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 |
|---|---|---|---|---|---|---|
Wedderburn_sum_id:= (Wedderburn_sum_id_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use Wedderburn_is_id_pchar instead.")] | 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 | Wedderburn_sum_id | |
Wedderburn_is_id:= (Wedderburn_is_id_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use Wedderburn_closed_pchar instead.")] | 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 | Wedderburn_is_id | |
Wedderburn_closed:= (Wedderburn_closed_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use Wedderburn_is_ring_pchar instead.")] | 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 | Wedderburn_closed | |
Wedderburn_is_ring:= (Wedderburn_is_ring_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use Wedderburn_min_ideal_pchar instead.")] | 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 | Wedderburn_is_ring | |
Wedderburn_min_ideal:= (Wedderburn_min_ideal_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use rsim_irr_comp_pchar instead.")] | 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 | Wedderburn_min_ideal | |
rsim_irr_comp:= (rsim_irr_comp_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use irr_comp'_op0_pchar instead.")] | 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 | rsim_irr_comp | |
irr_comp'_op0:= (irr_comp'_op0_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use irr_comp_envelop_pchar instead.")] | 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 | irr_comp'_op0 | |
irr_comp_envelop:= (irr_comp_envelop_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use ker_irr_comp_op_pchar instead.")] | 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 | irr_comp_envelop | |
ker_irr_comp_op:= (ker_irr_comp_op_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use regular_op_inj_pchar instead.")] | 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 | ker_irr_comp_op | |
regular_op_inj:= (regular_op_inj_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use rank_irr_comp_pchar instead.")] | 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 | regular_op_inj | |
rank_irr_comp:= (rank_irr_comp_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use irr_comp_rsim_pchar instead.")] | 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 | rank_irr_comp | |
irr_comp_rsim:= (irr_comp_rsim_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use irr_reprK_pchar instead.")] | 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 | irr_comp_rsim | |
irr_reprK:= (irr_reprK_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use irr_repr'_op0_pchar instead.")] | 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 | irr_reprK | |
irr_repr'_op0:= (irr_repr'_op0_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use op_Wedderburn_id_pchar instead.")] | 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 | irr_repr'_op0 | |
op_Wedderburn_id:= (op_Wedderburn_id_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use irr_comp_id_pchar instead.")] | 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 | op_Wedderburn_id | |
irr_comp_id:= (irr_comp_id_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use rank_Wedderburn_subring_pchar instead.")] | 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 | irr_comp_id | |
rank_Wedderburn_subring:= (rank_Wedderburn_subring_pchar)
(only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use sum_irr_degree_pchar instead.")] | 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 | rank_Wedderburn_subring | |
sum_irr_degree:= (sum_irr_degree_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use irr_mx_mult_pchar instead.")] | 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 | sum_irr_degree | |
irr_mx_mult:= (irr_mx_mult_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use mxtrace_regular_pchar instead.")] | 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 | irr_mx_mult | |
mxtrace_regular:= (mxtrace_regular_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use linear_irr_comp_pchar instead.")] | 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 | mxtrace_regular | |
linear_irr_comp:= (linear_irr_comp_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use Wedderburn_subring_center_pchar instead.")] | 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 | linear_irr_comp | |
Wedderburn_subring_center:= (Wedderburn_subring_center_pchar)
(only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use Wedderburn_center_pchar instead.")] | 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 | Wedderburn_subring_center | |
Wedderburn_center:= (Wedderburn_center_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use card_irr_pchar instead.")] | 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 | Wedderburn_center | |
card_irr:= (card_irr_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use cycle_repr_structure_pchar instead.")] | 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 | card_irr | |
cycle_repr_structure:= (cycle_repr_structure_pchar) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use splitting_cyclic_primitive_root_pchar instead.")] | 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 | cycle_repr_structure | |
splitting_cyclic_primitive_root:=
(splitting_cyclic_primitive_root_pchar) (only parsing).
Arguments rfix_mx {F gT G%_g n%_N} rG H%_g.
Arguments gset_mx F {gT} G%_g A%_g.
Arguments classg_base F {gT} G%_g _%_g : extra scopes.
Arguments irrType F {gT} G%_g.
Arguments mxmoduleP {F gT G n rG m U}.
Arguments envelop_mxP {F gT G n rG A}.
Arguments hom_mxP {F gT G n rG m f W}.
