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 |
|---|---|---|---|---|---|---|
gen_dim_ub_proofnA :
[exists B : 'rV_nA, row_free (subbase B)] -> (nA <= n)%N.
Proof.
case/existsP=> B /eqnP def_nAd.
by rewrite (leq_trans _ (rank_leq_col (subbase B))) // def_nAd leq_pmulr.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_dim_ub_proof | |
gen_dim:= ex_maxn gen_dim_ex_proof gen_dim_ub_proof. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_dim | |
nA:= gen_dim. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | nA | |
gen_base: 'rV_nA := odflt 0 [pick B | row_free (subbase 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_base | |
base:= subbase gen_base. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | base | |
base_free: row_free base.
Proof.
rewrite /base /gen_base /nA; case: pickP => //; case: ex_maxnP => nA_max.
by case/existsP=> B Bfree _ no_free; rewrite no_free in Bfree.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | base_free | |
base_full: row_full base.
Proof.
rewrite /row_full (eqnP base_free) /nA; case: ex_maxnP => nA.
case/existsP=> /= B /eqnP Bfree nA_max; rewrite -Bfree eqn_leq rank_leq_col.
rewrite -{1}(mxrank1 F n) mxrankS //; apply/row_subP=> j; set u := row _ _.
move/implyP: {nA_max}(nA_max nA.+1); rewrite ltnn implybF.
apply: contraR => nBj; apply/existsP.
exists (row_mx (const_mx j : 'M_1) B); rewrite -row_leq_rank.
pose Bj := Ad *m lin1_mx (mulmx u \o vec_mx).
have rBj: \rank Bj = d.
apply/eqP; rewrite eqn_leq rank_leq_row -subn_eq0 -mxrank_ker mxrank_eq0 /=.
apply/rowV0P=> v /sub_kermxP; rewrite mulmxA mul_rV_lin1 /=.
rewrite -horner_rVpoly; pose x := inFA v; rewrite -/(mxval x).
have [[] // | nzx /(congr1 (mulmx^~ (mxval x^-1)))] := eqVneq x 0.
rewrite mul0mx /= -mulmxA -mxvalM divff // mxval1 mulmx1.
by move/rowP/(_ j)/eqP; rewrite !mxE !eqxx oner_eq0.
rewrite {1}mulSn -Bfree -{1}rBj {rBj} -mxrank_disjoint_sum.
rewrite mxrankS // addsmx_sub -[nA.+1]/(1 + nA)%N; apply/andP; split.
apply/row_subP=> k; rewrite row_mul mul_rV_lin1 /=.
apply: eq_row_sub (mxvec_index (lshift _ 0) k) _.
by rewrite !rowK mxvecK mxvecE mxE row_mxEl mxE -row_mul mul1mx.
apply/row_subP; case/mxvec_indexP=> i k.
apply: eq_row_sub (mxvec_index (rshift 1 i) k) _.
by rewrite !rowK !mxvecE 2!mxE row_mxEr.
apply/eqP/rowV0P=> v; rewrite sub_capmx => /andP[/submxP[w]].
set x := inFA w; rewrite {Bj}mulmxA mul_rV_lin1 /= -horner_rVpoly -/(mxval x).
have [-> | nzx ->] := eqVneq x 0; first by rewrite
... | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | base_full | |
gen_dim_factor: (nA * d)%N = n.
Proof. by rewrite -(eqnP base_free) (eqnP base_full). 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_dim_factor | |
gen_dim_gt0: nA > 0.
Proof. by case: posnP gen_dim_factor => // ->. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_dim_gt0 | |
in_gen(W : 'M[F]_(m, n)) : 'M[FA]_(m, nA) :=
\matrix_(i, j) inFA (row j (vec_mx (row i W *m pinvmx base))). | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_gen | |
val_gen(W : 'M[FA]_(m, nA)) : 'M[F]_(m, n) :=
\matrix_i (mxvec (\matrix_j val (W i j)) *m base). | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_gen | |
in_genK: cancel in_gen val_gen.
Proof.
move=> W; apply/row_matrixP=> i; rewrite rowK; set w := row i W.
have b_w: (w <= base)%MS by rewrite submx_full ?base_full.
rewrite -{b_w}(mulmxKpV b_w); congr (_ *m _).
by apply/rowP; case/mxvec_indexP=> j k; rewrite mxvecE !mxE.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_genK | |
val_genK: cancel val_gen in_gen.
Proof.
move=> W; apply/matrixP=> i j; apply: val_inj; rewrite mxE /= rowK.
case/row_freeP: base_free => B' BB'; rewrite -[_ *m _]mulmx1 -BB' mulmxA.
by rewrite mulmxKpV ?submxMl // -mulmxA BB' mulmx1 mxvecK rowK.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_genK | |
in_gen0: in_gen 0 = 0.
Proof. by apply/matrixP=> i j; rewrite !mxE !(mul0mx, linear0). 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_gen0 | |
val_gen0: val_gen 0 = 0.
Proof. by apply: (canLR in_genK); rewrite in_gen0. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_gen0 | |
in_genD: {morph in_gen : U V / U + V}.
Proof.
by move=> U V; apply/matrixP=> i j; rewrite !mxE 4!(mulmxDl, linearD).
