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 |
|---|---|---|---|---|---|---|
cfnorm_orthogonal: '[\sum_(xi <- S) nu xi] = \sum_(xi <- S) '[xi].
Proof.
rewrite -(eq_bigr _ (fun _ _ => scale1r _)) cfnorm_sum_orthogonal.
by apply: eq_bigr => xi; rewrite normCK conjC1 !mul1r.
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/vcharacter.v | cfnorm_orthogonal | |
orthogonal_spanS phi :
pairwise_orthogonal S -> phi \in <<S>>%VS ->
{z | z = fun xi => '[phi, xi] / '[xi] & phi = \sum_(xi <- S) z xi *: xi}.
Proof.
move=> oSS /free_span[|c -> _]; first exact: orthogonal_free.
set z := fun _ => _ : algC; exists z => //; apply: eq_big_seq => u Su.
rewrite /z cfproj_sum_orthogonal // mulfK // cfnorm_eq0.
by rewrite (memPn _ u Su); case/andP: oSS.
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/vcharacter.v | orthogonal_span | |
map_orthonormal: orthonormal (map nu S).
Proof.
rewrite !orthonormalE map_pairwise_orthogonal // andbT.
by apply/allP=> _ /mapP[xi Sxi ->]; rewrite /= Inu ?nS1 // mem_zchar.
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/vcharacter.v | map_orthonormal | |
cfproj_sum_orthonormalz phi :
phi \in S -> '[\sum_(xi <- S) z xi *: nu xi, nu phi] = z phi.
Proof. by move=> Sphi; rewrite cfproj_sum_orthogonal // nS1 // 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/vcharacter.v | cfproj_sum_orthonormal | |
cfdot_sum_orthonormalz1 z2 :
'[\sum_(xi <- S) z1 xi *: xi, \sum_(xi <- S) z2 xi *: xi]
= \sum_(xi <- S) z1 xi * (z2 xi)^*.
Proof.
rewrite cfdot_sum_orthogonal //; apply: eq_big_seq => phi /nS1->.
by rewrite 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/vcharacter.v | cfdot_sum_orthonormal | |
cfnorm_sum_orthonormalz :
'[\sum_(xi <- S) z xi *: nu xi] = \sum_(xi <- S) `|z xi| ^+ 2.
Proof.
rewrite cfnorm_sum_orthogonal //.
by apply: eq_big_seq => xi /nS1->; rewrite 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/vcharacter.v | cfnorm_sum_orthonormal | |
cfnorm_map_orthonormal: '[\sum_(xi <- S) nu xi] = (size S)%:R.
Proof.
by rewrite cfnorm_orthogonal // (eq_big_seq _ nS1) big_tnth sumr_const card_ord.
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/vcharacter.v | cfnorm_map_orthonormal | |
orthonormal_spanphi :
phi \in <<S>>%VS ->
{z | z = fun xi => '[phi, xi] & phi = \sum_(xi <- S) z xi *: xi}.
Proof.
case/orthogonal_span=> // _ -> {2}->; set z := fun _ => _ : algC.
by exists z => //; apply: eq_big_seq => xi /nS1->; rewrite divr1.
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/vcharacter.v | orthonormal_span | |
cfnorm_orthonormalS :
orthonormal S -> '[\sum_(xi <- S) xi] = (size S)%:R.
Proof. exact: cfnorm_map_orthonormal. 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/vcharacter.v | cfnorm_orthonormal | |
vchar_orthonormalPS :
{subset S <= 'Z[irr G]} ->
reflect (exists I : {set Iirr G}, exists b : Iirr G -> bool,
perm_eq S [seq (-1) ^+ b i *: 'chi_i | i in I])
(orthonormal S).
Proof.
move=> vcS; apply: (equivP orthonormalP).
split=> [[uniqS oSS] | [I [b defS]]]; last first.
split=> [|xi1 xi2]; rewrite ?(perm_mem defS).
rewrite (perm_uniq defS) map_inj_uniq ?enum_uniq // => i j /eqP.
by rewrite eq_signed_irr => /andP[_ /eqP].
case/mapP=> [i _ ->] /mapP[j _ ->]; rewrite eq_signed_irr.
rewrite cfdotZl cfdotZr rmorph_sign mulrA cfdot_irr -signr_addb mulr_natr.
by rewrite mulrb andbC; case: eqP => //= ->; rewrite addbb eqxx.
pose I := [set i | ('chi_i \in S) || (- 'chi_i \in S)].
pose b i := - 'chi_i \in S; exists I, b.
apply: uniq_perm => // [|xi].
rewrite map_inj_uniq ?enum_uniq // => i j /eqP.
by rewrite eq_signed_irr => /andP[_ /eqP].
apply/idP/mapP=> [Sxi | [i Ii ->{xi}]]; last first.
move: Ii; rewrite mem_enum inE orbC -/(b i).
by case b_i: (b i); rewrite (scale1r, scaleN1r).
have: '[xi] = 1 by rewrite oSS ?eqxx.
have vc_xi := vcS _ Sxi; rewrite cfdot_sum_irr.
case/natr_sum_eq1 => [i _ | i [_ /eqP norm_xi_i xi_i'_0]].
by rewrite -normCK rpredX // natr_norm_int ?Cint_cfdot_vchar_irr.
suffices def_xi: xi = (-1) ^+ b i *: 'chi_i.
exists i; rewrite // mem_enum inE -/(b i) orbC.
by case: (b i) def_xi Sxi => // ->; rewrite scale1r.
move: Sxi; rewrite [xi]cfun_sum_cfdot (bigD1 i) //.
rewrite big1 //= ?addr0 => [|j ne_ji]; last first.
apply/eq
... | 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/vcharacter.v | vchar_orthonormalP | |
vchar_norm1Pphi :
phi \in 'Z[irr G] -> '[phi] = 1 ->
exists b : bool, exists i : Iirr G, phi = (-1) ^+ b *: 'chi_i.