Arguments mx_Maschke_pchar [F gT G n] rG _ [U].
Arguments rfix_mxP {F gT G n rG m W}.
Arguments cyclic_mxP {F gT G n rG u v}.
Arguments annihilator_mxP {F gT G n rG u A}.
Arguments row_hom_mxP {F gT G n rG u v}.
Arguments mxsimple_isoP {F gT G n rG U V}.
Arguments socle_exists [F gT G n].
Arguments socleP {F gT G n rG sG0 W W'}.
Arguments mx_abs_irrP {F gT G n rG}.
Arguments socle_rsimP {F gT G n rG sG W1 W2}.
Arguments val_submod {F n U m} W.
Arguments in_submod {F n} U {m} W.
Arguments val_submodK {F n U m} W : rename.
Arguments in_submodK {F n U m} [W] sWU.
Arguments val_submod_inj {F n U m} [W1 W2] : rename.
Arguments val_factmod {F n U m} W.
Arguments in_factmod {F n} U {m} W.
Arguments val_factmodK {F n U m} W : rename.
Arguments in_factmodK {F n} U {m} [W] sWU.
Arguments val_factmod_inj {F n U m} [W1 W2] : rename. | 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 | splitting_cyclic_primitive_root | |
mxmodule_form(U : 'M[term F]_n) :=
\big[And/True]_(x in G) submx_form (mulmx_term U (mx_term (rG x))) 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 | mxmodule_form | |
mxmodule_form_qfU : qf_form (mxmodule_form U).
Proof.
by rewrite (morphAnd (@qf_form _)) ?big1 //= => x _; rewrite submx_form_qf.
Qed. | Lemma | character | [
"From HB Require 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_form_qf | |
eval_mxmoduleU e :
qf_eval e (mxmodule_form U) = mxmodule rG (eval_mx e U).
Proof.
rewrite (morphAnd (qf_eval e)) //= big_andE /=.
apply/forallP/mxmoduleP=> Umod x; move/implyP: (Umod x);
by rewrite eval_submx eval_mulmx eval_mx_term.
Qed. | Lemma | character | [
"From HB Require 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_mxmodule | |
mxnonsimple_form(U : 'M[term F]_n) :=
let V := vec_mx (row_var F (n * n) 0) in
let nzV := (~ mxrank_form 0 V)%T in
let properVU := (submx_form V U /\ ~ submx_form U V)%T in
(Exists_row_form (n * n) 0 (mxmodule_form V /\ nzV /\ properVU))%T. | 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 | mxnonsimple_form | |
mxnonsimple_satU :=
GRing.sat (@row_env _ (n * n) [::]) (mxnonsimple_form rG (mx_term 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 | mxnonsimple_sat | |
mxnonsimplePU :
U != 0 -> reflect (mxnonsimple rG U) (mxnonsimple_sat U).
Proof.
rewrite /mxnonsimple_sat {1}/mxnonsimple_form; set Vt := vec_mx _ => /= nzU.
pose nsim V := [&& mxmodule rG V, (V <= U)%MS, V != 0 & \rank V < \rank U].
set nsimUt := (_ /\ _)%T; have: qf_form nsimUt.
by rewrite /= mxmodule_form_qf !mxrank_form_qf !submx_form_qf.
move/GRing.qf_evalP; set qev := @GRing.qf_eval _ => qevP.
have qev_nsim u: qev (row_env [:: u]) nsimUt = nsim n (vec_mx u).
rewrite /nsim -mxrank_eq0 /qev /= eval_mxmodule eval_mxrank.
rewrite !eval_submx eval_mx_term eval_vec_mx eval_row_var /=.
do 2!bool_congr; apply: andb_id2l => sUV.
by rewrite ltn_neqAle andbC !mxrank_leqif_sup.
have n2gt0: n ^ 2 > 0.
by move: nzU; rewrite muln_gt0 -mxrank_eq0 unlock; case: posnP (U) => // ->.
apply: (iffP satP) => [|[V nsimV]].
by case/Exists_rowP=> // v; move/qevP; rewrite qev_nsim; exists (vec_mx v).
apply/Exists_rowP=> //; exists (mxvec V); apply/qevP.
by rewrite qev_nsim 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 | mxnonsimpleP | |
dec_mxsimple_exists(U : 'M_n) :
mxmodule rG U -> U != 0 -> {V | mxsimple rG V & V <= U}%MS.