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_genD | |
val_genD: {morph val_gen : U V / U + V}.
Proof. by move=> U V; apply: (canLR in_genK); rewrite in_genD !val_genK. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_genD | |
in_genN: {morph in_gen : W / - W}.
Proof. by move=> W; apply/esym/addr0_eq; rewrite -in_genD subrr in_gen0. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_genN | |
val_genN: {morph val_gen : W / - W}.
Proof. by move=> W; apply: (canLR in_genK); rewrite in_genN val_genK. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_genN | |
in_gen_sum:= big_morph in_gen in_genD in_gen0. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_gen_sum | |
val_gen_sum:= big_morph val_gen val_genD val_gen0. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_gen_sum | |
in_genZa : {morph in_gen : W / a *: W >-> gen a *: W}.
Proof.
move=> W; apply/matrixP=> i j; apply: mxval_inj.
rewrite !mxE mxvalM genK ![mxval _]horner_rVpoly /=.
by rewrite mul_scalar_mx !(I, scalemxAl, linearZ).
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_genZ | |
val_gen_rV(w : 'rV_nA) :
val_gen w = mxvec (\matrix_j val (w 0 j)) *m base.
Proof. by apply/rowP=> j /[1!mxE]. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_gen_rV | |
val_gen_rowW (i : 'I_m) : val_gen (row i W) = row i (val_gen W).
Proof.
rewrite val_gen_rV rowK; congr (mxvec _ *m _).
by apply/matrixP=> j k /[!mxE].
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_gen_row | |
in_gen_rowW (i : 'I_m) : in_gen (row i W) = row i (in_gen W).
Proof. by apply: (canLR val_genK); rewrite val_gen_row in_genK. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_gen_row | |
row_gen_sum_mxvalW (i : 'I_m) :
row i (val_gen W) = \sum_j row (gen_base 0 j) (mxval (W i j)).
Proof.
rewrite -val_gen_row [row i W]row_sum_delta val_gen_sum.
apply: eq_bigr => /= j _ /[1!mxE]; move: {W i}(W i j) => x.
have ->: x = \sum_k gen (val x 0 k) * inFA (delta_mx 0 k).
case: x => u; apply: mxval_inj; rewrite {1}[u]row_sum_delta.
rewrite mxval_sum [mxval _]horner_rVpoly mulmx_suml linear_sum /=.
apply: eq_bigr => k _; rewrite mxvalM genK [mxval _]horner_rVpoly /=.
by rewrite mul_scalar_mx -scalemxAl linearZ.
rewrite scaler_suml val_gen_sum mxval_sum linear_sum; apply: eq_bigr => k _.
rewrite mxvalM genK mul_scalar_mx linearZ [mxval _]horner_rVpoly /=.
rewrite -scalerA; apply: (canLR in_genK); rewrite in_genZ; congr (_ *: _).
apply: (canRL val_genK); transitivity (row (mxvec_index j k) base); last first.
by rewrite -rowE rowK mxvecE mxE rowK mxvecK.
rewrite rowE -mxvec_delta -[val_gen _](row_id 0) rowK /=; congr (mxvec _ *m _).
apply/row_matrixP=> j'; rewrite rowK !mxE mulr_natr rowE mul_delta_mx_cond.
by rewrite !mulrb (fun_if rVval).
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | row_gen_sum_mxval | |
val_genZx : {morph @val_gen m : W / x *: W >-> W *m mxval x}.
Proof.
move=> W; apply/row_matrixP=> i; rewrite row_mul !row_gen_sum_mxval.
by rewrite mulmx_suml; apply: eq_bigr => j _; rewrite mxE mulrC mxvalM row_mul.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_genZ | |
submx_in_genm1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U <= V -> in_gen U <= in_gen V)%MS.
Proof.
move=> sUV; apply/row_subP=> i; rewrite -in_gen_row.
case/submxP: (row_subP sUV i) => u ->{i}.
rewrite mulmx_sum_row in_gen_sum summx_sub // => j _.
by rewrite in_genZ in_gen_row scalemx_sub ?row_sub.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | submx_in_gen | |
submx_in_gen_eqm1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(V *m A <= V -> (in_gen U <= in_gen V) = (U <= V))%MS.
Proof.
move=> sVA_V; apply/idP/idP=> siUV; last exact: submx_in_gen.
apply/row_subP=> i; rewrite -[row i U]in_genK in_gen_row.
case/submxP: (row_subP siUV i) => u ->{i U siUV}.
rewrite mulmx_sum_row val_gen_sum summx_sub // => j _.
rewrite val_genZ val_gen_row in_genK rowE -mulmxA mulmx_sub //.
rewrite [mxval _]horner_poly mulmx_sumr summx_sub // => [[k _]] _ /=.
rewrite mulmxA mul_mx_scalar -scalemxAl scalemx_sub {u j}//.
elim: k => [|k IHk]; first by rewrite mulmx1.
by rewrite exprSr mulmxA (submx_trans (submxMr A IHk)).
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | submx_in_gen_eq | |
gen_mxg := \matrix_i in_gen (row (gen_base 0 i) (rG g)).
Let val_genJmx m :
{in G, forall g, {morph @val_gen m : W / W *m gen_mx g >-> W *m rG g}}.