Proof.
move=> Zphi phiN1.
have: orthonormal phi by rewrite /orthonormal/= phiN1 eqxx.
case/vchar_orthonormalP=> [xi /predU1P[->|] // | I [b def_phi]].
have: phi \in (phi : seq _) := mem_head _ _.
by rewrite (perm_mem def_phi) => /mapP[i _ ->]; exists (b i), i.
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/vcharacter.v | vchar_norm1P | |
zchar_small_normphi n :
phi \in 'Z[irr G] -> '[phi] = n%:R -> (n < 4)%N ->
{S : n.-tuple 'CF(G) |
[/\ orthonormal S, {subset S <= 'Z[irr G]} & phi = \sum_(xi <- S) xi]}.
Proof.
move=> Zphi def_n lt_n_4.
pose S := [seq '[phi, 'chi_i] *: 'chi_i | i in irr_constt phi].
have def_phi: phi = \sum_(xi <- S) xi.
rewrite big_image big_mkcond {1}[phi]cfun_sum_cfdot.
by apply: eq_bigr => i _; rewrite if_neg; case: eqP => // ->; rewrite scale0r.
have orthS: orthonormal S.
apply/orthonormalP; split=> [|_ _ /mapP[i phi_i ->] /mapP[j _ ->]].
rewrite map_inj_in_uniq ?enum_uniq // => i j; rewrite mem_enum => phi_i _.
by move/eqP; rewrite eq_scaled_irr (negbTE phi_i) => /andP[_ /= /eqP].
rewrite eq_scaled_irr cfdotZl cfdotZr cfdot_irr mulrA mulr_natr mulrb.
rewrite mem_enum in phi_i; rewrite (negbTE phi_i) andbC; case: eqP => // <-.
have /natrP[m def_m] := natr_norm_int (Cint_cfdot_vchar_irr i Zphi).
apply/eqP; rewrite eqxx /= -normCK def_m -natrX eqr_nat eqn_leq lt0n.
rewrite expn_eq0 andbT -eqC_nat -def_m normr_eq0 [~~ _]phi_i andbT.
rewrite (leq_exp2r _ 1) // -ltnS -(@ltn_exp2r _ _ 2) //.
apply: leq_ltn_trans lt_n_4; rewrite -leC_nat -def_n natrX.
rewrite cfdot_sum_irr (bigD1 i) //= -normCK def_m addrC -subr_ge0 addrK.
by rewrite sumr_ge0 // => ? _; apply: mul_conjC_ge0.
have <-: size S = n.
by apply/eqP; rewrite -eqC_nat -def_n def_phi cfnorm_orthonormal.
exists (in_tuple S); split=> // _ /mapP[i _ ->].
by rewrite scale_zchar ?irr_vchar // Cint_cfdot_vchar_irr.
... | 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/vcharacter.v | zchar_small_norm | |
vchar_norm2phi :
phi \in 'Z[irr G, G^#] -> '[phi] = 2 ->
exists i, exists2 j, j != i & phi = 'chi_i - 'chi_j.
Proof.
rewrite zchar_split cfunD1E => /andP[Zphi phi1_0].
case/zchar_small_norm => // [[[|chi [|xi [|?]]] //= S2]].
case=> /andP[/and3P[Nchi Nxi _] /= ochi] /allP/and3P[Zchi Zxi _].
rewrite big_cons big_seq1 => def_phi.
have [b [i def_chi]] := vchar_norm1P Zchi (eqP Nchi).
have [c [j def_xi]] := vchar_norm1P Zxi (eqP Nxi).
have neq_ji: j != i.
apply: contraTneq ochi; rewrite !andbT def_chi def_xi => ->.
rewrite cfdotZl cfdotZr rmorph_sign cfnorm_irr mulr1 -signr_addb.
by rewrite signr_eq0.
have neq_bc: b != c.
apply: contraTneq phi1_0; rewrite def_phi def_chi def_xi => ->.
rewrite -scalerDr !cfunE mulf_eq0 signr_eq0 eq_le lt_geF //.
by rewrite ltr_pDl ?irr1_gt0.
rewrite {}def_phi {}def_chi {}def_xi !scaler_sign.
case: b c neq_bc => [|] [|] // _; last by exists i, j.
by exists j, i; rewrite 1?eq_sym // addrC.
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/vcharacter.v | vchar_norm2 | |
Zisometry_of_cfnorm(tauS : seq 'CF(G)) :
pairwise_orthogonal S -> pairwise_orthogonal tauS ->
map cfnorm tauS = map cfnorm S -> {subset tauS <= 'Z[irr G]} ->
{tau : {linear 'CF(L) -> 'CF(G)} | map tau S = tauS
& {in 'Z[S], isometry tau, to 'Z[irr G]}}.
Proof.
move=> oSS oTT /isometry_of_cfnorm[||tau defT Itau] // Z_T; exists tau => //.
split=> [|_ /zchar_nth_expansion[u Zu ->]].
by apply: sub_in2 Itau; apply: zchar_span.
rewrite big_seq linear_sum rpred_sum // => xi Sxi.
by rewrite linearZ scale_zchar ?Z_T // -defT map_f ?mem_nth.
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/vcharacter.v | Zisometry_of_cfnorm | |
Zisometry_of_isof :
free S -> {in S, isometry f, to 'Z[irr G]} ->
{tau : {linear 'CF(L) -> 'CF(G)} | {in S, tau =1 f}
& {in 'Z[S], isometry tau, to 'Z[irr G]}}.
Proof.
move=> freeS [If Zf]; have [tau Dtau Itau] := isometry_of_free freeS If.
exists tau => //; split; first by apply: sub_in2 Itau; apply: zchar_span.
move=> _ /zchar_nth_expansion[a Za ->]; rewrite linear_sum rpred_sum // => i _.
by rewrite linearZ rpredZ_int ?Dtau ?Zf ?mem_nth.
