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 |
|---|---|---|---|---|---|---|
rstabs_sub: rstabs \subset G.
Proof. by apply/subsetP=> x /setIdP[]. 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_sub | |
rstabs_group_set: group_set rstabs.
Proof.
apply/group_setP; rewrite inE group1 repr_mx1 mulmx1.
split=> //= x y /setIdP[Gx nUx] /setIdP[Gy]; rewrite inE repr_mxM ?groupM //.
by apply: submx_trans; rewrite mulmxA submxMr.
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_group_set | |
rstabs_group:= Group rstabs_group_set. | 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 | rstabs_group | |
rstab_actx m1 (W : 'M_(m1, n)) :
x \in rstab rG U -> (W <= U)%MS -> W *m rG x = W.
Proof. by case/setIdP=> _ /eqP cUx /submxP[w ->]; rewrite -mulmxA cUx. 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_act | |
rstabs_actx m1 (W : 'M_(m1, n)) :
x \in rstabs -> (W <= U)%MS -> (W *m rG x <= U)%MS.
Proof.
by case/setIdP=> [_ nUx] sWU; apply: submx_trans nUx; apply: submxMr.
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_act | |
mxmodule:= G \subset rstabs. | 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 | mxmodule | |
mxmoduleP: reflect {in G, forall x, U *m rG x <= U}%MS mxmodule.
Proof.
by apply: (iffP subsetP) => modU x Gx; have:= modU x Gx; rewrite !inE ?Gx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxmoduleP | |
rstabSm1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U <= V)%MS -> rstab rG V \subset rstab rG U.
Proof.
case/submxP=> u ->; apply/subsetP=> x.
by rewrite !inE => /andP[-> /= /eqP cVx]; rewrite -mulmxA cVx.
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 | |
eqmx_rstabm1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U :=: V)%MS -> rstab rG U = rstab rG V.
Proof. by move=> eqUV; apply/eqP; rewrite eqEsubset !rstabS ?eqUV. 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 | eqmx_rstab | |
eqmx_rstabsm1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U :=: V)%MS -> rstabs U = rstabs V.
Proof. by move=> eqUV; apply/setP=> x; rewrite !inE eqUV (eqmxMr _ eqUV). 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 | eqmx_rstabs | |
eqmx_modulem1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U :=: V)%MS -> mxmodule U = mxmodule V.
Proof. by move=> eqUV; rewrite /mxmodule (eqmx_rstabs eqUV). 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 | eqmx_module | |
mxmodule0m : mxmodule (0 : 'M_(m, n)).
Proof. by apply/mxmoduleP=> x _; rewrite mul0mx. 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 | mxmodule0 | |
mxmodule1: mxmodule 1%:M.
Proof. by apply/mxmoduleP=> x _; rewrite submx1. 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 | mxmodule1 | |
mxmodule_transm1 m2 (U : 'M_(m1, n)) (W : 'M_(m2, n)) x :
mxmodule U -> x \in G -> (W <= U -> W *m rG x <= U)%MS.
Proof.
by move=> modU Gx sWU; apply: submx_trans (mxmoduleP modU x Gx); apply: submxMr.
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_trans | |
mxmodule_eigenvectorm (U : 'M_(m, n)) :
mxmodule U -> \rank U = 1 ->
{u : 'rV_n & {a | (U :=: u)%MS & {in G, forall x, u *m rG x = a x *: u}}}.
Proof.
move=> modU linU; set u := nz_row U; exists u.
have defU: (U :=: u)%MS.
apply/eqmxP; rewrite andbC -(geq_leqif (mxrank_leqif_eq _)) ?nz_row_sub //.
by rewrite linU lt0n mxrank_eq0 nz_row_eq0 -mxrank_eq0 linU.
pose a x := (u *m rG x *m pinvmx u) 0 0; exists a => // x Gx.
by rewrite -mul_scalar_mx -mx11_scalar mulmxKpV // -defU mxmodule_trans ?defU.
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_eigenvector | |
addsmx_modulem1 m2 U V :
@mxmodule m1 U -> @mxmodule m2 V -> mxmodule (U + V)%MS.
Proof.
move=> modU modV; apply/mxmoduleP=> x Gx.
by rewrite addsmxMr addsmxS ?(mxmoduleP _ x Gx).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | addsmx_module | |
sumsmx_moduleI r (P : pred I) U :
(forall i, P i -> mxmodule (U i)) -> mxmodule (\sum_(i <- r | P i) U i)%MS.