Proof.
move=> g Gg /= W; apply/row_matrixP=> i; rewrite -val_gen_row !row_mul.
rewrite mulmx_sum_row val_gen_sum row_gen_sum_mxval mulmx_suml.
apply: eq_bigr => /= j _; rewrite val_genZ rowK in_genK mxE -!row_mul.
by rewrite (centgmxP (mxval_centg _)).
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_mx | |
gen_mx_repr: mx_repr G gen_mx.
Proof.
split=> [|g h Gg Gh]; apply: (can_inj val_genK).
by rewrite -[gen_mx 1]mul1mx val_genJmx // repr_mx1 mulmx1.
rewrite {1}[val_gen]lock -[gen_mx g]mul1mx !val_genJmx // -mulmxA -repr_mxM //.
by rewrite -val_genJmx ?groupM ?mul1mx -?lock.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_mx_repr | |
gen_repr:= MxRepresentation gen_mx_repr.
Local Notation rGA := gen_repr. | Canonical | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_repr | |
val_genJm :
{in G, forall g, {morph @val_gen m : W / W *m rGA g >-> W *m rG g}}.
Proof. exact: val_genJmx. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_genJ | |
in_genJm :
{in G, forall g, {morph @in_gen m : v / v *m rG g >-> v *m rGA g}}.
Proof.
by move=> g Gg /= v; apply: (canLR val_genK); rewrite val_genJ ?in_genK.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_genJ | |
rfix_gen(H : {set gT}) :
H \subset G -> (rfix_mx rGA H :=: in_gen (rfix_mx rG H))%MS.
Proof.
move/subsetP=> sHG; apply/eqmxP/andP; split; last first.
by apply/rfix_mxP=> g Hg; rewrite -in_genJ ?sHG ?rfix_mx_id.
rewrite -[rfix_mx rGA H]val_genK; apply: submx_in_gen.
by apply/rfix_mxP=> g Hg; rewrite -val_genJ ?rfix_mx_id ?sHG.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rfix_gen | |
rowval_genm U :=
<<\matrix_ik
mxvec (\matrix_(i < m, k < d) (row i (val_gen U) *m A ^+ k)) 0 ik>>%MS. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rowval_gen | |
submx_rowval_genm1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, nA)) :
(U <= rowval_gen V)%MS = (in_gen U <= V)%MS.
Proof.
rewrite genmxE; apply/idP/idP=> sUV.
apply: submx_trans (submx_in_gen sUV) _.
apply/row_subP; case/mxvec_indexP=> i k; rewrite -in_gen_row rowK mxvecE mxE.
rewrite -mxval_grootXn -val_gen_row -val_genZ val_genK scalemx_sub //.
exact: row_sub.
rewrite -[U]in_genK; case/submxP: sUV => u ->{U}.
apply/row_subP=> i0; rewrite -val_gen_row row_mul; move: {i0 u}(row _ u) => u.
rewrite mulmx_sum_row val_gen_sum summx_sub // => i _.
rewrite val_genZ [mxval _]horner_rVpoly [_ *m Ad]mulmx_sum_row.
rewrite !linear_sum summx_sub // => k _.
rewrite 2!linearZ scalemx_sub {u}//= rowK mxvecK val_gen_row.
by apply: (eq_row_sub (mxvec_index i k)); rewrite rowK mxvecE mxE.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | submx_rowval_gen | |
rowval_genKm (U : 'M_(m, nA)) : (in_gen (rowval_gen U) :=: U)%MS.
Proof.
apply/eqmxP; rewrite -submx_rowval_gen submx_refl /=.
by rewrite -{1}[U]val_genK submx_in_gen // submx_rowval_gen val_genK.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rowval_genK | |
rowval_gen_stablem (U : 'M_(m, nA)) :
(rowval_gen U *m A <= rowval_gen U)%MS.
Proof.
rewrite -[A]mxval_groot -{1}[_ U]in_genK -val_genZ.
by rewrite submx_rowval_gen val_genK scalemx_sub // rowval_genK.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rowval_gen_stable | |
rstab_in_genm (U : 'M_(m, n)) : rstab rGA (in_gen U) = rstab rG U.
Proof.
apply/setP=> x /[!inE]; case Gx: (x \in G) => //=.
by rewrite -in_genJ // (inj_eq (can_inj in_genK)).
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstab_in_gen | |
rstabs_in_genm (U : 'M_(m, n)) :
rstabs rG U \subset rstabs rGA (in_gen U).
Proof.
by apply/subsetP=> x /[!inE] /andP[Gx nUx]; rewrite -in_genJ Gx // submx_in_gen.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_in_gen | |
rstabs_rowval_genm (U : 'M_(m, nA)) :
rstabs rG (rowval_gen U) = rstabs rGA U.
Proof.
apply/setP=> x /[!inE]; case Gx: (x \in G) => //=.
by rewrite submx_rowval_gen in_genJ // (eqmxMr _ (rowval_genK U)).
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_rowval_gen | |
mxmodule_rowval_genm (U : 'M_(m, nA)) :
mxmodule rG (rowval_gen U) = mxmodule rGA U.
Proof. by rewrite /mxmodule rstabs_rowval_gen. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxmodule_rowval_gen | |
gen_mx_irr: mx_irreducible rGA.