Proof.
have [m] := ubnP (\rank U); elim: m U => // m IHm U leUm modU nzU.
have [nsimU | simU] := mxnonsimpleP nzU; last first.
by exists U; first apply/mxsimpleP.
move: (xchooseP nsimU); move: (xchoose _) => W /and4P[modW sWU nzW ltWU].
case: (IHm W) => // [|V simV sVW]; first exact: leq_trans ltWU _.
by exists V; last apply: submx_trans sVW sWU.
Qed. | Lemma | character | [
"From HB Require 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 | dec_mxsimple_exists | |
dec_mx_reducible_semisimpleU :
mxmodule rG U -> mx_completely_reducible rG U -> mxsemisimple rG U.
Proof.
have [m] := ubnP (\rank U); elim: m U => // m IHm U leUm modU redU.
have [U0 | nzU] := eqVneq U 0.
have{} U0: (\sum_(i < 0) 0 :=: U)%MS by rewrite big_ord0 U0.
by apply: (intro_mxsemisimple U0); case.
have [V simV sVU] := dec_mxsimple_exists modU nzU; have [modV nzV _] := simV.
have [W modW defVW dxVW] := redU V modV sVU.
have [||I W_ /= simW defW _] := IHm W _ modW.
- rewrite ltnS in leUm; apply: leq_trans leUm.
by rewrite -defVW (mxdirectP dxVW) /= -add1n leq_add2r lt0n mxrank_eq0.
- by apply: mx_reducibleS redU; rewrite // -defVW addsmxSr.
suffices defU: (\sum_i oapp W_ V i :=: U)%MS.
by apply: (intro_mxsemisimple defU) => [] [|i] //=.
apply: eqmx_trans defVW; rewrite (bigD1 None) //=; apply/eqmxP.
have [i0 _ | I0] := pickP I.
by rewrite (reindex some) ?addsmxS ?defW //; exists (odflt i0) => //; case.
rewrite big_pred0 //; last by case=> // /I0.
by rewrite !addsmxS ?sub0mx // -defW big_pred0.
Qed. | Lemma | character | [
"From HB Require 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 | dec_mx_reducible_semisimple | |
DecSocleType: socleType rG.
Proof.
have [n0 | n_gt0] := posnP n.
by exists [::] => // M [_]; rewrite -mxrank_eq0 -leqn0 -n0 rank_leq_row.
have n2_gt0: n ^ 2 > 0 by rewrite muln_gt0 n_gt0.
pose span Ms := (\sum_(M <- Ms) component_mx rG M)%MS.
have: {in [::], forall M, mxsimple rG M} by [].
have [m] := ubnP (n - \rank (span [::])).
elim: m [::] => // m IHm Ms /ltnSE-Ms_ge_n simMs.
pose V := span Ms; pose Vt := mx_term V.
pose Ut i := vec_mx (row_var F (n * n) i); pose Zt := mx_term (0 : 'M[F]_n).
pose exU i f := Exists_row_form (n * n) i (~ submx_form (Ut i) Zt /\ f (Ut i)).
pose meetUVf U := exU 1 (fun W => submx_form W Vt /\ submx_form W U)%T.
pose mx_sat := GRing.sat (@row_env F (n * n) [::]).
have ev_sub0 := GRing.qf_evalP _ (submx_form_qf _ Zt).
have ev_mod := GRing.qf_evalP _ (mxmodule_form_qf rG _).
pose ev := (eval_mxmodule, eval_submx, eval_vec_mx, eval_row_var, eval_mx_term).
case haveU: (mx_sat (exU 0 (fun U => mxmodule_form rG U /\ ~ meetUVf _ U)%T)).
have [U modU]: {U : 'M_n | mxmodule rG U & (U != 0) && ((U :&: V)%MS == 0)}.
apply: sig2W; case/Exists_rowP: (satP haveU) => //= u [nzU [modU tiUV]].
exists (vec_mx u); first by move/ev_mod: modU; rewrite !ev.
set W := (_ :&: V)%MS; move/ev_sub0: nzU; rewrite !ev -!submx0 => -> /=.
apply/idPn=> nzW; case: tiUV; apply/Exists_rowP=> //; exists (mxvec W).
apply/GRing.qf_evalP; rewrite /= ?submx_form_qf // !ev mxvecK nzW /=.
by rewrite andbC -sub_capmx.
case/andP=> nzU tiUV; have [M simM sMU] := dec_mxsi
... | Lemma | character | [
"From HB Require 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 | DecSocleType | |
map_rfix_mxH : (rfix_mx rG H)^f = rfix_mx rGf H.