Proof.
by move=> modU; elim/big_ind: _; [apply: mxmodule0 | apply: addsmx_module | ].
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 | sumsmx_module | |
capmx_modulem1 m2 U V :
@mxmodule m1 U -> @mxmodule m2 V -> mxmodule (U :&: V)%MS.
Proof.
move=> modU modV; apply/mxmoduleP=> x Gx.
by rewrite sub_capmx !mxmodule_trans ?capmxSl ?capmxSr.
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 | capmx_module | |
bigcapmx_moduleI r (P : pred I) U :
(forall i, P i -> mxmodule (U i)) -> mxmodule (\bigcap_(i <- r | P i) U i)%MS.
Proof.
by move=> modU; elim/big_ind: _; [apply: mxmodule1 | apply: capmx_module | ].
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 | bigcapmx_module | |
val_submodm : 'M_(m, \rank U) -> 'M_(m, n) := mulmxr (row_base U). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_submod | |
in_submodm : 'M_(m, n) -> 'M_(m, \rank U) :=
mulmxr (invmx (row_ebase U) *m pid_mx (\rank U)).
HB.instance Definition _ m := GRing.Linear.on (@val_submod m).
HB.instance Definition _ m := GRing.Linear.on (@in_submod m). | 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_submod | |
val_submodEm W : @val_submod m W = W *m val_submod 1%:M.
Proof. by rewrite mulmxA mulmx1. 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_submodE | |
in_submodEm W : @in_submod m W = W *m in_submod 1%:M.
Proof. by rewrite mulmxA mulmx1. 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_submodE | |
val_submod1: (val_submod 1%:M :=: U)%MS.
Proof. by rewrite /val_submod /= mul1mx; apply: eq_row_base. 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_submod1 | |
val_submodPm W : (@val_submod m W <= U)%MS.
Proof. by rewrite mulmx_sub ?eq_row_base. 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_submodP | |
val_submodKm : cancel (@val_submod m) (@in_submod m).
Proof.
move=> W; rewrite /in_submod /= -!mulmxA mulKVmx ?row_ebase_unit //.
by rewrite pid_mx_id ?rank_leq_row // pid_mx_1 mulmx1.
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_submodK | |
val_submod_injm : injective (@val_submod m).
Proof. exact: can_inj (@val_submodK m). 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_submod_inj | |
val_submodSm1 m2 (V : 'M_(m1, \rank U)) (W : 'M_(m2, \rank U)) :
(val_submod V <= val_submod W)%MS = (V <= W)%MS.
Proof.
apply/idP/idP=> sVW; last exact: submxMr.
by rewrite -[V]val_submodK -[W]val_submodK submxMr.
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_submodS | |
in_submodKm W : (W <= U)%MS -> val_submod (@in_submod m W) = W.
Proof.
case/submxP=> w ->; rewrite /val_submod /= -!mulmxA.
congr (_ *m _); rewrite -{1}[U]mulmx_ebase !mulmxA mulmxK ?row_ebase_unit //.
by rewrite -2!(mulmxA (col_ebase U)) !pid_mx_id ?rank_leq_row // mulmx_ebase.
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_submodK | |
val_submod_eq0m W : (@val_submod m W == 0) = (W == 0).
Proof. by rewrite -!submx0 -val_submodS linear0 !(submx0, eqmx0). 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_submod_eq0 | |
in_submod_eq0m W : (@in_submod m W == 0) = (W <= U^C)%MS.
Proof.
apply/eqP/submxP=> [W_U0 | [w ->{W}]].
exists (W *m invmx (row_ebase U)).
rewrite mulmxA mulmxBr mulmx1 -(pid_mx_id _ _ _ (leqnn _)).
rewrite mulmxA -(mulmxA W) [W *m (_ *m _)]W_U0 mul0mx subr0.
by rewrite mulmxKV ?row_ebase_unit.
rewrite /in_submod /= -!mulmxA mulKVmx ?row_ebase_unit //.
by rewrite mul_copid_mx_pid ?rank_leq_row ?mulmx0.
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_submod_eq0 | |
mxrank_in_submodm (W : 'M_(m, n)) :
(W <= U)%MS -> \rank (in_submod W) = \rank W.
Proof.
by move=> sWU; apply/eqP; rewrite eqn_leq -{3}(in_submodK sWU) !mxrankM_maxl.
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 | mxrank_in_submod | |
val_factmodm : _ -> 'M_(m, n) :=
mulmxr (row_base (cokermx U) *m row_ebase U). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_factmod | |
in_factmodm : 'M_(m, n) -> _ := mulmxr (col_base (cokermx U)).