Proof.
apply/mx_irrP; split=> [|U Umod nzU]; first exact: gen_dim_gt0.
rewrite -sub1mx -rowval_genK -submx_rowval_gen submx_full //.
case/mx_irrP: irrG => _; apply; first by rewrite mxmodule_rowval_gen.
rewrite -(inj_eq (can_inj in_genK)) in_gen0.
by rewrite -mxrank_eq0 rowval_genK mxrank_eq0.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_mx_irr | |
rker_gen: rker rGA = rker rG.
Proof.
apply/setP=> g; rewrite !inE !mul1mx; case Gg: (g \in G) => //=.
apply/eqP/eqP=> g1; apply/row_matrixP=> i.
by apply: (can_inj in_genK); rewrite rowE in_genJ //= g1 mulmx1 row1.
by apply: (can_inj val_genK); rewrite rowE val_genJ //= g1 mulmx1 row1.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rker_gen | |
gen_mx_faithful: mx_faithful rGA = mx_faithful rG.
Proof. by rewrite /mx_faithful rker_gen. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_mx_faithful | |
eval_mulTe u v :
eval_mx e (mulT u v) = val (inFA (eval_mx e u) * inFA (eval_mx e v)).
Proof.
rewrite !(eval_mulmx, eval_mxvec) !eval_mxT eval_mx_term.
by apply: (can_inj rVpolyK); rewrite -mxvalM [rVpoly _]horner_rVpolyK.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | eval_mulT | |
gen_termt := match t with
| 'X_k => row_var _ d k
| x%:T => mx_term (val (x : FA))
| n1%:R => mx_term (val (n1%:R : FA))%R
| t1 + t2 => \row_i (gen_term t1 0%R i + gen_term t2 0%R i)
| - t1 => \row_i (- gen_term t1 0%R i)
| t1 *+ n1 => mulmx_term (mx_term n1%:R%:M)%R (gen_term t1)
| t1 * t2 => mulT (gen_term t1) (gen_term t2)
| t1^-1 => gen_term t1
| t1 ^+ n1 => iter n1 (mulT (gen_term t1)) (mx_term (val (1%R : FA)))
end%T. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_term | |
gen_env(e : seq FA) := row_env (map val e). | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_env | |
nth_map_rVval(e : seq FA) j : (map val e)`_j = val e`_j.
Proof.
case: (ltnP j (size e)) => [| leej]; first exact: (nth_map 0 0).
by rewrite !nth_default ?size_map.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | nth_map_rVval | |
set_nth_map_rVval(e : seq FA) j v :
set_nth 0 (map val e) j v = map val (set_nth 0 e j (inFA v)).
Proof.
apply: (@eq_from_nth _ 0) => [|k _]; first by rewrite !(size_set_nth, size_map).
by rewrite !(nth_map_rVval, nth_set_nth) /= nth_map_rVval [rVval _]fun_if.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | set_nth_map_rVval | |
eval_gen_terme t :
GRing.rterm t -> eval_mx (gen_env e) (gen_term t) = val (GRing.eval e t).
Proof.
elim: t => //=.
- by move=> k _; apply/rowP=> i; rewrite !mxE /= nth_row_env nth_map_rVval.
- by move=> x _; rewrite eval_mx_term.
- by move=> x _; rewrite eval_mx_term.
- by move=> t1 + t2 + /andP[rt1 rt2] => <-// <-//; apply/rowP => k /[!mxE].
- by move=> t1 + rt1 => <-//; apply/rowP=> k /[!mxE].
- move=> t1 IH1 n1 rt1; rewrite eval_mulmx eval_mx_term mul_scalar_mx.
by rewrite scaler_nat {}IH1 //; elim: n1 => //= n1 IHn1; rewrite !mulrS IHn1.
- by move=> t1 IH1 t2 IH2 /andP[rt1 rt2]; rewrite eval_mulT IH1 ?IH2.
move=> t1 + n1 => /[apply] IH1.
elim: n1 => [|n1 IHn1] /=; first by rewrite eval_mx_term.
by rewrite eval_mulT exprS IH1 IHn1.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | eval_gen_term | |
gen_formf := match f with
| Bool b => Bool b
| t1 == t2 => mxrank_form 0 (gen_term (t1 - t2))
| GRing.Unit t1 => mxrank_form 1 (gen_term t1)
| f1 /\ f2 => gen_form f1 /\ gen_form f2
| f1 \/ f2 => gen_form f1 \/ gen_form f2
| f1 ==> f2 => gen_form f1 ==> gen_form f2
| ~ f1 => ~ gen_form f1
| ('exists 'X_k, f1) => Exists_row_form d k (gen_form f1)
| ('forall 'X_k, f1) => ~ Exists_row_form d k (~ (gen_form f1))
end%T. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_form | |
sat_gen_forme f : GRing.rformula f ->
reflect (GRing.holds e f) (GRing.sat (gen_env e) (gen_form f)).
Proof.
have ExP := Exists_rowP; have set_val := set_nth_map_rVval.
elim: f e => //.
- by move=> b e _; apply: (iffP satP).