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/vcharacter.v | Zisometry_of_iso | |
Zisometry_injA nu :
{in 'Z[S, A] &, isometry nu} -> {in 'Z[S, A] &, injective nu}.
Proof. by move/isometry_raddf_inj; apply; apply: rpredB. 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/vcharacter.v | Zisometry_inj | |
isometry_in_zcharnu : {in S &, isometry nu} -> {in 'Z[S] &, isometry nu}.
Proof.
move=> Inu _ _ /zchar_nth_expansion[u Zu ->] /zchar_nth_expansion[v Zv ->].
rewrite !raddf_sum; apply: eq_bigr => j _ /=.
rewrite !cfdot_suml; apply: eq_bigr => i _.
by rewrite !raddfZ_int //= !cfdotZl !cfdotZr Inu ?mem_nth.
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/vcharacter.v | isometry_in_zchar | |
cfAut_zcharS A psi :
cfAut_closed u S -> psi \in 'Z[S, A] -> psi^u \in 'Z[S, A].
Proof.
rewrite zchar_split => SuS /andP[/zchar_nth_expansion[z Zz Dpsi] Apsi].
rewrite zchar_split cfAut_on {}Apsi {psi}Dpsi rmorph_sum rpred_sum //= => i _.
by rewrite cfAutZ_Cint // scale_zchar // mem_zchar ?SuS ?mem_nth.
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/vcharacter.v | cfAut_zchar | |
cfAut_vcharA psi : psi \in 'Z[irr G, A] -> psi^u \in 'Z[irr G, A].
Proof. by apply: cfAut_zchar; apply: irr_aut_closed. 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/vcharacter.v | cfAut_vchar | |
sub_aut_zcharS A psi :
{subset S <= 'Z[irr G]} -> psi \in 'Z[S, A] -> psi^u \in 'Z[S, A] ->
psi - psi^u \in 'Z[S, A^#].
Proof.
move=> Z_S Spsi Spsi_u; rewrite zcharD1 !cfunE subr_eq0 rpredB //=.
by rewrite aut_intr // Cint_vchar1 // (zchar_trans Z_S) ?(zcharW Spsi).
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/vcharacter.v | sub_aut_zchar | |
conjC_vcharAutchi x : chi \in 'Z[irr G] -> (u (chi x))^* = u (chi x)^*.
Proof.
case/vcharP=> chi1 Nchi1 [chi2 Nchi2 ->].
by rewrite !cfunE !rmorphB /= !conjC_charAut.
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/vcharacter.v | conjC_vcharAut | |
cfdot_aut_vcharphi chi :
chi \in 'Z[irr G] -> '[phi^u , chi^u] = u '[phi, chi].
Proof.
by case/vcharP=> chi1 Nchi1 [chi2 Nchi2 ->]; rewrite !raddfB /= !cfdot_aut_char.
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/vcharacter.v | cfdot_aut_vchar | |
vchar_autA chi : (chi^u \in 'Z[irr G, A]) = (chi \in 'Z[irr G, A]).
Proof.
rewrite !(zchar_split _ A) cfAut_on; congr (_ && _).
apply/idP/idP=> [Zuchi|]; last exact: cfAut_vchar.
rewrite [chi]cfun_sum_cfdot rpred_sum // => i _.
rewrite scale_zchar ?irr_vchar //.
by rewrite -(intr_aut u) -cfdot_aut_irr -aut_IirrE Cint_cfdot_vchar_irr.
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/vcharacter.v | vchar_aut | |
cfConjC_vchar:= cfAut_vchar Num.conj. | 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/vcharacter.v | cfConjC_vchar | |
cfRes_vcharphi : phi \in 'Z[irr G] -> 'Res[H] phi \in 'Z[irr H].
Proof.
case/vcharP=> xi1 Nx1 [xi2 Nxi2 ->].
by rewrite raddfB rpredB ?char_vchar ?cfRes_char.
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/vcharacter.v | cfRes_vchar | |
cfRes_vchar_onA phi :
H \subset G -> phi \in 'Z[irr G, A] -> 'Res[H] phi \in 'Z[irr H, A].
Proof.
rewrite zchar_split => sHG /andP[Zphi Aphi]; rewrite zchar_split cfRes_vchar //.
apply/cfun_onP=> x /(cfun_onP Aphi); rewrite !cfunElock !genGid sHG => ->.
exact: mul0rn.
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/vcharacter.v | cfRes_vchar_on | |
cfInd_vcharphi : phi \in 'Z[irr H] -> 'Ind[G] phi \in 'Z[irr G].
Proof.
move=> /vcharP[xi1 Nx1 [xi2 Nxi2 ->]].
by rewrite raddfB rpredB ?char_vchar ?cfInd_char.
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/vcharacter.v | cfInd_vchar | |
sub_conjC_vcharA phi :
phi \in 'Z[irr G, A] -> phi - (phi^*)%CF \in 'Z[irr G, A^#].
Proof.
move=> Zphi; rewrite sub_aut_zchar ?cfAut_zchar // => _ /irrP[i ->].
exact: irr_vchar.
exact: cfConjC_irr.
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/vcharacter.v | sub_conjC_vchar | |
Frobenius_kernel_exists:
[Frobenius G with complement H] -> {K : {group gT} | [Frobenius G = K ><| H]}.