HB.instance Definition _ m := GRing.Linear.on (@val_factmod m).
HB.instance Definition _ m := GRing.Linear.on (@in_factmod m). | 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_factmod | |
val_factmodEm W : @val_factmod m W = W *m val_factmod 1%:M.
Proof. by rewrite mulmxA mulmx1. 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_factmodE | |
in_factmodEm W : @in_factmod m W = W *m in_factmod 1%:M.
Proof. by rewrite mulmxA mulmx1. 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_factmodE | |
val_factmodPm W : (@val_factmod m W <= U^C)%MS.
Proof.
by rewrite mulmx_sub {m W}// (eqmxMr _ (eq_row_base _)) -mulmxA submxMl.
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_factmodP | |
val_factmodKm : cancel (@val_factmod m) (@in_factmod m).
Proof.
move=> W /=; rewrite /in_factmod /=; set Uc := cokermx U.
apply: (row_free_inj (row_base_free Uc)); rewrite -mulmxA mulmx_base.
rewrite /val_factmod /= 2!mulmxA -/Uc mulmxK ?row_ebase_unit //.
have /submxP[u ->]: (row_base Uc <= Uc)%MS by rewrite eq_row_base.
by rewrite -!mulmxA copid_mx_id ?rank_leq_row.
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_factmodK | |
val_factmod_injm : injective (@val_factmod m).
Proof. exact: can_inj (@val_factmodK m). 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_factmod_inj | |
val_factmodSm1 m2 (V : 'M_(m1, _)) (W : 'M_(m2, _)) :
(val_factmod V <= val_factmod W)%MS = (V <= W)%MS.
Proof.
apply/idP/idP=> sVW; last exact: submxMr.
by rewrite -[V]val_factmodK -[W]val_factmodK submxMr.
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_factmodS | |
val_factmod_eq0m W : (@val_factmod m W == 0) = (W == 0).
Proof. by rewrite -!submx0 -val_factmodS linear0 !(submx0, eqmx0). 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_factmod_eq0 | |
in_factmod_eq0m (W : 'M_(m, n)) : (in_factmod W == 0) = (W <= U)%MS.
Proof.
rewrite submxE -!mxrank_eq0 -{2}[_ U]mulmx_base mulmxA.
by rewrite (mxrankMfree _ (row_base_free _)).
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_factmod_eq0 | |
in_factmodKm (W : 'M_(m, n)) :
(W <= U^C)%MS -> val_factmod (in_factmod W) = W.
Proof.
case/submxP=> w ->{W}; rewrite /val_factmod /= -2!mulmxA.
congr (_ *m _); rewrite (mulmxA (col_base _)) mulmx_base -2!mulmxA.
by rewrite mulKVmx ?row_ebase_unit // mulmxA copid_mx_id ?rank_leq_row.
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_factmodK | |
in_factmod_addsKm (W : 'M_(m, n)) :
(in_factmod (U + W)%MS :=: in_factmod W)%MS.
Proof.
apply: eqmx_trans (addsmxMr _ _ _) _.
by rewrite ((_ *m _ =P 0) _) ?in_factmod_eq0 //; apply: adds0mx.
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_factmod_addsK | |
add_sub_fact_modm (W : 'M_(m, n)) :
val_submod (in_submod W) + val_factmod (in_factmod W) = W.
Proof.
rewrite /val_submod /val_factmod /= -!mulmxA -mulmxDr.
rewrite addrC ) pid_mx_id //.
rewrite (mulmxA (col_ebase _)) (mulmxA _ _ (row_ebase _)) mulmx_ebase.
rewrite (mulmxA (pid_mx _)) pid_mx_id // mulmxA -mulmxDl -mulmxDr.
by rewrite subrK mulmx1 mulmxA mulmxKV ?row_ebase_unit.
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 | add_sub_fact_mod | |
proj_factmodSm (W : 'M_(m, n)) :
(val_factmod (in_factmod W) <= U + W)%MS.
Proof.
by rewrite -{2}[W]add_sub_fact_mod addsmx_addKl ?val_submodP ?addsmxSr.
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 | proj_factmodS | |
in_factmodsKm (W : 'M_(m, n)) :
(U <= W)%MS -> (U + val_factmod (in_factmod W) :=: W)%MS.
Proof.
move/addsmx_idPr; apply: eqmx_trans (eqmx_sym _).
by rewrite -{1}[W]add_sub_fact_mod; apply: addsmx_addKl; apply: val_submodP.