- rewrite /gen_form => t1 t2 e rt_t; set t := (_ - _)%T.
have:= GRing.qf_evalP (gen_env e) (mxrank_form_qf 0 (gen_term t)).
rewrite eval_mxrank mxrank_eq0 eval_gen_term // => tP.
by rewrite (sameP satP tP) /= subr_eq0 val_eqE; apply: eqP.
- move=> f1 IH1 f2 IH2 s /= /andP[/(IH1 s)f1P /(IH2 s)f2P].
by apply: (iffP satP) => [[/satP/f1P ? /satP/f2P] | [/f1P/satP ? /f2P/satP]].
- move=> f1 IH1 f2 IH2 s /= /andP[/(IH1 s)f1P /(IH2 s)f2P].
by apply: (iffP satP) => /= [] [];
try move/satP; do [move/f1P | move/f2P]; try move/satP; auto.
- move=> f1 IH1 f2 IH2 s /= /andP[/(IH1 s)f1P /(IH2 s)f2P].
by apply: (iffP satP) => /= implP;
try move/satP; move/f1P; try move/satP; move/implP;
try move/satP; move/f2P; try move/satP.
- move=> f1 IH1 s /= /(IH1 s) f1P.
by apply: (iffP satP) => /= notP; try move/satP; move/f1P; try move/satP.
- move=> k f1 IHf1 s /IHf1 f1P; apply: (iffP satP) => /= [|[[v f1v]]].
by case/ExP=> // x /satP; rewrite set_val => /f1P; exists (inFA x).
by apply/ExP=> //; exists v; rewrite set_val; apply/satP/f1P.
move=> i f1 IHf1 s /IHf1 f1P; apply: (iffP satP) => /= allf1 => [[v]|].
apply/f1P; case: satP => // notf1x; case: allf1; apply/ExP=> //.
by exists v; rewrite set_val.
by case/ExP=> //= v []; apply/satP; rewrite set_val
... | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | sat_gen_form | |
gen_sate f := GRing.sat (gen_env e) (gen_form (GRing.to_rform f)). | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_sat | |
gen_satP:
GRing.MathCompCompatDecidableField.DecidableField.axiom gen_sat.
Proof.
move=> e f; have [tor rto] := GRing.to_rformP e f.
exact: (iffP (sat_gen_form e (GRing.to_rform_rformula f))).
Qed.
#[export] HB.instance Definition _ := GRing.Field_isDecField.Build FA gen_satP. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_satP | |
FA:= (gen_of irrG cGA).
#[export] HB.instance Definition _ := [Finite of FA by <:].
#[export] HB.instance Definition _ := [finGroupMixin of FA for +%R]. | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | FA | |
card_gen: #|{:FA}| = (#|F| ^ degree_mxminpoly A)%N.
Proof. by rewrite card_sub card_mx mul1n. 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | card_gen | |
group_splitting_field_existsgT (G : {group gT}) F :
classically {Fs : fieldType & {rmorphism F -> Fs}
& group_splitting_field Fs G}.
Proof.
move: F => F0 [] // nosplit; pose nG := #|G|; pose aG F := regular_repr F G.
pose m := nG.+1; pose F := F0; pose U : seq 'M[F]_nG := [::].
suffices: size U + m <= nG by rewrite ltnn.
have: mx_subseries (aG F) U /\ path ltmx 0 U by [].
pose f : {rmorphism F0 -> F} := idfun.
elim: m F U f => [|m IHm] F U f [modU ltU].
by rewrite addn0 (leq_trans (max_size_mx_series ltU)) ?rank_leq_row.
rewrite addnS ltnNge -implybF; apply/implyP=> le_nG_Um; apply: nosplit.
exists F => //; case=> [|n] rG irrG; first by case/mx_irrP: irrG.
apply/idPn=> nabsG; pose cG := ('C(enveloping_algebra_mx rG))%MS.
have{nabsG} [A]: exists2 A, (A \in cG)%MS & ~~ is_scalar_mx A.
apply/has_non_scalar_mxP; rewrite ?scalar_mx_cent // ltnNge.
by apply: contra nabsG; apply: cent_mx_scalar_abs_irr.
rewrite {cG}memmx_cent_envelop -mxminpoly_linear_is_scalar -ltnNge => cGA.
move/(non_linear_gen_reducible irrG cGA). | 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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | group_splitting_field_exists | |
group_closure_field_existsgT F :
classically {Fs : fieldType & {rmorphism F -> Fs}
& group_closure_field Fs gT}.
Proof.
set n := #|{group gT}|.
suffices: classically {Fs : fieldType & {rmorphism F -> Fs}
& forall G : {group gT}, enum_rank G < n -> group_splitting_field Fs G}.
- apply: classic_bind => [[Fs f splitFs]] _ -> //.
by exists Fs => // G; apply: splitFs.
elim: (n) => [|i IHi]; first by move=> _ -> //; exists F => //; exists id.
apply: classic_bind IHi => [[F' f splitF']].
have [le_n_i _ -> // | lt_i_n] := leqP n i.
by exists F' => // G _; apply: splitF'; apply: leq_trans le_n_i.
have:= @group_splitting_field_exists _ (enum_val (Ordinal lt_i_n)) F'.
apply: classic_bind => [[Fs f' splitFs]] _ -> //.
exists Fs => [|G]; first exact: (f' \o f).
rewrite ltnS leq_eqVlt -{1}[i]/(val (Ordinal lt_i_n)) val_eqE.
case/predU1P=> [defG | ltGi]; first by rewrite -[G]enum_rankK defG.
by apply: (extend_group_splitting_field f'); apply: splitF'.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | group_closure_field_exists | |
group_closure_closed_field(F : closedFieldType) gT :
group_closure_field F gT.