Proof.
rewrite map_kermx //; congr (kermx _); apply: map_lin1_mx => //= v.
rewrite map_mxvec map_mxM; congr (mxvec (_ *m _)); last first.
by apply: map_lin1_mx => //= u; rewrite map_mxM map_vec_mx.
by apply/row_matrixP=> i; rewrite -map_row !rowK map_mxvec map_mxB map_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 | map_rfix_mx | |
rcent_mapA : rcent rGf A^f = rcent rG A.
Proof.
by apply/setP=> x; rewrite !inE -!map_mxM inj_eq //; apply: map_mx_inj.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rcent_map | |
rstab_mapm (U : 'M_(m, n)) : rstab rGf U^f = rstab rG U.
Proof.
by apply/setP=> x; rewrite !inE -!map_mxM inj_eq //; apply: map_mx_inj.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstab_map | |
rstabs_mapm (U : 'M_(m, n)) : rstabs rGf U^f = rstabs rG U.
Proof. by apply/setP=> x; rewrite !inE -!map_mxM ?map_submx. Qed. | Lemma | character | [
"From HB Require 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_map | |
centgmx_mapA : centgmx rGf A^f = centgmx rG A.
Proof. by rewrite /centgmx rcent_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 | centgmx_map | |
mxmodule_mapm (U : 'M_(m, n)) : mxmodule rGf U^f = mxmodule rG U.
Proof. by rewrite /mxmodule rstabs_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 | mxmodule_map | |
mxsimple_map(U : 'M_n) : mxsimple rGf U^f -> mxsimple rG U.
Proof.
case; rewrite map_mx_eq0 // mxmodule_map // => modU nzU minU.
split=> // V modV sVU nzV; rewrite -(map_submx f).
by rewrite (minU V^f) //= ?mxmodule_map ?map_mx_eq0 // map_submx.
Qed. | Lemma | character | [
"From HB Require 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 | mxsimple_map | |
mx_irr_map: mx_irreducible rGf -> mx_irreducible rG.
Proof. by move=> irrGf; apply: mxsimple_map; rewrite map_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 | mx_irr_map | |
rker_map: rker rGf = rker rG.
Proof. by rewrite /rker -rstab_map map_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 | rker_map | |
map_mx_faithful: mx_faithful rGf = mx_faithful rG.
Proof. by rewrite /mx_faithful rker_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 | map_mx_faithful | |
map_mx_abs_irr:
mx_absolutely_irreducible rGf = mx_absolutely_irreducible rG.
Proof.
by rewrite /mx_absolutely_irreducible -map_enveloping_algebra_mx row_full_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 | map_mx_abs_irr | |
mx_rsim_mapn1 n2 rG1 rG2 :
@mx_rsim _ _ G n1 rG1 n2 rG2 -> mx_rsim (map_repr f rG1) (map_repr f rG2).
Proof.
case=> g eqn12 inj_g hom_g.
by exists g^f => // [|x Gx]; rewrite ?row_free_map // -!map_mxM ?hom_g.
Qed. | Lemma | character | [
"From HB Require 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_rsim_map | |
map_section_reprn (rG : mx_representation aF G n) rGf U V
(modU : mxmodule rG U) (modV : mxmodule rG V)
(modUf : mxmodule rGf U^f) (modVf : mxmodule rGf V^f) :
map_repr f rG =1 rGf ->
mx_rsim (map_repr f (section_repr modU modV)) (section_repr modUf modVf).
Proof.
move=> def_rGf; set VU := <<_>>%MS.
pose valUV := val_factmod (val_submod (1%:M : 'M[aF]_(\rank VU))).
have sUV_Uf: (valUV^f <= U^f + V^f)%MS.
rewrite -map_addsmx map_submx; apply: submx_trans (proj_factmodS _ _).
by rewrite val_factmodS val_submod1 genmxE.
exists (in_submod _ (in_factmod U^f valUV^f)) => [||x Gx].
- rewrite !genmxE -(mxrank_map f) map_mxM map_col_base.
by case: (\rank (cokermx U)) / (mxrank_map _ _); rewrite map_cokermx.