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_factmodsK | |
mxrank_in_factmodm (W : 'M_(m, n)) :
(\rank (in_factmod W) + \rank U)%N = \rank (U + W).
Proof.
rewrite -in_factmod_addsK in_factmodE; set fU := in_factmod 1%:M.
suffices <-: ((U + W) :&: kermx fU :=: U)%MS by rewrite mxrank_mul_ker.
apply: eqmx_trans (capmx_idPr (addsmxSl U W)).
apply: cap_eqmx => //; apply/eqmxP/rV_eqP => u.
by rewrite (sameP sub_kermxP eqP) -in_factmodE in_factmod_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 | mxrank_in_factmod | |
submod_mxof mxmodule U :=
fun x => in_submod (val_submod 1%:M *m rG x). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | submod_mx | |
factmod_mxof mxmodule U :=
fun x => in_factmod (val_factmod 1%:M *m rG x).
Hypothesis Umod : mxmodule U. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | factmod_mx | |
in_submodJm (W : 'M_(m, n)) x :
(W <= U)%MS -> in_submod (W *m rG x) = in_submod W *m submod_mx Umod x.
Proof.
move=> sWU; rewrite mulmxA; congr (in_submod _).
by rewrite mulmxA -val_submodE in_submodK.
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_submodJ | |
val_submodJm (W : 'M_(m, \rank U)) x :
x \in G -> val_submod (W *m submod_mx Umod x) = val_submod W *m rG x.
Proof.
move=> Gx; rewrite 2!(mulmxA W) -val_submodE in_submodK //.
by rewrite mxmodule_trans ?val_submodP.
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_submodJ | |
submod_mx_repr: mx_repr G (submod_mx Umod).
Proof.
rewrite /submod_mx; split=> [|x y Gx Gy /=].
by rewrite repr_mx1 mulmx1 val_submodK.
rewrite -in_submodJ; first by rewrite repr_mxM ?mulmxA.
by rewrite mxmodule_trans ?val_submodP.
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 | submod_mx_repr | |
submod_repr:= MxRepresentation submod_mx_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 | submod_repr | |
in_factmodJm (W : 'M_(m, n)) x :
x \in G -> in_factmod (W *m rG x) = in_factmod W *m factmod_mx Umod x.
Proof.
move=> Gx; rewrite -{1}[W]add_sub_fact_mod mulmxDl linearD /=.
apply: (canLR (subrK _)); apply: etrans (_ : 0 = _).
apply/eqP; rewrite in_factmod_eq0 (submx_trans _ (mxmoduleP Umod x Gx)) //.
by rewrite submxMr ?val_submodP.
by rewrite /in_factmod /val_factmod /= !mulmxA mulmx1 ?subrr.
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_factmodJ | |
val_factmodJm (W : 'M_(m, \rank (cokermx U))) x :
x \in G ->
val_factmod (W *m factmod_mx Umod x) =
val_factmod (in_factmod (val_factmod W *m rG x)).
Proof. by move=> Gx; rewrite -{1}[W]val_factmodK -in_factmodJ. 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_factmodJ | |
factmod_mx_repr: mx_repr G (factmod_mx Umod).
Proof.
split=> [|x y Gx Gy /=].
by rewrite /factmod_mx repr_mx1 mulmx1 val_factmodK.
by rewrite -in_factmodJ // -mulmxA -repr_mxM.
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 | factmod_mx_repr | |
factmod_repr:= MxRepresentation factmod_mx_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 | factmod_repr | |
mxtrace_sub_fact_modx :
\tr (submod_repr x) + \tr (factmod_repr x) = \tr (rG x).
Proof.
rewrite -[submod_repr x]mulmxA mxtrace_mulC -val_submodE addrC.
rewrite -[factmod_repr x]mulmxA mxtrace_mulC -val_factmodE addrC.
by rewrite -mxtraceD add_sub_fact_mod.
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 | mxtrace_sub_fact_mod | |
envelop_mx_idx : x \in G -> (rG x \in E_G)%MS.
Proof.
by move=> Gx; rewrite (eq_row_sub (enum_rank_in Gx x)) // rowK enum_rankK_in.
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 | envelop_mx_id | |
envelop_mx1: (1%:M \in E_G)%MS.
Proof. by rewrite -(repr_mx1 rG) envelop_mx_id. 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 | envelop_mx1 | |
envelop_mxPA :
reflect (exists a, A = \sum_(x in G) a x *: rG x) (A \in E_G)%MS.
