neq0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | neq0CiG | |
gt0CGG : 0 < #|G|%:R :> algC. Proof. exact: natrG_gt0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | gt0CG | |
gt0CiGG B : 0 < #|G : B|%:R :> algC. Proof. exact: natr_indexg_gt0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | gt0CiG | |
algC'G_pcharG : [pchar algC]^'.-group G.
Proof. by apply/pgroupP=> p _; rewrite inE /= pchar_num. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | algC'G_pchar | |
algC'G:= (algC'G_pchar) (only parsing). | Notation | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | algC'G | |
is_class_fun(B : {set gT}) (f : {ffun gT -> algC}) :=
[forall x, forall y in B, f (x ^ y) == f x] && (support f \subset B). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | is_class_fun | |
intro_class_fun(G : {group gT}) f :
{in G &, forall x y, f (x ^ y) = f x} ->
(forall x, x \notin G -> f x = 0) ->
is_class_fun G (finfun f).
Proof.
move=> fJ Gf; apply/andP; split; last first.
by apply/supportP=> x notAf; rewrite ffunE Gf.
apply/'forall_eqfun_inP=> x y Gy; rewrite !ffunE.
by have [/fJ-> // | notGx] := boolP (x \in G); rewrite !Gf ?groupJr.
Qed.
Variable B : {set gT}.
Local Notation G := <<B>>. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | intro_class_fun | |
classfun: predArgType :=
Classfun {cfun_val; _ : is_class_fun G cfun_val}.
Implicit Types phi psi xi : classfun. | Record | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | classfun | |
Cfun:= locked_with classfun_key (fun flag : nat => Classfun).
HB.instance Definition _ := [isSub for cfun_val].
HB.instance Definition _ := [Choice of classfun by <:]. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | Cfun | |
cfun_eqType: eqType := classfun. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_eqType | |
fun_of_cfunphi := cfun_val phi : gT -> algC. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | fun_of_cfun | |
fun_of_cfun: classfun >-> Funclass. | Coercion | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | fun_of_cfun | |
cfunElockk f fP : @Cfun k (finfun f) fP =1 f.
Proof. by rewrite locked_withE; apply: ffunE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfunElock | |
cfunEf fP : @Cfun 0 (finfun f) fP =1 f.
Proof. exact: cfunElock. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfunE | |
cfunPphi psi : phi =1 psi <-> phi = psi.
Proof. by split=> [/ffunP/val_inj | ->]. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfunP | |
cfun0genphi x : x \notin G -> phi x = 0.
Proof. by case: phi => f fP; case: (andP fP) => _ /supportP; apply. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun0gen | |
cfun_in_genPphi psi : {in G, phi =1 psi} -> phi = psi.
Proof.
move=> eq_phi; apply/cfunP=> x.
by have [/eq_phi-> // | notAx] := boolP (x \in G); rewrite !cfun0gen.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_in_genP | |
cfunJgenphi x y : y \in G -> phi (x ^ y) = phi x.
Proof.
case: phi => f fP Gy; apply/eqP.
by case: (andP fP) => /'forall_forall_inP->.
Qed.
Fact cfun_zero_subproof : is_class_fun G (0 : {ffun _}).
Proof. exact: intro_class_fun. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfunJgen | |
cfun_zero:= Cfun 0 cfun_zero_subproof.
Fact cfun_comp_subproof f phi :
f 0 = 0 -> is_class_fun G [ffun x => f (phi x)].
Proof.
by move=> f0; apply: intro_class_fun => [x y _ /cfunJgen | x /cfun0gen] ->.
Qed. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_zero | |
cfun_compf f0 phi := Cfun 0 (@cfun_comp_subproof f phi f0). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_comp | |
cfun_opp:= cfun_comp (oppr0 _).
Fact cfun_add_subproof phi psi : is_class_fun G [ffun x => phi x + psi x].
Proof.
apply: intro_class_fun => [x y Gx Gy | x notGx]; rewrite ?cfunJgen //.
by rewrite !cfun0gen ?add0r.
Qed. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_opp | |
cfun_addphi psi := Cfun 0 (cfun_add_subproof phi psi).
Fact cfun_indicator_subproof (A : {set gT}) :
is_class_fun G [ffun x => ((x \in G) && (x ^: G \subset A))%:R].