Proof.
move=> frobG; have [_ ntiHG] := andP frobG.
have [[_ sHG regGH][_ tiHG /eqP defNH]] := (normedTI_memJ_P ntiHG, and3P ntiHG).
suffices /sigW[K defG]: exists K, gval K ><| H == G by exists K; apply/andP.
pose K1 := G :\: cover (H^# :^: G).
have oK1: #|K1| = #|G : H|.
rewrite cardsD (setIidPr _); last first.
rewrite cover_imset; apply/bigcupsP=> x Gx.
by rewrite sub_conjg conjGid ?groupV // (subset_trans (subsetDl _ _)).
rewrite (cover_partition (partition_normedTI ntiHG)) -(Lagrange sHG).
by rewrite (card_support_normedTI ntiHG) (cardsD1 1%g) group1 mulSn addnK.
suffices extG i: {j | {in H, 'chi[G]_j =1 'chi[H]_i} & K1 \subset cfker 'chi_j}.
pose K := [group of \bigcap_i cfker 'chi_(s2val (extG i))].
have nKH: H \subset 'N(K).
by apply/norms_bigcap/bigcapsP=> i _; apply: subset_trans (cfker_norm _).
have tiKH: K :&: H = 1%g.
apply/trivgP; rewrite -(TI_cfker_irr H) /= setIC; apply/bigcapsP=> i _.
apply/subsetP=> x /setIP[Hx /bigcapP/(_ i isT)/=]; rewrite !cfkerEirr !inE.
by case: (extG i) => /= j def_j _; rewrite !def_j.
exists K; rewrite sdprodE // eqEcard TI_cardMg // mul_subG //=; last first.
by rewrite (bigcap_min (0 : Iirr H)) ?cfker_sub.
rewrite -(Lagrange sHG) mulnC leq_pmul2r // -oK1 subset_leq_card //.
by apply/bigcapsP=> i _; case: (extG i).
case i0: (i == 0).
exists 0 => [x Hx|]; last by rewrite irr0 cfker_cfun1 subsetDl.
by rewri
... | 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/vcharacter.v | Frobenius_kernel_exists | |
dirr(gT : finGroupType) (B : {set gT}) : {pred 'CF(B)} :=
[pred f | (f \in irr B) || (- f \in irr B)].
Arguments dirr {gT}. | 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/vcharacter.v | dirr | |
Definition_ := GRing.isOppClosed.Build (classfun G) (dirr G)
dirr_oppr_closed. | HB.instance | 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/vcharacter.v | Definition | |
dirr_oppv : (- v \in dirr G) = (v \in dirr G). Proof. exact: rpredN. 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/vcharacter.v | dirr_opp | |
dirr_signn v : ((-1)^+ n *: v \in dirr G) = (v \in dirr G).
Proof. exact: rpredZsign. 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/vcharacter.v | dirr_sign | |
irr_dirri : 'chi_i \in dirr G.
Proof. by rewrite !inE mem_irr. 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/vcharacter.v | irr_dirr | |
dirrPf :
reflect (exists b : bool, exists i, f = (-1) ^+ b *: 'chi_i) (f \in dirr G).
Proof.
apply: (iffP idP) => [| [b [i ->]]]; last by rewrite dirr_sign irr_dirr.
case/orP=> /irrP[i Hf]; first by exists false, i; rewrite scale1r.
by exists true, i; rewrite scaleN1r -Hf opprK.
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/vcharacter.v | dirrP | |
dirrEphi : phi \in dirr G = (phi \in 'Z[irr G]) && ('[phi] == 1).
Proof.
apply/dirrP/andP=> [[b [i ->]] | [Zphi /eqP/vchar_norm1P]]; last exact.
by rewrite rpredZsign irr_vchar cfnorm_sign cfnorm_irr.
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/vcharacter.v | dirrE | |
cfdot_dirrf g : f \in dirr G -> g \in dirr G ->
'[f, g] = (if f == - g then -1 else (f == g)%:R).
Proof.
case/dirrP=> [b1 [i1 ->]] /dirrP[b2 [i2 ->]].
rewrite cfdotZl cfdotZr rmorph_sign mulrA -signr_addb cfdot_irr.
rewrite -scaleNr -signrN !eq_scaled_irr signr_eq0 !(inj_eq signr_inj) /=.
by rewrite -!negb_add addbN mulr_sign -mulNrn mulrb; case: ifP.
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/vcharacter.v | cfdot_dirr | |
dirr_norm1phi : phi \in 'Z[irr G] -> '[phi] = 1 -> phi \in dirr G.
Proof. by rewrite dirrE => -> -> /=. 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/vcharacter.v | dirr_norm1 | |
dirr_autu phi : (cfAut u phi \in dirr G) = (phi \in dirr G).
Proof.
rewrite !dirrE vchar_aut; apply: andb_id2l => /cfdot_aut_vchar->.
exact: fmorph_eq1.
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/vcharacter.v | dirr_aut | |
dIirr(B : {set gT}) := (bool * (Iirr B))%type. | 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/vcharacter.v | dIirr | |
dirr1(B : {set gT}) : dIirr B := (false, 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/vcharacter.v | dirr1 | |
ndirr(B : {set gT}) (i : dIirr B) : dIirr B :=
(~~ i.1, i.2). | 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/vcharacter.v | ndirr | |
ndirr_diff(i : dIirr G) : ndirr i != i.
Proof. by case: i => [] [|] i. 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/vcharacter.v | ndirr_diff | |
ndirrK: involutive (@ndirr G).
Proof. by move=> [b i]; rewrite /ndirr /= negbK. 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/vcharacter.v | ndirrK | |
ndirr_inj: injective (@ndirr G).
Proof. exact: (inv_inj ndirrK). 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/vcharacter.v | ndirr_inj | |
dchi(B : {set gT}) (i : dIirr B) : 'CF(B) := (-1)^+ i.1 *: 'chi_i.2. | 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/vcharacter.v | dchi | |
dchi1: dchi (dirr1 G) = 1.
Proof. by rewrite /dchi scale1r irr0. 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/vcharacter.v | dchi1 | |
dirr_dchii : dchi i \in dirr G.
Proof. by apply/dirrP; exists i.1; exists i.2. 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/vcharacter.v | dirr_dchi | |
dIrrPphi : reflect (exists i, phi = dchi i) (phi \in dirr G).