Proof.
have G_1 := group1 G; have bijG := enum_val_bij_in G_1.
set h := enum_val in bijG; have Gh: h _ \in G by apply: enum_valP.
apply: (iffP submxP) => [[u defA] | [a ->]].
exists (fun x => u 0 (enum_rank_in G_1 x)); apply: (can_inj mxvecK).
rewrite defA mulmx_sum_row linear_sum (reindex h) //=.
by apply: eq_big => [i | i _]; rewrite ?Gh // rowK linearZ enum_valK_in.
exists (\row_i a (h i)); rewrite mulmx_sum_row linear_sum (reindex h) //=.
by apply: eq_big => [i | i _]; rewrite ?Gh // mxE rowK 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 | envelop_mxP | |
envelop_mxMA B : (A \in E_G -> B \in E_G -> A *m B \in E_G)%MS.
Proof.
move=> {A B} /envelop_mxP[a ->] /envelop_mxP[b ->].
rewrite mulmx_suml !linear_sum summx_sub //= => x Gx.
rewrite !linear_sum summx_sub //= => y Gy.
rewrite -scalemxAl 3!linearZ !scalemx_sub//= -repr_mxM //.
by rewrite envelop_mx_id ?groupM.
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 | envelop_mxM | |
mxmodule_envelopm1 m2 (U : 'M_(m1, n)) (W : 'M_(m2, n)) A :
(mxmodule U -> mxvec A <= E_G -> W <= U -> W *m A <= U)%MS.
Proof.
move=> modU /envelop_mxP[a ->] sWU; rewrite linear_sum summx_sub //= => x Gx.
by rewrite -scalemxAr scalemx_sub ?mxmodule_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",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxmodule_envelop | |
dom_hom_mxf : 'M_n :=
kermx (lin1_mx (mxvec \o mulmx (cent_mx_fun E_G f) \o lin_mul_row)). | 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 | dom_hom_mx | |
hom_mxPm f (W : 'M_(m, n)) :
reflect (forall x, x \in G -> W *m rG x *m f = W *m f *m rG x)
(W <= dom_hom_mx f)%MS.
Proof.
apply: (iffP row_subP) => [cGf x Gx | cGf i].
apply/row_matrixP=> i; apply/eqP; rewrite -subr_eq0 -!mulmxA -!linearB /=.
have:= sub_kermxP (cGf i); rewrite mul_rV_lin1 /=.
move/(canRL mxvecK)/row_matrixP/(_ (enum_rank_in Gx x))/eqP; rewrite !linear0.
by rewrite !row_mul rowK mul_vec_lin /= mul_vec_lin_row enum_rankK_in.
apply/sub_kermxP; rewrite mul_rV_lin1 /=; apply: (canLR vec_mxK).
apply/row_matrixP=> j; rewrite !row_mul rowK mul_vec_lin /= mul_vec_lin_row.
by rewrite -!row_mul mulmxBr !mulmxA cGf ?enum_valP // subrr !linear0.
Qed.
Arguments hom_mxP {m f W}. | 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 | hom_mxP | |
hom_envelop_mxCm f (W : 'M_(m, n)) A :
(W <= dom_hom_mx f -> A \in E_G -> W *m A *m f = W *m f *m A)%MS.
Proof.
move/hom_mxP=> cWfG /envelop_mxP[a ->]; rewrite !linear_sum mulmx_suml.
by apply: eq_bigr => x Gx /=; rewrite -2!scalemxAr -scalemxAl cWfG.
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 | hom_envelop_mxC | |
dom_hom_invmxf :
f \in unitmx -> (dom_hom_mx (invmx f) :=: dom_hom_mx f *m f)%MS.
Proof.
move=> injf; set U := dom_hom_mx _; apply/eqmxP.
rewrite -{1}[U](mulmxKV injf) submxMr; apply/hom_mxP=> x Gx.
by rewrite -[_ *m rG x](hom_mxP _) ?mulmxK.
by rewrite -[_ *m rG x](hom_mxP _) ?mulmxKV.
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 | dom_hom_invmx | |
dom_hom_mx_modulef : mxmodule (dom_hom_mx f).
Proof.
apply/mxmoduleP=> x Gx; apply/hom_mxP=> y Gy.
rewrite -[_ *m rG y]mulmxA -repr_mxM // 2?(hom_mxP _) ?groupM //.
by rewrite repr_mxM ?mulmxA.
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 | dom_hom_mx_module | |
hom_mxmodulem (U : 'M_(m, n)) f :
(U <= dom_hom_mx f)%MS -> mxmodule U -> mxmodule (U *m f).