- rewrite -kermx_eq0 -submx0; apply/rV_subP=> u.
rewrite (sameP sub_kermxP eqP) submx0 -val_submod_eq0.
rewrite val_submodE -mulmxA -val_submodE in_submodK; last first.
by rewrite genmxE -(in_factmod_addsK _ V^f) submxMr.
rewrite in_factmodE mulmxA -in_factmodE in_factmod_eq0.
move/(submxMr (in_factmod U 1%:M *m in_submod VU 1%:M)^f).
rewrite -mulmxA -!map_mxM //; do 2!rewrite mulmxA -in_factmodE -in_submodE.
rewrite val_factmodK val_submodK map_mx1 mulmx1.
have ->: in_factmod U U = 0 by apply/eqP; rewrite in_factmod_eq0.
by rewrite linear0 map_mx0 eqmx0 submx0.
rewrite {1}in_submodE mulmxA -in_submodE -in_submodJ; last first.
by rewrite genmxE -(in_factmod_addsK _ V^f) submxMr.
congr (in_submod _ _); rewrite -in_factm
... | Lemma | character | [
"From HB Require 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 | map_section_repr | |
map_regular_subseriesU i (modU : mx_subseries (regular_repr aF G) U)
(modUf : mx_subseries (regular_repr rF G) [seq M^f | M <- U]) :
mx_rsim (map_repr f (subseries_repr i modU)) (subseries_repr i modUf).
Proof.
set mf := map _ in modUf *; rewrite /subseries_repr.
do 2!move: (mx_subseries_module' _ _) (mx_subseries_module _ _).
have mf_i V: nth 0^f (mf V) i = (V`_i)^f.
case: (ltnP i (size V)) => [ltiV | leVi]; first exact: nth_map.
by rewrite !nth_default ?size_map.
rewrite -(map_mx0 f) mf_i (mf_i (0 :: U)) => modUi'f modUif modUi' modUi.
by apply: map_section_repr; apply: map_regular_repr.
Qed. | Lemma | character | [
"From HB Require 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 | map_regular_subseries | |
extend_group_splitting_field:
group_splitting_field aF G -> group_splitting_field rF G.
Proof.
move=> splitG n rG irrG.
have modU0: all ((mxmodule (regular_repr aF G)) #|G|) [::] by [].
apply: (mx_Schreier modU0 _) => // [[U [compU lastU _]]]; have [modU _]:= compU.
pose Uf := map (map_mx f) U.
have{lastU} lastUf: (last 0 Uf :=: 1%:M)%MS.
by rewrite -(map_mx0 f) -(map_mx1 f) last_map; apply/map_eqmx.
have modUf: mx_subseries (regular_repr rF G) Uf.
rewrite /mx_subseries all_map; apply: etrans modU; apply: eq_all => Ui /=.
rewrite -mxmodule_map; apply: eq_subset_r => x.
by rewrite !inE map_regular_repr.
have absUf i: i < size U -> mx_absolutely_irreducible (subseries_repr i modUf).
move=> lt_i_U; rewrite -(mx_rsim_abs_irr (map_regular_subseries i modU _)).
rewrite map_mx_abs_irr; apply: splitG.
by apply: mx_rsim_irr (mx_series_repr_irr compU lt_i_U); apply: section_eqmx.
have compUf: mx_composition_series (regular_repr rF G) Uf.
split=> // i; rewrite size_map => ltiU.
move/max_submodP: (mx_abs_irrW (absUf i ltiU)); apply.
rewrite -{2}(map_mx0 f) -map_cons !(nth_map 0) ?leqW //.
by rewrite map_submx // ltmxW // (pathP _ (mx_series_lt compU)).
have [[i ltiU] simUi] := rsim_regular_series irrG compUf lastUf.
have{} simUi: mx_rsim rG (subseries_repr i modUf).
by apply: mx_rsim_trans simUi _; apply: section_eqmx.
by rewrite (mx_rsim_abs_irr simUi) absUf; rewrite size_map in ltiU.
Qed. | Lemma | character | [
"From HB Require 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 | extend_group_splitting_field | |
gen_of{F : fieldType} {gT : finGroupType} {G : {group gT}} {n' : nat}
{rG : mx_representation F G n'.+1} {A : 'M[F]_n'.+1}
(irrG : mx_irreducible rG) (cGA : centgmx rG A) :=
Gen {rVval : 'rV[F]_(degree_mxminpoly A)}.