Proof.
by apply: (iffP idP)=> [/dirrP[b]|] [i ->]; [exists (b, i) | apply: dirr_dchi].
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/vcharacter.v | dIrrP | |
dchi_ndirrE(i : dIirr G) : dchi (ndirr i) = - dchi i.
Proof. by case: i => [b i]; rewrite /ndirr /dchi signrN scaleNr. 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/vcharacter.v | dchi_ndirrE | |
cfdot_dchi(i j : dIirr G) :
'[dchi i, dchi j] = (i == j)%:R - (i == ndirr j)%:R.
Proof.
case: i => bi i; case: j => bj j; rewrite cfdot_dirr ?dirr_dchi // !xpair_eqE.
rewrite -dchi_ndirrE !eq_scaled_irr signr_eq0 !(inj_eq signr_inj) /=.
by rewrite -!negb_add addbN negbK; case: andP => [[->]|]; rewrite ?subr0 ?add0r.
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/vcharacter.v | cfdot_dchi | |
dchi_vchari : dchi i \in 'Z[irr G].
Proof. by case: i => b i; rewrite rpredZsign irr_vchar. 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/vcharacter.v | dchi_vchar | |
cfnorm_dchi(i : dIirr G) : '[dchi i] = 1.
Proof. by case: i => b i; rewrite cfnorm_sign cfnorm_irr. 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/vcharacter.v | cfnorm_dchi | |
dirr_inj: injective (@dchi G).
Proof.
case=> b1 i1 [b2 i2] /eqP; rewrite eq_scaled_irr (inj_eq signr_inj) /=.
by rewrite signr_eq0 -xpair_eqE => /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/vcharacter.v | dirr_inj | |
dirr_dIirr(B : {set gT}) J (f : J -> 'CF(B)) j : dIirr B :=
odflt (dirr1 B) [pick i | dchi i == f j]. | 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/vcharacter.v | dirr_dIirr | |
dirr_dIirrPEJ (f : J -> 'CF(G)) (P : pred J) :
(forall j, P j -> f j \in dirr G) ->
forall j, P j -> dchi (dirr_dIirr f j) = f j.
Proof.
rewrite /dirr_dIirr => dirrGf j Pj; case: pickP => [i /eqP //|].
by have /dIrrP[i-> /(_ i)/eqP] := dirrGf j Pj.
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/vcharacter.v | dirr_dIirrPE | |
dirr_dIirrEJ (f : J -> 'CF(G)) :
(forall j, f j \in dirr G) -> forall j, dchi (dirr_dIirr f j) = f j.
Proof. by move=> dirrGf j; apply: (@dirr_dIirrPE _ _ xpredT). 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/vcharacter.v | dirr_dIirrE | |
dirr_constt(B : {set gT}) (phi: 'CF(B)) : {set (dIirr B)} :=
[set i | 0 < '[phi, dchi i]]. | 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/vcharacter.v | dirr_constt | |
dirr_consttE(phi : 'CF(G)) (i : dIirr G) :
(i \in dirr_constt phi) = (0 < '[phi, dchi i]).
Proof. by 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/vcharacter.v | dirr_consttE | |
Cnat_dirr(phi : 'CF(G)) i :
phi \in 'Z[irr G] -> i \in dirr_constt phi -> '[phi, dchi i] \in Num.nat.
Proof.
move=> PiZ; rewrite natrEint dirr_consttE andbC => /ltW -> /=.
by case: i => b i; rewrite cfdotZr rmorph_sign rpredMsign Cint_cfdot_vchar_irr.
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/vcharacter.v | Cnat_dirr | |
dirr_constt_oppr(i : dIirr G) (phi : 'CF(G)) :
(i \in dirr_constt (-phi)) = (ndirr i \in dirr_constt phi).
Proof. by rewrite !dirr_consttE dchi_ndirrE cfdotNl cfdotNr. 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/vcharacter.v | dirr_constt_oppr | |
dirr_constt_oppI(phi: 'CF(G)) :
dirr_constt phi :&: dirr_constt (-phi) = set0.
Proof.
apply/setP=> i; rewrite inE !dirr_consttE cfdotNl inE.
apply/idP=> /andP [L1 L2]; have := ltr_pDl L1 L2.
by rewrite subrr lt_def eqxx.
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/vcharacter.v | dirr_constt_oppI | |
dirr_constt_oppl(phi: 'CF(G)) i :
i \in dirr_constt phi -> (ndirr i) \notin dirr_constt phi.
Proof.
by rewrite !dirr_consttE dchi_ndirrE cfdotNr oppr_gt0 => /ltW /le_gtF ->.
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/vcharacter.v | dirr_constt_oppl | |
to_dirr(B : {set gT}) (phi : 'CF(B)) (i : Iirr B) : dIirr B :=
('[phi, 'chi_i] < 0, i). | 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/vcharacter.v | to_dirr | |
of_irr(B : {set gT}) (i : dIirr B) : Iirr B := i.2. | 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/vcharacter.v | of_irr | |
irr_constt_to_dirr(phi: 'CF(G)) i : phi \in 'Z[irr G] ->
(i \in irr_constt phi) = (to_dirr phi i \in dirr_constt phi).
Proof.
move=> Zphi; rewrite irr_consttE dirr_consttE cfdotZr rmorph_sign /=.
by rewrite -real_normrEsign ?normr_gt0 ?Rreal_int // Cint_cfdot_vchar_irr.
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/vcharacter.v | irr_constt_to_dirr | |
to_dirrK(phi: 'CF(G)) : cancel (to_dirr phi) (@of_irr G).
Proof. by []. 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/vcharacter.v | to_dirrK | |
of_irrK(phi: 'CF(G)) :
{in dirr_constt phi, cancel (@of_irr G) (to_dirr phi)}.