Proof.
move/hom_mxP=> cGfU modU; apply/mxmoduleP=> x Gx.
by rewrite -cGfU // submxMr // (mxmoduleP modU).
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 | hom_mxmodule | |
kermx_hom_modulem (U : 'M_(m, n)) f :
(U <= dom_hom_mx f)%MS -> mxmodule U -> mxmodule (U :&: kermx f)%MS.
Proof.
move=> homUf modU; apply/mxmoduleP=> x Gx.
rewrite sub_capmx mxmodule_trans ?capmxSl //=.
apply/sub_kermxP; rewrite (hom_mxP _) ?(submx_trans (capmxSl _ _)) //.
by rewrite (sub_kermxP (capmxSr _ _)) mul0mx.
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 | kermx_hom_module | |
scalar_mx_homa m (U : 'M_(m, n)) : (U <= dom_hom_mx a%:M)%MS.
Proof. by apply/hom_mxP=> x Gx; rewrite -!mulmxA scalar_mxC. 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 | scalar_mx_hom | |
proj_mx_hom(U V : 'M_n) :
(U :&: V = 0)%MS -> mxmodule U -> mxmodule V ->
(U + V <= dom_hom_mx (proj_mx U V))%MS.
Proof.
move=> dxUV modU modV; apply/hom_mxP=> x Gx.
rewrite -{1}(add_proj_mx dxUV (submx_refl _)) !mulmxDl addrC.
rewrite {1}[_ *m _]proj_mx_0 ?add0r //; last first.
by rewrite mxmodule_trans ?proj_mx_sub.
by rewrite [_ *m _](proj_mx_id dxUV) // mxmodule_trans ?proj_mx_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 | proj_mx_hom | |
rfix_mx(H : {set gT}) :=
let commrH := \matrix_(i < #|H|) mxvec (rG (enum_val i) - 1%:M) in
kermx (lin1_mx (mxvec \o mulmx commrH \o lin_mul_row)). | 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 | rfix_mx | |
rfix_mxPm (W : 'M_(m, n)) (H : {set gT}) :
reflect (forall x, x \in H -> W *m rG x = W) (W <= rfix_mx H)%MS.
Proof.
rewrite /rfix_mx; set C := \matrix_i _.
apply: (iffP row_subP) => [cHW x Hx | cHW j].
apply/row_matrixP=> j; apply/eqP; rewrite -subr_eq0 row_mul.
move/sub_kermxP: {cHW}(cHW j); rewrite mul_rV_lin1 /=; move/(canRL mxvecK).
move/row_matrixP/(_ (enum_rank_in Hx x)); rewrite row_mul rowK !linear0.
by rewrite enum_rankK_in // mul_vec_lin_row mulmxBr mulmx1 => ->.
apply/sub_kermxP; rewrite mul_rV_lin1 /=; apply: (canLR vec_mxK).
apply/row_matrixP=> i; rewrite row_mul rowK mul_vec_lin_row -row_mul.
by rewrite mulmxBr mulmx1 cHW ?enum_valP // subrr !linear0.
Qed.
Arguments rfix_mxP {m W}. | 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_mxP | |
rfix_mx_id(H : {set gT}) x : x \in H -> rfix_mx H *m rG x = rfix_mx H.
Proof. exact/rfix_mxP. 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_mx_id | |
rfix_mxS(H K : {set gT}) : H \subset K -> (rfix_mx K <= rfix_mx H)%MS.
Proof.
by move=> sHK; apply/rfix_mxP=> x Hx; apply: rfix_mxP (subsetP sHK x Hx).
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_mxS | |
rfix_mx_conjsg(H : {set gT}) x :
x \in G -> H \subset G -> (rfix_mx (H :^ x) :=: rfix_mx H *m rG x)%MS.
Proof.
move=> Gx sHG; pose rf y := rfix_mx (H :^ y).
suffices{x Gx} IH: {in G &, forall y z, rf y *m rG z <= rf (y * z)%g}%MS.
apply/eqmxP; rewrite -/(rf x) -[H]conjsg1 -/(rf 1%g).
rewrite -{4}[x] mul1g -{1}[rf x](repr_mxKV rG Gx) -{1}(mulgV x).
by rewrite submxMr IH ?groupV.
move=> x y Gx Gy; apply/rfix_mxP=> zxy; rewrite actM => /imsetP[zx Hzx ->].
have Gzx: zx \in G by apply: subsetP Hzx; rewrite conj_subG.
rewrite -mulmxA -repr_mxM ?groupM ?groupV // -conjgC repr_mxM // mulmxA.
by rewrite rfix_mx_id.