Proof.
apply: intro_class_fun => [x y Gx Gy | x /negbTE/= -> //].
by rewrite groupJr ?classGidl.
Qed. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_add | |
cfun_indicatorA := Cfun 1 (cfun_indicator_subproof A).
Local Notation "''1_' A" := (cfun_indicator A) : ring_scope. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_indicator | |
cfun1Egenx : '1_G x = (x \in G)%:R.
Proof. by rewrite cfunElock andb_idr // => /class_subG->. Qed.
Fact cfun_mul_subproof phi psi : is_class_fun G [ffun x => phi x * psi x].
Proof.
apply: intro_class_fun => [x y Gx Gy | x notGx]; rewrite ?cfunJgen //.
by rewrite cfun0gen ?mul0r.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun1Egen | |
cfun_mulphi psi := Cfun 0 (cfun_mul_subproof phi psi). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_mul | |
cfun_unit:= [pred phi : classfun | [forall x in G, phi x != 0]]. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_unit | |
cfun_invphi :=
if phi \in cfun_unit then cfun_comp (invr0 _) phi else phi. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_inv | |
cfun_scalea := cfun_comp (mulr0 a).
Fact cfun_addA : associative cfun_add.
Proof. by move=> phi psi xi; apply/cfunP=> x; rewrite !cfunE addrA. Qed.
Fact cfun_addC : commutative cfun_add.
Proof. by move=> phi psi; apply/cfunP=> x; rewrite !cfunE addrC. Qed.
Fact cfun_add0 : left_id cfun_zero cfun_add.
Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE add0r. Qed.
Fact cfun_addN : left_inverse cfun_zero cfun_opp cfun_add.
Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE addNr. Qed.
HB.instance Definition _ := GRing.isZmodule.Build classfun
cfun_addA cfun_addC cfun_add0 cfun_addN. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_scale | |
muln_cfunEphi n x : (phi *+ n) x = phi x *+ n.
Proof. by elim: n => [|n IHn]; rewrite ?mulrS !cfunE ?IHn. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | muln_cfunE | |
sum_cfunEI r (P : pred I) (phi : I -> classfun) x :
(\sum_(i <- r | P i) phi i) x = \sum_(i <- r | P i) (phi i) x.
Proof. by elim/big_rec2: _ => [|i _ psi _ <-]; rewrite cfunE. Qed.
Fact cfun_mulA : associative cfun_mul.
Proof. by move=> phi psi xi; apply/cfunP=> x; rewrite !cfunE mulrA. Qed.
Fact cfun_mulC : commutative cfun_mul.
Proof. by move=> phi psi; apply/cfunP=> x; rewrite !cfunE mulrC. Qed.
Fact cfun_mul1 : left_id '1_G cfun_mul.
Proof.
by move=> phi; apply: cfun_in_genP => x Gx; rewrite !cfunE cfun1Egen Gx mul1r.
Qed.
Fact cfun_mulD : left_distributive cfun_mul cfun_add.
Proof. by move=> phi psi xi; apply/cfunP=> x; rewrite !cfunE mulrDl. Qed.
Fact cfun_nz1 : '1_G != 0.
Proof.
by apply/eqP=> /cfunP/(_ 1%g)/eqP; rewrite cfun1Egen cfunE group1 oner_eq0.
Qed.
HB.instance Definition _ := GRing.Zmodule_isComNzRing.Build classfun
cfun_mulA cfun_mulC cfun_mul1 cfun_mulD cfun_nz1. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | sum_cfunE | |
cfun_nzRingType: nzRingType := classfun.
#[deprecated(since="mathcomp 2.4.0",
note="Use cfun_nzRingType instead.")] | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_nzRingType | |
cfun_ringType:= (cfun_nzRingType) (only parsing). | Notation | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_ringType | |
expS_cfunEphi n x : (phi ^+ n.+1) x = phi x ^+ n.+1.
Proof. by elim: n => //= n IHn; rewrite !cfunE IHn. Qed.
Fact cfun_mulV : {in cfun_unit, left_inverse 1 cfun_inv *%R}.
Proof.
move=> phi Uphi; rewrite /cfun_inv Uphi; apply/cfun_in_genP=> x Gx.
by rewrite !cfunE cfun1Egen Gx mulVf ?(forall_inP Uphi).