Proof.
move=> G [|n] rG irrG; first by case/mx_irrP: irrG.
apply: cent_mx_scalar_abs_irr => //; rewrite leqNgt.
apply/(has_non_scalar_mxP (scalar_mx_cent _ _)) => [[A cGA nscalA]].
have [a]: exists a, eigenvalue A a.
pose P := mxminpoly A; pose d := degree_mxminpoly A.
have Pd1: P`_d = 1.
by rewrite -(eqP (mxminpoly_monic A)) /lead_coef size_mxminpoly.
have d_gt0: d > 0 := mxminpoly_nonconstant A.
have [a def_ad] := solve_monicpoly (nth 0 (- P)) d_gt0.
exists a; rewrite eigenvalue_root_min -/P /root -oppr_eq0 -hornerN.
rewrite horner_coef size_polyN size_mxminpoly -/d big_ord_recr -def_ad.
by rewrite coefN Pd1 mulN1r /= subrr.
case/negP; rewrite kermx_eq0 row_free_unit (mx_Schur irrG) ?subr_eq0 //.
by rewrite -memmx_cent_envelop -raddfN linearD addmx_sub ?scalar_mx_cent.
by apply: contraNneq nscalA => ->; exact: scalar_mx_is_scalar.
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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | group_closure_closed_field | |
Zchar: {pred 'CF(B)} :=
[pred phi in 'CF(B, A) | dec_Cint_span (in_tuple S) 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/vcharacter.v | Zchar | |
cfun0_zchar: 0 \in Zchar.
Proof.
rewrite inE mem0v; apply/sumboolP; exists 0.
by rewrite big1 // => i _; rewrite ffunE.
Qed.
Fact Zchar_zmod : zmod_closed Zchar.
Proof.
split; first exact: cfun0_zchar.
move=> phi xi /andP[Aphi /sumboolP[a Da]] /andP[Axi /sumboolP[b Db]].
rewrite inE rpredB // Da Db -sumrB; apply/sumboolP; exists (a - b).
by apply: eq_bigr => i _; rewrite -mulrzBr !ffunE.
Qed.
HB.instance Definition _ := GRing.isZmodClosed.Build (classfun B) Zchar
Zchar_zmod. | 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 | cfun0_zchar | |
scale_zchara phi : a \in Num.int -> phi \in Zchar -> a *: phi \in Zchar.
Proof. by case/intrP=> m -> Zphi; rewrite scaler_int rpredMz. 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 | scale_zchar | |
zchar_splitS A phi :
phi \in 'Z[S, A] = (phi \in 'Z[S]) && (phi \in 'CF(G, A)).
Proof. by rewrite !inE cfun_onT andbC. 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 | zchar_split | |
zcharD1Ephi S : (phi \in 'Z[S, G^#]) = (phi \in 'Z[S]) && (phi 1%g == 0).
Proof. by rewrite zchar_split cfunD1E. 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 | zcharD1E | |
zcharD1phi S A :
(phi \in 'Z[S, A^#]) = (phi \in 'Z[S, A]) && (phi 1%g == 0).
Proof. by rewrite zchar_split cfun_onD1 andbA -zchar_split. 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 | zcharD1 | |
zcharWS A : {subset 'Z[S, A] <= 'Z[S]}.
Proof. by move=> phi; rewrite zchar_split => /andP[]. 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 | zcharW | |
zchar_onS A : {subset 'Z[S, A] <= 'CF(G, A)}.
Proof. by move=> phi /andP[]. 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 | zchar_on | |
zchar_onSA B S : A \subset B -> {subset 'Z[S, A] <= 'Z[S, B]}.
Proof.
move=> sAB phi; rewrite zchar_split (zchar_split _ B) => /andP[->].
exact: cfun_onS.
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 | zchar_onS | |
zchar_onGS : 'Z[S, G] =i 'Z[S].
Proof. by move=> phi; rewrite zchar_split cfun_onG andbT. 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 | zchar_onG | |
irr_vchar_onA : {subset 'Z[irr G, A] <= 'CF(G, A)}.
Proof. exact: zchar_on. 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_vchar_on | |
support_zcharS A phi : phi \in 'Z[S, A] -> support phi \subset A.
Proof. by move/zchar_on; 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/vcharacter.v | support_zchar | |
mem_zchar_onS A phi :
phi \in 'CF(G, A) -> phi \in S -> phi \in 'Z[S, A].
Proof.
move=> Aphi /(@tnthP _ _ (in_tuple S))[i Dphi]; rewrite inE /= {}Aphi {phi}Dphi.
apply/sumboolP; exists [ffun j => (j == i)%:Z].
rewrite (bigD1 i) //= ffunE eqxx (tnth_nth 0) big1 ?addr0 // => j i'j.
by rewrite ffunE (negPf i'j).
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 | mem_zchar_on | |
mem_zcharS phi : phi \in S -> phi \in 'Z[S].