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_mx_conjsg | |
norm_sub_rstabs_rfix_mx(H : {set gT}) :
H \subset G -> 'N_G(H) \subset rstabs (rfix_mx H).
Proof.
move=> sHG; apply/subsetP=> x /setIP[Gx nHx]; rewrite inE Gx.
apply/rfix_mxP=> y Hy; have Gy := subsetP sHG y Hy.
have Hyx: (y ^ x^-1)%g \in H by rewrite memJ_norm ?groupV.
rewrite -mulmxA -repr_mxM // conjgCV repr_mxM ?(subsetP sHG _ Hyx) // mulmxA.
by rewrite (rfix_mx_id Hyx).
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 | norm_sub_rstabs_rfix_mx | |
normal_rfix_mx_moduleH : H <| G -> mxmodule (rfix_mx H).
Proof.
case/andP=> sHG nHG.
by rewrite /mxmodule -{1}(setIidPl nHG) norm_sub_rstabs_rfix_mx.
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 | normal_rfix_mx_module | |
rfix_mx_module: mxmodule (rfix_mx G).
Proof. exact: normal_rfix_mx_module. 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_mx_module | |
rfix_mx_rstabC(H : {set gT}) m (U : 'M[F]_(m, n)) :
H \subset G -> (H \subset rstab rG U) = (U <= rfix_mx H)%MS.
Proof.
move=> sHG; apply/subsetP/rfix_mxP=> cHU x Hx.
by rewrite (rstab_act (cHU x Hx)).
by rewrite !inE (subsetP sHG) //= cHU.
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_mx_rstabC | |
cyclic_mxu := <<E_G *m lin_mul_row u>>%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 | cyclic_mx | |
cyclic_mxPu v :
reflect (exists2 A, A \in E_G & v = u *m A)%MS (v <= cyclic_mx u)%MS.
Proof.
rewrite genmxE; apply: (iffP submxP) => [[a] | [A /submxP[a defA]]] -> {v}.
exists (vec_mx (a *m E_G)); last by rewrite mulmxA mul_rV_lin1.
by rewrite vec_mxK submxMl.
by exists a; rewrite mulmxA mul_rV_lin1 /= -defA mxvecK.
Qed.
Arguments cyclic_mxP {u v}. | 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 | cyclic_mxP | |
cyclic_mx_idu : (u <= cyclic_mx u)%MS.
Proof. by apply/cyclic_mxP; exists 1%:M; rewrite ?mulmx1 ?envelop_mx1. 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 | cyclic_mx_id | |
cyclic_mx_eq0u : (cyclic_mx u == 0) = (u == 0).
Proof.
rewrite -!submx0; apply/idP/idP.
by apply: submx_trans; apply: cyclic_mx_id.
move/submx0null->; rewrite genmxE; apply/row_subP=> i.
by rewrite row_mul mul_rV_lin1 /= mul0mx ?sub0mx.
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 | cyclic_mx_eq0 | |
cyclic_mx_moduleu : mxmodule (cyclic_mx u).
Proof.
apply/mxmoduleP=> x Gx; apply/row_subP=> i; rewrite row_mul.
have [A E_A ->{i}] := @cyclic_mxP u _ (row_sub i _); rewrite -mulmxA.
by apply/cyclic_mxP; exists (A *m rG x); rewrite ?envelop_mxM ?envelop_mx_id.
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 | cyclic_mx_module | |
cyclic_mx_subm u (W : 'M_(m, n)) :
mxmodule W -> (u <= W)%MS -> (cyclic_mx u <= W)%MS.
Proof.
move=> modU Wu; rewrite genmxE; apply/row_subP=> i.
by rewrite row_mul mul_rV_lin1 /= mxmodule_envelop // vec_mxK 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 | cyclic_mx_sub | |
hom_cyclic_mxu f :
(u <= dom_hom_mx f)%MS -> (cyclic_mx u *m f :=: cyclic_mx (u *m f))%MS.
Proof.
move=> domf_u; apply/eqmxP; rewrite !(eqmxMr _ (genmxE _)).
apply/genmxP; rewrite genmx_id; congr <<_>>%MS; apply/row_matrixP=> i.
by rewrite !row_mul !mul_rV_lin1 /= hom_envelop_mxC // vec_mxK 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 | hom_cyclic_mx | |
annihilator_mxu := (E_G :&: kermx (lin_mul_row u))%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 | annihilator_mx | |
annihilator_mxPu A :
reflect (A \in E_G /\ u *m A = 0)%MS (A \in annihilator_mx u)%MS.