Qed.
Fact cfun_unitP phi psi : psi * phi = 1 -> phi \in cfun_unit.
Proof.
move/cfunP=> phiK; apply/forall_inP=> x Gx; rewrite -unitfE; apply/unitrP.
by exists (psi x); have:= phiK x; rewrite !cfunE cfun1Egen Gx mulrC.
Qed.
Fact cfun_inv0id : {in [predC cfun_unit], cfun_inv =1 id}.
Proof. by rewrite /cfun_inv => phi /negbTE/= ->. Qed.
HB.instance Definition _ :=
GRing.ComNzRing_hasMulInverse.Build classfun cfun_mulV cfun_unitP cfun_inv0id.
Fact cfun_scaleA a b phi :
cfun_scale a (cfun_scale b phi) = cfun_scale (a * b) phi.
Proof. by apply/cfunP=> x; rewrite !cfunE mulrA. Qed.
Fact cfun_scale1 : left_id 1 cfun_scale.
Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE mul1r. Qed.
Fact cfun_scaleDr : right_distributive cfun_scale +%R.
Proof. by move=> a phi psi; apply/cfunP=> x; rewrite !cfunE mulrDr. Qed.
Fact cfun_scaleDl phi : {morph cfun_scale^~ phi : a b / a + b}.
Proof. by move=> a b; apply/cfunP=> x; rewrite !cfunE mulrDl. Qed.
HB.instance Definition _ := GRing.Zmodule_isLmodule.Build algC classfun
cfun_scaleA cfun_scale1 cfun_scaleDr cfun_scaleDl.
Fact cfun_scaleAl a phi psi : a *: (phi * psi) = (a *: phi) * psi.
Proof. by apply/cfunP=> x; rewrite !cfunE mulrA. Qed.
Fact cf
... | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | expS_cfunE | |
cfAut:= cfun_comp (rmorph0 u). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut | |
cfAut_cfun1iA : cfAut '1_A = '1_A.
Proof. by apply/cfunP=> x; rewrite !cfunElock rmorph_nat. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut_cfun1i | |
cfAutZa phi : cfAut (a *: phi) = u a *: cfAut phi.
Proof. by apply/cfunP=> x; rewrite !cfunE rmorphM. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAutZ | |
cfAut_is_zmod_morphism: zmod_morphism cfAut.
Proof.
by move=> phi psi; apply/cfunP=> x; rewrite ?cfAut_cfun1i // !cfunE /= rmorphB.
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `cfAut_is_zmod_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut_is_zmod_morphism | |
cfAut_is_additive:= cfAut_is_zmod_morphism. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut_is_additive | |
cfAut_is_monoid_morphism: monoid_morphism cfAut.
Proof.
by split=> [|phi psi]; apply/cfunP=> x; rewrite ?cfAut_cfun1i // !cfunE rmorphM.
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `cfAut_is_monoid_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut_is_monoid_morphism | |
cfAut_is_multiplicative:=
(fun g => (g.2,g.1)) cfAut_is_monoid_morphism.
HB.instance Definition _ := GRing.isZmodMorphism.Build classfun classfun cfAut
cfAut_is_zmod_morphism.
HB.instance Definition _ := GRing.isMonoidMorphism.Build classfun classfun cfAut
cfAut_is_monoid_morphism. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut_is_multiplicative | |
cfAut_cfun1: cfAut 1 = 1. Proof. exact: rmorph1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut_cfun1 | |
cfAut_scalable: scalable_for (u \; *:%R) cfAut.
Proof. by move=> a phi; apply/cfunP=> x; rewrite !cfunE rmorphM. Qed.
HB.instance Definition _ :=
GRing.isScalable.Build algC classfun classfun (u \; *:%R) cfAut
cfAut_scalable. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut_scalable | |
cfAut_closed(S : seq classfun) :=
{in S, forall phi, cfAut phi \in S}. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfAut_closed | |
cfRealphi := cfAut conjC phi == phi. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfReal | |
cfConjC_subset(S1 S2 : seq classfun) :=
[/\ uniq S1, {subset S1 <= S2} & cfAut_closed conjC S1].