Proof.
case=> b i; rewrite dirr_consttE cfdotZr rmorph_sign /= /to_dirr mulr_sign.
by rewrite fun_if oppr_gt0; case: b => [|/ltW/le_gtF] ->.
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/vcharacter.v | of_irrK | |
cfdot_todirrE(phi: 'CF(G)) i (phi_i := dchi (to_dirr phi i)) :
'[phi, phi_i] *: phi_i = '[phi, 'chi_i] *: 'chi_i.
Proof. by rewrite cfdotZr rmorph_sign mulrC -scalerA signrZK. 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/vcharacter.v | cfdot_todirrE | |
cfun_sum_dconstt(phi : 'CF(G)) :
phi \in 'Z[irr G] ->
phi = \sum_(i in dirr_constt phi) '[phi, dchi i] *: dchi i.
Proof.
move=> PiZ; rewrite [LHS]cfun_sum_constt.
rewrite (reindex (to_dirr phi))=> [/= |]; last first.
by exists (@of_irr _)=> //; apply: of_irrK .
by apply: eq_big => i; rewrite ?irr_constt_to_dirr // cfdot_todirrE.
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/vcharacter.v | cfun_sum_dconstt | |
cnorm_dconstt(phi : 'CF(G)) :
phi \in 'Z[irr G] ->
'[phi] = \sum_(i in dirr_constt phi) '[phi, dchi i] ^+ 2.
Proof.
move=> PiZ; rewrite {1 2}(cfun_sum_dconstt PiZ).
rewrite cfdot_suml; apply: eq_bigr=> i IiD.
rewrite cfdot_sumr (bigD1 i) //= big1 ?addr0 => [|j /andP [JiD IdJ]].
rewrite cfdotZr cfdotZl cfdot_dchi eqxx eq_sym (negPf (ndirr_diff i)).
by rewrite subr0 mulr1 aut_natr ?Cnat_dirr.
rewrite cfdotZr cfdotZl cfdot_dchi eq_sym (negPf IdJ) -natrB ?mulr0 //.
by rewrite (negPf (contraNneq _ (dirr_constt_oppl JiD))) => // <-.
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/vcharacter.v | cnorm_dconstt | |
dirr_small_norm(phi : 'CF(G)) n :
phi \in 'Z[irr G] -> '[phi] = n%:R -> (n < 4)%N ->
[/\ #|dirr_constt phi| = n, dirr_constt phi :&: dirr_constt (- phi) = set0 &
phi = \sum_(i in dirr_constt phi) dchi i].
Proof.
move=> PiZ Pln; rewrite ltnNge -leC_nat => Nl4.
suffices Fd i: i \in dirr_constt phi -> '[phi, dchi i] = 1.
split; last 2 [by apply/setP=> u; rewrite !inE cfdotNl oppr_gt0 lt_asym].
apply/eqP; rewrite -eqC_nat -sumr_const -Pln (cnorm_dconstt PiZ).
by apply/eqP/eq_bigr=> i Hi; rewrite Fd // expr1n.
rewrite {1}[phi]cfun_sum_dconstt //.
by apply: eq_bigr => i /Fd->; rewrite scale1r.
move=> IiD; apply: contraNeq Nl4 => phi_i_neq1.
rewrite -Pln cnorm_dconstt // (bigD1 i) ?ler_wpDr ?sumr_ge0 //=.
by move=> j /andP[JiD _]; rewrite exprn_ge0 ?natr_ge0 ?Cnat_dirr.
have /natrP[m Dm] := Cnat_dirr PiZ IiD; rewrite Dm -natrX ler_nat (leq_sqr 2).
by rewrite ltn_neqAle eq_sym -eqC_nat -ltC_nat -Dm phi_i_neq1 -dirr_consttE.
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/vcharacter.v | dirr_small_norm | |
cfdot_sum_dchi(phi1 phi2 : 'CF(G)) :
'[\sum_(i in dirr_constt phi1) dchi i,
\sum_(i in dirr_constt phi2) dchi i] =
#|dirr_constt phi1 :&: dirr_constt phi2|%:R -
#|dirr_constt phi1 :&: dirr_constt (- phi2)|%:R.
Proof.
rewrite addrC (big_setID (dirr_constt (- phi2))) /= cfdotDl; congr (_ + _).
rewrite cfdot_suml -sumr_const -sumrN; apply: eq_bigr => i /setIP[p1i p2i].
rewrite cfdot_sumr (bigD1 (ndirr i)) -?dirr_constt_oppr //= dchi_ndirrE.
rewrite cfdotNr cfnorm_dchi big1 ?addr0 // => j /andP[p2j i'j].
rewrite cfdot_dchi -(inv_eq ndirrK) [in rhs in - rhs]eq_sym (negPf i'j) subr0.
rewrite (negPf (contraTneq _ p2i)) // => ->.
by rewrite dirr_constt_oppr dirr_constt_oppl.
rewrite cfdot_sumr (big_setID (dirr_constt phi1)) setIC /= addrC.
rewrite big1 ?add0r => [|j /setDP[p2j p1'j]]; last first.
rewrite cfdot_suml big1 // => i /setDP[p1i p2'i].
rewrite cfdot_dchi (negPf (contraTneq _ p1i)) => [|-> //].
rewrite (negPf (contraNneq _ p2'i)) ?subrr // => ->.
by rewrite dirr_constt_oppr ndirrK.
rewrite -sumr_const; apply: eq_bigr => i /setIP[p1i p2i]; rewrite cfdot_suml.
rewrite (bigD1 i) /=; last by rewrite inE dirr_constt_oppr dirr_constt_oppl.
rewrite cfnorm_dchi big1 ?addr0 // => j /andP[/setDP[p1j _] i'j].
rewrite cfdot_dchi (negPf i'j) (negPf (contraTneq _ p1j)) ?subrr // => ->.
exact: dirr_constt_oppl.