Proof.
rewrite sub_capmx; apply: (iffP andP) => [[-> /sub_kermxP]|[-> uA0]].
by rewrite mul_rV_lin1 /= mxvecK.
by split=> //; apply/sub_kermxP; rewrite mul_rV_lin1 /= mxvecK.
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 | annihilator_mxP | |
row_hom_mxu :=
(\bigcap_j kermx (vec_mx (row j (annihilator_mx u))))%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 | row_hom_mx | |
row_hom_mxPu v :
reflect (exists2 f, u <= dom_hom_mx f & u *m f = v)%MS (v <= row_hom_mx u)%MS.
Proof.
apply: (iffP sub_bigcapmxP) => [iso_uv | [f hom_uf <-] i _].
have{iso_uv} uv0 A: (A \in E_G)%MS /\ u *m A = 0 -> v *m A = 0.
move/annihilator_mxP=> /submxP[a defA].
rewrite -[A]mxvecK {A}defA [a *m _]mulmx_sum_row !linear_sum big1 // => i _.
by rewrite !linearZ /= (sub_kermxP _) ?scaler0 ?iso_uv.
pose U := E_G *m lin_mul_row u; pose V := E_G *m lin_mul_row v.
pose f := pinvmx U *m V.
have hom_uv_f x: x \in G -> u *m rG x *m f = v *m rG x.
move=> Gx; apply/eqP; rewrite 2!mulmxA mul_rV_lin1 -subr_eq0 -mulmxBr /=.
rewrite uv0 // linearB /= mulmxBr vec_mxK; split.
by rewrite addmx_sub ?submxMl // eqmx_opp envelop_mx_id.
have Uux: (u *m rG x <= U)%MS.
by rewrite -(genmxE U) mxmodule_trans ?cyclic_mx_id ?cyclic_mx_module.
by rewrite -{2}(mulmxKpV Uux) [_ *m U]mulmxA mul_rV_lin1 subrr.
have def_uf: u *m f = v.
by rewrite -[u]mulmx1 -[v]mulmx1 -(repr_mx1 rG) hom_uv_f.
by exists f => //; apply/hom_mxP=> x Gx; rewrite def_uf hom_uv_f.
apply/sub_kermxP; set A := vec_mx _.
have: (A \in annihilator_mx u)%MS by rewrite vec_mxK row_sub.
by case/annihilator_mxP => E_A uA0; rewrite -hom_envelop_mxC // uA0 mul0mx.
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_hom_mxP | |
mx_iso(U V : 'M_n) : Prop :=
MxIso f of f \in unitmx & (U <= dom_hom_mx f)%MS & (U *m f :=: V)%MS. | Variant | character | [
"From HB Require 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 | mx_iso | |
eqmx_isoU V : (U :=: V)%MS -> mx_iso U V.
Proof.
by move=> eqUV; exists 1%:M; rewrite ?unitmx1 ?scalar_mx_hom ?mulmx1.
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 | eqmx_iso | |
mx_iso_reflU : mx_iso U U.
Proof. exact: eqmx_iso. 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 | mx_iso_refl | |
mx_iso_symU V : mx_iso U V -> mx_iso V U.
Proof.
case=> f injf homUf defV; exists (invmx f); first by rewrite unitmx_inv.
by rewrite dom_hom_invmx // -defV submxMr.
by rewrite -[U](mulmxK injf); apply: eqmxMr (eqmx_sym _).
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 | mx_iso_sym | |
mx_iso_transU V W : mx_iso U V -> mx_iso V W -> mx_iso U W.
Proof.
case=> f injf homUf defV [g injg homVg defW].
exists (f *m g); first by rewrite unitmx_mul injf.
by apply/hom_mxP=> x Gx; rewrite !mulmxA 2?(hom_mxP _) ?defV.
by rewrite mulmxA; apply: eqmx_trans (eqmxMr g defV) defW.
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 | mx_iso_trans | |
mxrank_isoU V : mx_iso U V -> \rank U = \rank V.
Proof. by case=> f injf _ <-; rewrite mxrankMfree ?row_free_unit. 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 | mxrank_iso | |
mx_iso_moduleU V : mx_iso U V -> mxmodule U -> mxmodule V.
Proof.
by case=> f _ homUf defV; rewrite -(eqmx_module defV); apply: hom_mxmodule.
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 | mx_iso_module |
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