Fact cfun_vect_iso : Vector.axiom #|classes G| classfun.
Proof.
exists (fun phi => \row_i phi (repr (enum_val i))) => [a phi psi|].
by apply/rowP=> i; rewrite !(mxE, cfunE).
set n := #|_|; pose eK x : 'I_n := enum_rank_in (classes1 _) (x ^: G).
have rV2vP v : is_class_fun G [ffun x => v (eK x) *+ (x \in G)].
apply: intro_class_fun => [x y Gx Gy | x /negbTE/=-> //].
by rewrite groupJr // /eK classGidl.
exists (fun v : 'rV_n => Cfun 0 (rV2vP (v 0))) => [phi | v].
apply/cfun_in_genP=> x Gx; rewrite cfunE Gx mxE enum_rankK_in ?mem_classes //.
by have [y Gy ->] := repr_class <<B>> x; rewrite cfunJgen.
apply/rowP=> i; rewrite mxE cfunE; have /imsetP[x Gx def_i] := enum_valP i.
rewrite def_i; have [y Gy ->] := repr_class <<B>> x.
by rewrite groupJ // /eK classGidl // -def_i enum_valK_in.
Qed.
HB.instance Definition _ := Lmodule_hasFinDim.Build algC classfun cfun_vect_iso. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfConjC_subset | |
cfun_vectType: vectType _ := classfun. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_vectType | |
cfun_baseA : #|classes B ::&: A|.-tuple classfun :=
[tuple of [seq '1_xB | xB in classes B ::&: A]]. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_base | |
classfun_onA := <<cfun_base A>>%VS. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | classfun_on | |
cfdotphi psi := #|B|%:R^-1 * \sum_(x in B) phi x * (psi x)^*. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfdot | |
cfdotrpsi phi := cfdot phi psi. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfdotr | |
cfnormphi := cfdot phi phi. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfnorm | |
seq_of_cfunphi := [:: phi]. | Coercion | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | seq_of_cfun | |
cforderphi := \big[lcmn/1]_(x in <<B>>) #[phi x]%C. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cforder | |
cfConjC_closed:= (cfAut_closed conjC).
Prenex Implicits cfReal. | Notation | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfConjC_closed | |
eqcfP:= (@eqP (cfun_eqType _) _ _) (only parsing). | Notation | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | eqcfP | |
cfkerphi := [set x in D | [forall y, phi (x * y)%g == phi y]]. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfker | |
cfaithfulphi := cfker phi \subset [1]. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfaithful | |
ortho_recS1 S2 :=
all [pred phi | all [pred psi | '[phi, psi] == 0] S2] S1. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | ortho_rec | |
orthogonal:= ortho_rec.
Arguments orthogonal : simpl never. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | orthogonal | |
pair_ortho_recS :=
if S is psi :: S' then ortho_rec psi S' && pair_ortho_rec S' else true. | Fixpoint | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | pair_ortho_rec | |
pairwise_orthogonalS := (0 \notin S) && pair_ortho_rec S. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | pairwise_orthogonal | |
orthonormalS := all [pred psi | '[psi] == 1] S && pair_ortho_rec S. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | orthonormal | |
isometrytau := forall phi psi, '[tau phi, tau psi] = '[phi, psi]. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | isometry | |
isometry_from_tomCFD tau mCFR :=
prop_in2 mCFD (inPhantom (isometry tau))
/\ prop_in1 mCFD (inPhantom (forall phi, in_mem (tau phi) mCFR)). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | isometry_from_to | |
cfun0phi x : x \notin G -> phi x = 0.
Proof. by rewrite -{1}(genGid G) => /(cfun0gen phi). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun0 | |
support_cfunphi : support phi \subset G.
Proof. by apply/subsetP=> g; apply: contraR => /cfun0->. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | support_cfun | |
cfunJphi x y : y \in G -> phi (x ^ y) = phi x.
Proof. by rewrite -{1}(genGid G) => /(cfunJgen phi)->. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfunJ | |
cfun_reprphi x : phi (repr (x ^: G)) = phi x.
Proof. by have [y Gy ->] := repr_class G x; apply: cfunJ. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_repr | |
cfun_inPphi psi : {in G, phi =1 psi} -> phi = psi.