Proof. by move=> Sphi; rewrite mem_zchar_on ?cfun_onT. 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 | mem_zchar | |
zchar_nth_expansionS A phi :
phi \in 'Z[S, A] ->
{z | forall i, z i \in Num.int & phi = \sum_(i < size S) z i *: S`_i}.
Proof.
case/andP=> _ /sumboolP/sig_eqW[/= z ->]; exists (intr \o z) => //=.
by apply: eq_bigr => i _; rewrite scaler_int.
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 | zchar_nth_expansion | |
zchar_tuple_expansionn (S : n.-tuple 'CF(G)) A phi :
phi \in 'Z[S, A] ->
{z | forall i, z i \in Num.int & phi = \sum_(i < n) z i *: S`_i}.
Proof. by move/zchar_nth_expansion; rewrite 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/vcharacter.v | zchar_tuple_expansion | |
zchar_expansionS A phi : uniq S ->
phi \in 'Z[S, A] ->
{z | forall xi, z xi \in Num.int & phi = \sum_(xi <- S) z xi *: xi}.
Proof.
move=> Suniq /zchar_nth_expansion[z Zz ->] /=.
pose zS xi := oapp z 0 (insub (index xi S)).
exists zS => [xi | ]; rewrite {}/zS; first by case: (insub _) => /=.
rewrite (big_nth 0) big_mkord; apply: eq_bigr => i _; congr (_ *: _).
by rewrite index_uniq // valK.
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 | zchar_expansion | |
zchar_spanS A : {subset 'Z[S, A] <= <<S>>%VS}.
Proof.
move=> _ /zchar_nth_expansion[z Zz ->] /=.
by apply: rpred_sum => i _; rewrite rpredZ // memv_span ?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 | zchar_span | |
zchar_transS1 S2 A B :
{subset S1 <= 'Z[S2, B]} -> {subset 'Z[S1, A] <= 'Z[S2, A]}.
Proof.
move=> sS12 phi; rewrite !(zchar_split _ A) andbC => /andP[->]; rewrite andbT.
case/zchar_nth_expansion=> z Zz ->; apply: rpred_sum => i _.
by rewrite scale_zchar // (@zcharW _ B) ?sS12 ?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 | zchar_trans | |
zchar_trans_onS1 S2 A :
{subset S1 <= 'Z[S2, A]} -> {subset 'Z[S1] <= 'Z[S2, A]}.
Proof.
move=> sS12 _ /zchar_nth_expansion[z Zz ->]; apply: rpred_sum => i _.
by rewrite scale_zchar // sS12 ?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 | zchar_trans_on | |
zchar_sub_irrS A :
{subset S <= 'Z[irr G]} -> {subset 'Z[S, A] <= 'Z[irr G, A]}.
Proof. exact: zchar_trans. 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 | zchar_sub_irr | |
zchar_subsetS1 S2 A :
{subset S1 <= S2} -> {subset 'Z[S1, A] <= 'Z[S2, A]}.
Proof.
move=> sS12; apply: zchar_trans setT _ => // f /sS12 S2f.
by rewrite 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 | zchar_subset | |
zchar_subseqS1 S2 A :
subseq S1 S2 -> {subset 'Z[S1, A] <= 'Z[S2, A]}.
Proof. by move/mem_subseq; apply: zchar_subset. 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 | zchar_subseq | |
zchar_filterS A (p : pred 'CF(G)) :
{subset 'Z[filter p S, A] <= 'Z[S, A]}.
Proof. by apply: zchar_subset=> f; apply/mem_subseq/filter_subseq. 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 | zchar_filter | |
char_vcharchi : chi \is a character -> chi \in 'Z[irr G].
Proof.
case/char_sum_irr=> r ->; apply: rpred_sum => i _.
by rewrite mem_zchar ?mem_tnth.
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 | char_vchar | |
irr_vchari : 'chi[G]_i \in 'Z[irr G].
Proof. exact/char_vchar/irr_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 | irr_vchar | |
cfun1_vchar: 1 \in 'Z[irr G]. Proof. by rewrite -irr0 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 | cfun1_vchar | |
vcharPphi :
reflect (exists2 chi1, chi1 \is a character
& exists2 chi2, chi2 \is a character & phi = chi1 - chi2)
(phi \in 'Z[irr G]).
Proof.
apply: (iffP idP) => [| [a Na [b Nb ->]]]; last by rewrite rpredB ?char_vchar.
case/zchar_tuple_expansion=> z Zz ->; rewrite (bigID (fun i => 0 <= z i)) /=.
set chi1 := \sum_(i | _) _; set nchi2 := \sum_(i | _) _.
exists chi1; last exists (- nchi2); last by rewrite opprK.
apply: rpred_sum => i zi_ge0; rewrite -tnth_nth rpredZ_nat ?irr_char //.
by rewrite natrEint Zz.
rewrite -sumrN rpred_sum // => i zi_lt0; rewrite -scaleNr -tnth_nth.
rewrite rpredZ_nat ?irr_char // natrEint rpredN Zz oppr_ge0 ltW //.
by rewrite real_ltNge ?Rreal_int.
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 | vcharP | |
Aint_vcharphi x : phi \in 'Z[irr G] -> phi x \in Aint.