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/vcharacter.v | cfdot_sum_dchi | |
cfdot_dirr_eq1:
{in dirr G &, forall phi psi, ('[phi, psi] == 1) = (phi == psi)}.
Proof.
move=> _ _ /dirrP[b1 [i1 ->]] /dirrP[b2 [i2 ->]].
rewrite eq_signed_irr cfdotZl cfdotZr rmorph_sign cfdot_irr mulrA -signr_addb.
rewrite pmulrn -rmorphMsign (eqr_int _ _ 1) -negb_add.
by case: (b1 (+) b2) (i1 == i2) => [] [].
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/vcharacter.v | cfdot_dirr_eq1 | |
cfdot_add_dirr_eq1:
{in dirr G & &, forall phi1 phi2 psi,
'[phi1 + phi2, psi] = 1 -> psi = phi1 \/ psi = phi2}.
Proof.
move=> _ _ _ /dirrP[b1 [i1 ->]] /dirrP[b2 [i2 ->]] /dirrP[c [j ->]] /eqP.
rewrite cfdotDl !cfdotZl !cfdotZr !rmorph_sign !cfdot_irr !mulrA -!signr_addb.
rewrite 2!{1}signrE !mulrBl !mul1r -!natrM addrCA -subr_eq0 -!addrA.
rewrite -!opprD addrA subr_eq0 -mulrSr -!natrD eqr_nat => eq_phi_psi.
apply/pred2P; rewrite /= !eq_signed_irr -!negb_add !(eq_sym j) !(addbC c).
by case: (i1 == j) eq_phi_psi; case: (i2 == j); do 2!case: (_ (+) c).
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/vcharacter.v | cfdot_add_dirr_eq1 | |
RecordisComplex L of GRing.ClosedField L := {
conj : {rmorphism L -> L};
conjK : involutive conj;
conj_nt : ~ conj =1 id
}.
HB.builders Context L of isComplex L. | HB.factory | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Record | |
nz2: 2 != 0 :> L.
Proof.
apply/eqP=> pchar2; apply: conj_nt => e; apply/eqP/idPn=> eJ.
have opp_id x: - x = x :> L.
by apply/esym/eqP; rewrite -addr_eq0 -mulr2n -mulr_natl pchar2 mul0r.
have{} pchar2: 2%N \in [pchar L] by apply/eqP.
without loss{eJ} eJ: e / conj e = e + 1.
move/(_ (e / (e + conj e))); apply.
rewrite fmorph_div rmorphD /= conjK -{1}[conj e](addNKr e) mulrDl.
by rewrite opp_id (addrC e) divff // addr_eq0 opp_id.
pose a := e * conj e; have aJ: conj a = a by rewrite rmorphM /= conjK mulrC.
have [w Dw] := @solve_monicpoly _ 2%N (nth 0 [:: e * a; - 1]) isT.
have{} Dw: w ^+ 2 + w = e * a.
by rewrite Dw !big_ord_recl big_ord0 /= mulr1 mulN1r addr0 subrK.
pose b := w + conj w; have bJ: conj b = b by rewrite rmorphD /= conjK addrC.
have Db2: b ^+ 2 + b = a.
rewrite -pFrobenius_autE // rmorphD addrACA Dw /= pFrobenius_autE -rmorphXn.
by rewrite -rmorphD Dw rmorphM /= aJ eJ -mulrDl -{1}[e]opp_id addKr mul1r.
have /eqP[] := oner_eq0 L; apply: (addrI b); rewrite addr0 -{2}bJ.
have: (b + e) * (b + conj e) == 0. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | nz2 | |
mul2I: injective (fun z : L => z *+ 2).
Proof.
have nz2 := nz2.
by move=> x y; rewrite /= -mulr_natl -(mulr_natl y) => /mulfI->.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | mul2I | |
sqrtx : L :=
sval (sig_eqW (@solve_monicpoly _ 2%N (nth 0 [:: x]) isT)). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | sqrt | |
sqrtKx: sqrt x ^+ 2 = x.
Proof.
rewrite /sqrt; case: sig_eqW => /= y ->.
by rewrite !big_ord_recl big_ord0 /= mulr1 mul0r !addr0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | sqrtK | |
sqrtEx y: y ^+ 2 = x -> {b : bool | y = (-1) ^+ b * sqrt x}.
Proof.
move=> Dx; exists (y != sqrt x); apply/eqP; rewrite mulr_sign if_neg.
by case: ifPn => //; apply/implyP; rewrite implyNb -eqf_sqr Dx sqrtK.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | sqrtE | |
i:= sqrt (- 1). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | i | |
sqrMix: (i * x) ^+ 2 = - x ^+ 2.
Proof. by rewrite exprMn sqrtK mulN1r. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | sqrMi | |
iJ: conj i = - i.
Proof.
have nz2 := nz2.
have /sqrtE[b]: conj i ^+ 2 = - 1 by rewrite -rmorphXn /= sqrtK rmorphN1.
rewrite mulr_sign -/i; case: b => // Ri.
case: conj_nt => z; wlog zJ: z / conj z = - z.
move/(_ (z - conj z)); rewrite !rmorphB conjK opprB => zJ.
by apply/mul2I/(canRL (subrK _)); rewrite -addrA zJ // addrC subrK.
have [-> | nz_z] := eqVneq z 0; first exact: rmorph0.
have [u Ru [v Rv Dz]]:
exists2 u, conj u = u & exists2 v, conj v = v & (u + z * v) ^+ 2 = z.
- pose y := sqrt z; exists ((y + conj y) / 2).
by rewrite fmorph_div rmorphD /= conjK addrC rmorph_nat.
exists ((y - conj y) / (z *+ 2)).
rewrite fmorph_div rmorphMn /= zJ mulNrn invrN mulrN -mulNr rmorphB opprB.
by rewrite conjK.
rewrite -(mulr_natl z) invfM (mulrC z) !mulrA divfK // -mulrDl addrACA. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | iJ | |
normx := sqrt x * conj (sqrt x). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | norm | |
normKx : norm x ^+ 2 = x * conj x.