Proof. by rewrite -{1}genGid => /cfun_in_genP. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_inP | |
cfuniEA x : A <| G -> '1_A x = (x \in A)%:R.
Proof.
case/andP=> sAG nAG; rewrite cfunElock genGid.
by rewrite class_sub_norm // andb_idl // => /(subsetP sAG).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfuniE | |
support_cfuniA : A <| G -> support '1_A =i A.
Proof. by move=> nsAG x; rewrite !inE cfuniE // pnatr_eq0 -lt0n lt0b. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | support_cfuni | |
eq_mul_cfuniA phi : A <| G -> {in A, phi * '1_A =1 phi}.
Proof. by move=> nsAG x Ax; rewrite cfunE cfuniE // Ax mulr1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | eq_mul_cfuni | |
eq_cfuniA : A <| G -> {in A, '1_A =1 (1 : 'CF(G))}.
Proof. by rewrite -['1_A]mul1r; apply: eq_mul_cfuni. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | eq_cfuni | |
cfuniG: '1_G = 1.
Proof. by rewrite -[G in '1_G]genGid. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfuniG | |
cfun1Eg : (1 : 'CF(G)) g = (g \in G)%:R.
Proof. by rewrite -cfuniG cfuniE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun1E | |
cfun11: (1 : 'CF(G)) 1%g = 1.
Proof. by rewrite cfun1E group1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun11 | |
prod_cfunEI r (P : pred I) (phi : I -> 'CF(G)) x :
x \in G -> (\prod_(i <- r | P i) phi i) x = \prod_(i <- r | P i) (phi i) x.
Proof.
by move=> Gx; elim/big_rec2: _ => [|i _ psi _ <-]; rewrite ?cfunE ?cfun1E ?Gx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | prod_cfunE | |
exp_cfunEphi n x : x \in G -> (phi ^+ n) x = phi x ^+ n.
Proof. by rewrite -[n]card_ord -!prodr_const; apply: prod_cfunE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | exp_cfunE | |
mul_cfuniA B : '1_A * '1_B = '1_(A :&: B) :> 'CF(G).
Proof.
apply/cfunP=> g; rewrite !cfunElock -natrM mulnb subsetI.
by rewrite andbCA !andbA andbb.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | mul_cfuni | |
cfun_classEx y : '1_(x ^: G) y = ((x \in G) && (y \in x ^: G))%:R.
Proof.
rewrite cfunElock genGid class_sub_norm ?class_norm //; congr (_ : bool)%:R.
by apply: andb_id2r => /imsetP[z Gz ->]; rewrite groupJr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_classE | |
cfun_on_sumA :
'CF(G, A) = (\sum_(xG in classes G | xG \subset A) <['1_xG]>)%VS.
Proof.
by rewrite ['CF(G, A)]span_def big_image; apply: eq_bigl => xG; rewrite !inE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_on_sum | |
cfun_onPA phi :
reflect (forall x, x \notin A -> phi x = 0) (phi \in 'CF(G, A)).
Proof.
apply: (iffP idP) => [/coord_span-> x notAx | Aphi].
set b := cfun_base G A; rewrite sum_cfunE big1 // => i _; rewrite cfunE.
have /mapP[xG]: b`_i \in b by rewrite -tnth_nth mem_tnth.
rewrite mem_enum => /setIdP[/imsetP[y Gy ->] Ay] ->.
by rewrite cfun_classE Gy (contraNF (subsetP Ay x)) ?mulr0.
suffices <-: \sum_(xG in classes G) phi (repr xG) *: '1_xG = phi.
apply: memv_suml => _ /imsetP[x Gx ->]; rewrite rpredZeq cfun_repr.
have [s_xG_A | /subsetPn[_ /imsetP[y Gy ->]]] := boolP (x ^: G \subset A).
by rewrite cfun_on_sum [_ \in _](sumv_sup (x ^: G)) ?mem_classes ?orbT.
by move/Aphi; rewrite cfunJ // => ->; rewrite eqxx.
apply/cfun_inP=> x Gx; rewrite sum_cfunE (bigD1 (x ^: G)) ?mem_classes //=.
rewrite cfunE cfun_repr cfun_classE Gx class_refl mulr1.
rewrite big1 ?addr0 // => _ /andP[/imsetP[y Gy ->]]; apply: contraNeq.
rewrite cfunE cfun_repr cfun_classE Gy mulf_eq0 => /norP[_].
by rewrite pnatr_eq0 -lt0n lt0b => /class_eqP->.