Local Arguments rVval {F gT G%_G n'%_N rG A%_R irrG cGA} x%_R : rename.
Bind Scope ring_scope with gen_of. | Record | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_of | |
FA:= (gen_of irrG cGA).
Let inFA := Gen irrG cGA.
#[export, hnf] HB.instance Definition _ := [isNew for rVval : FA -> 'rV_d].
#[export] HB.instance Definition _ := [Choice of FA by <:]. | 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 | |
gen0:= inFA 0. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen0 | |
genN(x : FA) := inFA (- val 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 | genN | |
genD(x y : FA) := inFA (val x + val y). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | genD | |
gen_addA: associative genD.
Proof. by move=> x y z; apply: val_inj; rewrite /= addrA. Qed. | Lemma | character | [
"From HB Require 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_addA | |
gen_addC: commutative genD.
Proof. by move=> x y; apply: val_inj; rewrite /= addrC. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_addC | |
gen_add0r: left_id gen0 genD.
Proof. by move=> x; apply: val_inj; rewrite /= add0r. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_add0r | |
gen_addNr: left_inverse gen0 genN genD.
Proof. by move=> x; apply: val_inj; rewrite /= addNr. Qed.
#[export] HB.instance Definition _ := GRing.isZmodule.Build FA
gen_addA gen_addC gen_add0r gen_addNr. | Lemma | character | [
"From HB Require 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_addNr | |
pval(x : FA) := rVpoly (val 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 | pval | |
mxval(x : FA) := horner_mx A (pval 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 | mxval | |
gen(x : F) := inFA (poly_rV x%:P). | 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 | |
genKx : mxval (gen x) = x%:M.
Proof.
by rewrite /mxval [pval _]poly_rV_K ?horner_mx_C // size_polyC; case: (x != 0).
Qed. | Lemma | character | [
"From HB Require 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 | genK | |
mxval_inj: injective mxval.
Proof. exact: inj_comp horner_rVpoly_inj val_inj. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxval_inj | |
mxval0: mxval 0 = 0.
Proof. by rewrite /mxval [pval _]raddf0 rmorph0. Qed. | Lemma | character | [
"From HB Require 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 | mxval0 | |
mxvalN: {morph mxval : x / - x}.
Proof. by move=> x; rewrite /mxval [pval _](@raddfN 'rV[F]_d) rmorphN. Qed. | Lemma | character | [
"From HB Require 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 | mxvalN | |
mxvalD: {morph mxval : x y / x + y}.
Proof. by move=> x y; rewrite /mxval [pval _]raddfD rmorphD. Qed. | Lemma | character | [
"From HB Require 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 | mxvalD | |
mxval_sum:= big_morph mxval mxvalD mxval0. | 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 | mxval_sum | |
gen1:= inFA (poly_rV 1). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen1 | |
genMx y := inFA (poly_rV (pval x * pval y %% pA)). | 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 | genM | |
genVx := inFA (poly_rV (mx_inv_horner A (mxval x)^-1)). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | genV | |
mxval_gen1: mxval gen1 = 1%:M.
Proof. by rewrite /mxval [pval _]poly_rV_K ?size_poly1 // horner_mx_C. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxval_gen1 | |
mxval_genM: {morph mxval : x y / genM x y >-> x *m y}.
Proof.
move=> x y; rewrite /mxval [pval _]poly_rV_K ?size_mod_mxminpoly //.
by rewrite -horner_mxK mx_inv_hornerK ?horner_mx_mem // rmorphM.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxval_genM | |
mxval_genV: {morph mxval : x / genV x >-> invmx x}.
Proof.
move=> x; rewrite /mxval [pval _]poly_rV_K ?size_poly ?mx_inv_hornerK //.
pose m B : 'M[F]_(n * n) := lin_mx (mulmxr B); set B := mxval x.
case uB: (B \is a GRing.unit); last by rewrite invr_out ?uB ?horner_mx_mem.
have defAd: Ad = Ad *m m B *m m B^-1.
apply/row_matrixP=> i.
by rewrite !row_mul mul_rV_lin /= mx_rV_lin /= mulmxK ?vec_mxK.
rewrite -[B^-1]mul1mx -(mul_vec_lin (mulmxr B^-1)) defAd submxMr //.
rewrite -mxval_gen1 (submx_trans (horner_mx_mem _ _)) // {1}defAd.
rewrite -(geq_leqif (mxrank_leqif_sup _)) ?mxrankM_maxl // -{}defAd.
apply/row_subP=> i; rewrite row_mul rowK mul_vec_lin /= -{2}[A]horner_mx_X.
by rewrite -rmorphXn mulmxE -rmorphM horner_mx_mem.