Proof. by rewrite exprMn -rmorphXn sqrtK. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | normK | |
normEx y : y ^+ 2 = x -> norm x = y * conj y.
Proof.
rewrite /norm => /sqrtE[b /(canLR (signrMK b)) <-].
by rewrite !rmorphM /= rmorph_sign mulrACA -mulrA signrMK.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | normE | |
norm_eq0x : norm x = 0 -> x = 0.
Proof.
by move/eqP; rewrite mulf_eq0 fmorph_eq0 -mulf_eq0 -expr2 sqrtK => /eqP.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | norm_eq0 | |
normMx y : norm (x * y) = norm x * norm y.
Proof.
by rewrite mulrACA -rmorphM; apply: normE; rewrite exprMn !sqrtK.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | normM | |
normNx : norm (- x) = norm x.
Proof.
by rewrite -mulN1r normM {1}/norm iJ mulrN -expr2 sqrtK opprK mul1r.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | normN | |
lex y := norm (y - x) == y - x. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | le | |
ltx y := (y != x) && le x y. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | lt | |
posEx: le 0 x = (norm x == x).
Proof. by rewrite /le subr0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | posE | |
leBx y: le x y = le 0 (y - x).
Proof. by rewrite posE. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | leB | |
posPx : reflect (exists y, x = y * conj y) (le 0 x).
Proof.
rewrite posE; apply: (iffP eqP) => [Dx | [y {x}->]]; first by exists (sqrt x).
by rewrite (normE (normK y)) rmorphM /= conjK (mulrC (conj _)) -expr2 normK.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | posP | |
posJx : le 0 x -> conj x = x.
Proof.
by case/posP=> {x}u ->; rewrite rmorphM /= conjK mulrC.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | posJ | |
pos_linearx y : le 0 x -> le 0 y -> le x y || le y x.
Proof.
move=> pos_x pos_y; rewrite leB -opprB orbC leB !posE normN -eqf_sqr.
by rewrite normK rmorphB !posJ ?subrr.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | pos_linear | |
sposDlx y : lt 0 x -> le 0 y -> lt 0 (x + y).
Proof.
have sqrtJ z : le 0 z -> conj (sqrt z) = sqrt z.
rewrite posE -{2}[z]sqrtK -subr_eq0 -mulrBr mulf_eq0 subr_eq0.
by case/pred2P=> ->; rewrite ?rmorph0.
case/andP=> nz_x /sqrtJ uJ /sqrtJ vJ.
set u := sqrt x in uJ; set v := sqrt y in vJ; pose w := u + i * v.
have ->: x + y = w * conj w.
rewrite rmorphD rmorphM /= iJ uJ vJ mulNr mulrC -subr_sqr sqrMi opprK.
by rewrite !sqrtK.
apply/andP; split; last by apply/posP; exists w.
rewrite -normK expf_eq0 //=; apply: contraNneq nz_x => /norm_eq0 w0.
rewrite -[x]sqrtK expf_eq0 /= -/u -(inj_eq mul2I) !mulr2n -{2}(rmorph0 conj).
by rewrite -w0 rmorphD rmorphM /= iJ uJ vJ mulNr addrACA subrr addr0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | sposDl | |
sposDx y : lt 0 x -> lt 0 y -> lt 0 (x + y).
Proof.
by move=> x_gt0 /andP[_]; apply: sposDl.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | sposD | |
normDx y : le (norm (x + y)) (norm x + norm y).
Proof.
have sposM u v: lt 0 u -> le 0 (u * v) -> le 0 v.
by rewrite /lt !posE normM andbC => /andP[/eqP-> /mulfI/inj_eq->].
have posD u v: le 0 u -> le 0 v -> le 0 (u + v).
have [-> | nz_u u_ge0 v_ge0] := eqVneq u 0; first by rewrite add0r.
by have /andP[]: lt 0 (u + v) by rewrite sposDl // /lt nz_u.
have le_sqr u v: conj u = u -> le 0 v -> le (u ^+ 2) (v ^+ 2) -> le u v.
case: (eqVneq u 0) => [-> //|nz_u Ru v_ge0].
have [u_gt0 | u_le0 _] := boolP (lt 0 u).
by rewrite leB (leB u) subr_sqr mulrC addrC; apply: sposM; apply: sposDl.
rewrite leB posD // posE normN -addr_eq0; apply/eqP.
rewrite /lt nz_u posE -subr_eq0 in u_le0; apply: (mulfI u_le0).
by rewrite mulr0 -subr_sqr normK Ru subrr.
have pos_norm z: le 0 (norm z) by apply/posP; exists (sqrt z).
rewrite le_sqr ?posJ ?posD // sqrrD !normK -normM rmorphD mulrDl !mulrDr.
rewrite addrA addrC !addrA -(addrC (y * conj y)) !addrA.
move: (y * _ + _) => u; rewrite -!addrA leB opprD addrACA {u}subrr add0r -leB.
rewrite {}le_sqr ?posD //.
by rewrite rmorphD !rmorphM /= !conjK addrC (mulrC x) (mulrC y).
rewrite -mulr2n -mulr_natr exprMn normK -natrX mulr_natr sqrrD mulrACA.
rewrite -rmorphM (mulrC y x) addrAC leB mulrnA mulr2n opprD addrACA.
rewrite subrr addr0 {2}(mulrC x) rmorphM mulrACA -opprB addrAC -sqrrB -sqrMi.
apply/posP; exists (i * (x * conj y - y * conj x)); congr (_ * _).
rewrite !(rmorphM, rmorphB) iJ !conjK mulN
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | normD |
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