Qed.
Arguments cfun_onP {A phi}. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_onP | |
cfun_on0A phi x : phi \in 'CF(G, A) -> x \notin A -> phi x = 0.
Proof. by move/cfun_onP; apply. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_on0 | |
sum_by_classes(R : nzRingType) (F : gT -> R) :
{in G &, forall g h, F (g ^ h) = F g} ->
\sum_(g in G) F g = \sum_(xG in classes G) #|xG|%:R * F (repr xG).
Proof.
move=> FJ; rewrite {1}(partition_big _ _ ((@mem_classes gT)^~ G)) /=.
apply: eq_bigr => _ /imsetP[x Gx ->]; have [y Gy ->] := repr_class G x.
rewrite mulr_natl -sumr_const FJ {y Gy}//; apply/esym/eq_big=> y /=.
apply/idP/andP=> [xGy | [Gy /eqP<-]]; last exact: class_refl.
by rewrite (class_eqP xGy) (subsetP (class_subG Gx (subxx _))).
by case/imsetP=> z Gz ->; rewrite FJ.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | sum_by_classes | |
cfun_base_freeA : free (cfun_base G A).
Proof.
have b_i (i : 'I_#|classes G ::&: A|) : (cfun_base G A)`_i = '1_(enum_val i).
by rewrite /enum_val -!tnth_nth tnth_map.
apply/freeP => s S0 i; move/cfunP/(_ (repr (enum_val i))): S0.
rewrite sum_cfunE (bigD1 i) //= big1 ?addr0 => [|j].
rewrite b_i !cfunE; have /setIdP[/imsetP[x Gx ->] _] := enum_valP i.
by rewrite cfun_repr cfun_classE Gx class_refl mulr1.
apply: contraNeq; rewrite b_i !cfunE mulf_eq0 => /norP[_].
rewrite -(inj_eq enum_val_inj).
have /setIdP[/imsetP[x _ ->] _] := enum_valP i; rewrite cfun_repr.
have /setIdP[/imsetP[y Gy ->] _] := enum_valP j; rewrite cfun_classE Gy.
by rewrite pnatr_eq0 -lt0n lt0b => /class_eqP->.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_base_free | |
dim_cfun: \dim 'CF(G) = #|classes G|.
Proof. by rewrite dimvf /dim /= genGid. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | dim_cfun | |
dim_cfun_onA : \dim 'CF(G, A) = #|classes G ::&: A|.
Proof. by rewrite (eqnP (cfun_base_free A)) size_tuple. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | dim_cfun_on | |
dim_cfun_on_abelianA : abelian G -> A \subset G -> \dim 'CF(G, A) = #|A|.
Proof.
move/abelian_classP=> cGG sAG; rewrite -(card_imset _ set1_inj) dim_cfun_on.
apply/eq_card=> xG; rewrite !inE.
apply/andP/imsetP=> [[/imsetP[x Gx ->] Ax] | [x Ax ->]] {xG}.
by rewrite cGG ?sub1set // in Ax *; exists x.
by rewrite -{1}(cGG x) ?mem_classes ?(subsetP sAG) ?sub1set.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | dim_cfun_on_abelian | |
cfuni_onA : '1_A \in 'CF(G, A).
Proof.
apply/cfun_onP=> x notAx; rewrite cfunElock genGid.
by case: andP => // [[_ s_xG_A]]; rewrite (subsetP s_xG_A) ?class_refl in notAx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfuni_on | |
mul_cfuni_onA phi : phi * '1_A \in 'CF(G, A).
Proof.
by apply/cfun_onP=> x /(cfun_onP (cfuni_on A)) Ax0; rewrite cfunE Ax0 mulr0.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | mul_cfuni_on | |
cfun_onEphi A : (phi \in 'CF(G, A)) = (support phi \subset A).
Proof. exact: (sameP cfun_onP supportP). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_onE | |
cfun_onTphi : phi \in 'CF(G, [set: gT]).
Proof. by rewrite cfun_onE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfun_onT |
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