Qed. | Lemma | character | [
"From HB Require 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 | mxval_genV | |
gen_mulA: associative genM.
Proof. by move=> x y z; apply: mxval_inj; rewrite !mxval_genM 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 | gen_mulA | |
gen_mulC: commutative genM.
Proof. by move=> x y; rewrite /genM mulrC. Qed. | Lemma | character | [
"From HB Require 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_mulC | |
gen_mul1r: left_id gen1 genM.
Proof. by move=> x; apply: mxval_inj; rewrite mxval_genM mxval_gen1 mul1mx. Qed. | Lemma | character | [
"From HB Require 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_mul1r | |
gen_mulDr: left_distributive genM +%R.
Proof.
by move=> x y z; apply: mxval_inj; rewrite !(mxvalD, mxval_genM) mulmxDl.
Qed. | Lemma | character | [
"From HB Require 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_mulDr | |
gen_ntriv: gen1 != 0.
Proof. by rewrite -(inj_eq mxval_inj) mxval_gen1 mxval0 oner_eq0. Qed.
#[export] HB.instance Definition _ := GRing.Zmodule_isComNzRing.Build FA
gen_mulA gen_mulC gen_mul1r gen_mulDr gen_ntriv. | Lemma | character | [
"From HB Require 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_ntriv | |
mxval1: mxval 1 = 1%:M. Proof. exact: mxval_gen1. Qed. | Lemma | character | [
"From HB Require 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 | mxval1 | |
mxvalM: {morph mxval : x y / x * y >-> x *m y}.
Proof. exact: mxval_genM. Qed. | Lemma | character | [
"From HB Require 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 | mxvalM | |
mxval_is_zmod_morphism: zmod_morphism mxval.
Proof. by move=> x y; rewrite mxvalD mxvalN. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `mxval_is_zmod_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxval_is_zmod_morphism | |
mxval_sub:= mxval_is_zmod_morphism.
#[export] HB.instance Definition _ :=
GRing.isZmodMorphism.Build FA 'M[F]_n mxval mxval_is_zmod_morphism. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxval_sub | |
mxval_is_monoid_morphism: monoid_morphism mxval.
Proof. by split; [apply: mxval1 | apply: mxvalM]. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `mxval_is_monoid_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxval_is_monoid_morphism | |
mxval_is_multiplicative:=
(fun g => (g.2,g.1)) mxval_is_monoid_morphism.
#[export] HB.instance Definition _ :=
GRing.isMonoidMorphism.Build FA 'M[F]_n mxval mxval_is_monoid_morphism. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxval_is_multiplicative | |
mxval_centgx : centgmx rG (mxval x).
Proof.
rewrite [mxval _]horner_rVpoly -memmx_cent_envelop vec_mxK {x}mulmx_sub //.
apply/row_subP=> k; rewrite rowK memmx_cent_envelop; apply/centgmxP => g Gg /=.
by rewrite !mulmxE commrX // /GRing.comm -mulmxE (centgmxP cGA).
Qed. | Lemma | character | [
"From HB Require 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 | mxval_centg | |
gen_mulVrx : x != 0 -> genV x * x = 1.
Proof.
rewrite -(inj_eq mxval_inj) mxval0.
move/(mx_Schur irrG (mxval_centg x)) => u_x.
by apply: mxval_inj; rewrite mxvalM mxval_genV mxval1 mulVmx.
Qed. | Lemma | character | [
"From HB Require 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_mulVr | |
gen_invr0: genV 0 = 0.
Proof. by apply: mxval_inj; rewrite mxval_genV !mxval0 -{2}invr0. Qed.
#[export] HB.instance Definition _ := GRing.ComNzRing_isField.Build FA
gen_mulVr gen_invr0. | Lemma | character | [
"From HB Require 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_invr0 | |
mxvalV: {morph mxval : x / x^-1 >-> invmx x}.
Proof. exact: mxval_genV. Qed. | Lemma | character | [
"From HB Require 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 | mxvalV | |
gen_is_zmod_morphism: zmod_morphism gen.