Proof.
case/vcharP=> [chi1 Nchi1 [chi2 Nchi2 ->]].
by rewrite !cfunE rpredB ?Aint_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 | Aint_vchar | |
Cint_vchar1phi : phi \in 'Z[irr G] -> phi 1%g \in Num.int.
Proof.
case/vcharP=> phi1 Nphi1 [phi2 Nphi2 ->].
by rewrite !cfunE rpredB // rpred_nat_num ?Cnat_char1.
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 | Cint_vchar1 | |
Cint_cfdot_vchar_irri phi :
phi \in 'Z[irr G] -> '[phi, 'chi_i] \in Num.int.
Proof.
case/vcharP=> chi1 Nchi1 [chi2 Nchi2 ->].
by rewrite cfdotBl rpredB // rpred_nat_num ?Cnat_cfdot_char_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 | Cint_cfdot_vchar_irr | |
cfdot_vchar_rphi psi :
psi \in 'Z[irr G] -> '[phi, psi] = \sum_i '[phi, 'chi_i] * '[psi, 'chi_i].
Proof.
move=> Zpsi; rewrite cfdot_sum_irr; apply: eq_bigr => i _; congr (_ * _).
by rewrite aut_intr ?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 | cfdot_vchar_r | |
Cint_cfdot_vchar:
{in 'Z[irr G] &, forall phi psi, '[phi, psi] \in Num.int}.
Proof.
move=> phi psi Zphi Zpsi; rewrite /= cfdot_vchar_r // rpred_sum // => k _.
by rewrite rpredM ?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 | Cint_cfdot_vchar | |
Cnat_cfnorm_vchar: {in 'Z[irr G], forall phi, '[phi] \in Num.nat}.
Proof. by move=> phi Zphi; rewrite /= natrEint cfnorm_ge0 Cint_cfdot_vchar. Qed.
Fact vchar_mulr_closed : mulr_closed 'Z[irr G].
Proof.
split; first exact: cfun1_vchar.
move=> _ _ /vcharP[xi1 Nxi1 [xi2 Nxi2 ->]] /vcharP[xi3 Nxi3 [xi4 Nxi4 ->]].
by rewrite mulrBl !mulrBr !(rpredB, rpredD) // char_vchar ?rpredM.
Qed.
HB.instance Definition _ := GRing.isMulClosed.Build (classfun G) 'Z[irr G]
vchar_mulr_closed. | 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_cfnorm_vchar | |
mul_vcharA :
{in 'Z[irr G, A] &, forall phi psi, phi * psi \in 'Z[irr G, A]}.
Proof.
move=> phi psi; rewrite zchar_split => /andP[Zphi Aphi] /zcharW Zpsi.
rewrite zchar_split rpredM //; apply/cfun_onP=> x A'x.
by rewrite cfunE (cfun_onP Aphi) ?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/vcharacter.v | mul_vchar | |
map_pairwise_orthogonal: pairwise_orthogonal (map nu S).
Proof.
have inj_nu: {in S &, injective nu}.
move=> phi psi Sphi Spsi /= eq_nu; apply: contraNeq (memPn notS0 _ Sphi).
by rewrite -cfnorm_eq0 -Inu ?Z_S // {2}eq_nu Inu ?Z_S // => /dotSS->.
have notSnu0: 0 \notin map nu S.
apply: contra notS0 => /mapP[phi Sphi /esym/eqP].
by rewrite -cfnorm_eq0 Inu ?Z_S // cfnorm_eq0 => /eqP <-.
apply/pairwise_orthogonalP; split; first by rewrite /= notSnu0 map_inj_in_uniq.
move=> _ _ /mapP[phi Sphi ->] /mapP[psi Spsi ->].
by rewrite (inj_in_eq inj_nu) // Inu ?Z_S //; apply: dotSS.
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_pairwise_orthogonal | |
cfproj_sum_orthogonalP z phi :
phi \in S ->
'[\sum_(xi <- S | P xi) z xi *: nu xi, nu phi]
= if P phi then z phi * '[phi] else 0.
Proof.
move=> Sphi; have defS := perm_to_rem Sphi.
rewrite cfdot_suml (perm_big _ defS) big_cons /= cfdotZl Inu ?Z_S //.
rewrite big1_seq ?addr0 // => xi; rewrite mem_rem_uniq ?inE //.
by case/and3P=> _ neq_xi Sxi; rewrite cfdotZl Inu ?Z_S // dotSS ?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/vcharacter.v | cfproj_sum_orthogonal | |
cfdot_sum_orthogonalz1 z2 :
'[\sum_(xi <- S) z1 xi *: nu xi, \sum_(xi <- S) z2 xi *: nu xi]
= \sum_(xi <- S) z1 xi * (z2 xi)^* * '[xi].
Proof.
rewrite cfdot_sumr; apply: eq_big_seq => phi Sphi.
by rewrite cfdotZr cfproj_sum_orthogonal // mulrCA mulrA.
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_orthogonal | |
cfnorm_sum_orthogonalz :
'[\sum_(xi <- S) z xi *: nu xi] = \sum_(xi <- S) `|z xi| ^+ 2 * '[xi].
Proof.
by rewrite cfdot_sum_orthogonal; apply: eq_bigr => xi _; rewrite normCK.
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_orthogonal |
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