Proof. by move=> x y; apply: mxval_inj; rewrite genK !rmorphB /= !genK. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `gen_is_zmod_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_is_zmod_morphism | |
gen_is_additive:= gen_is_zmod_morphism. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_is_additive | |
gen_is_monoid_morphism: monoid_morphism gen.
Proof. by split=> // x y; apply: mxval_inj; rewrite genK !rmorphM /= !genK. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `gen_is_monoid_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_is_monoid_morphism | |
gen_is_multiplicative:=
(fun g => (g.2,g.1)) gen_is_monoid_morphism.
#[export] HB.instance Definition _ := GRing.isZmodMorphism.Build F FA gen
gen_is_zmod_morphism.
#[export] HB.instance Definition _ := GRing.isMonoidMorphism.Build F FA gen
gen_is_monoid_morphism. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | gen_is_multiplicative | |
groot:= inFA (poly_rV ('X %% pA)). | 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 | groot | |
mxval_groot: mxval groot = A.
Proof.
rewrite /mxval [pval _]poly_rV_K ?size_mod_mxminpoly // -horner_mxK.
by rewrite mx_inv_hornerK ?horner_mx_mem // horner_mx_X.
Qed. | Lemma | character | [
"From HB Require 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 | mxval_groot | |
mxval_grootXnk : mxval (groot ^+ k) = A ^+ k.
Proof. by rewrite rmorphXn /= mxval_groot. Qed. | Lemma | character | [
"From HB Require 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 | mxval_grootXn | |
map_mxminpoly_groot: (map_poly gen pA).[groot] = 0.
Proof.
apply: mxval_inj; rewrite -horner_map [_ groot]/= mxval_groot mxval0.
rewrite -(mx_root_minpoly A); congr ((_ : {poly _}).[A]).
by apply/polyP=> i; rewrite 3!coef_map; apply: 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 | map_mxminpoly_groot | |
non_linear_gen_reducible: d > 1 -> mxnonsimple (map_repr gen rG) 1%:M.
Proof.
rewrite ltnNge mxminpoly_linear_is_scalar => Anscal.
pose Af := map_mx gen A; exists (kermx (Af - groot%:M)).
rewrite submx1 kermx_centg_module /=; last first.
apply/centgmxP=> z Gz; rewrite mulmxBl [RHS]mulmxBr [in RHS]scalar_mxC.
by rewrite -!map_mxM 1?(centgmxP cGA).
rewrite andbC mxrank_ker -subn_gt0 mxrank1 subKn ?rank_leq_row // lt0n.
rewrite mxrank_eq0 subr_eq0; case: eqP => [defAf | _].
rewrite -(map_mx_is_scalar gen) -/Af in Anscal.
by case/is_scalar_mxP: Anscal; exists groot.
rewrite -mxrank_eq0 mxrank_ker subn_eq0 row_leq_rank.
apply/row_freeP=> [[XA' XAK]].
have pAf0: (mxminpoly Af).[groot] == 0.
by rewrite mxminpoly_map ?map_mxminpoly_groot.
have{pAf0} [q def_pAf]:= factor_theorem _ _ pAf0.
have q_nz: q != 0.
case: eqP (congr1 (fun p : {poly _} => size p) def_pAf) => // ->.
by rewrite size_mxminpoly mul0r size_poly0.
have qAf0: horner_mx Af q = 0.
rewrite -[_ q]mulr1 -[1]XAK mulrA -{2}(horner_mx_X Af) -(horner_mx_C Af).
by rewrite -rmorphB -rmorphM -def_pAf /= mx_root_minpoly mul0r.
have{qAf0} := dvdp_leq q_nz (mxminpoly_min qAf0); rewrite def_pAf.
by rewrite size_Mmonic ?monicXsubC // polyseqXsubC addn2 ltnn.
Qed. | Lemma | character | [
"From HB Require 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 | non_linear_gen_reducible | |
subbasenA (B : 'rV_nA) : 'M_(nA * d, n) :=
\matrix_ik mxvec (\matrix_(i, k) (row (B 0 i) (A ^+ k))) 0 ik. | 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 | subbase | |
gen_dim_ex_proof: exists nA, [exists B : 'rV_nA, row_free (subbase B)].
Proof. by exists 0; apply/existsP; exists 0; rewrite /row_free unlock. Qed. | Lemma | character | [
"From HB Require 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_ex_proof |
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