statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
component_mx_iso V : mxsimple V -> (V <= compU)%MS -> mx_iso U V. | Proof.
move=> simV; rewrite -[compU]mulmx1.
exact: hom_component_mx_iso (scalar_mx_hom _ _).
Qed. | Lemma | component_mx_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"compU",
"hom_component_mx_iso",
"mulmx1",
"mx_iso",
"mxsimple",
"scalar_mx_hom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hom_component_mx f :
(compU <= dom_hom_mx f)%MS -> (compU *m f <= compU)%MS. | Proof.
move=> hom_f.
have [I W /= simW defW _] := hom_mxsemisimple component_mx_semisimple hom_f.
rewrite -defW; apply/sumsmx_subP=> i _; apply: mx_iso_component.
by apply: hom_component_mx_iso hom_f _ => //; rewrite -defW (sumsmx_sup i).
Qed. | Lemma | hom_component_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"compU",
"component_mx_semisimple",
"dom_hom_mx",
"hom_component_mx_iso",
"hom_mxsemisimple",
"mx_iso_component",
"sumsmx_subP",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_isoP U V :
mxsimple U -> mxsimple V ->
reflect (mx_iso U V) (component_mx U == component_mx V). | Proof.
move=> simU simV; apply: (iffP eqP) => isoUV.
by apply: component_mx_iso; rewrite ?isoUV ?component_mx_id.
rewrite -(genmx_component U) -(genmx_component V); apply/genmxP.
wlog suffices: U V simU simV isoUV / (component_mx U <= component_mx V)%MS.
by move=> IH; rewrite !IH //; apply: mx_iso_sym.
have [I [W i... | Lemma | component_mx_isoP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx",
"component_mx_def",
"component_mx_id",
"component_mx_iso",
"genmxP",
"genmx_component",
"mx_iso",
"mx_iso_component",
"mx_iso_sym",
"mx_iso_trans",
"mxsimple",
"simU",
"sumsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_disjoint U V :
mxsimple U -> mxsimple V -> component_mx U != component_mx V ->
(component_mx U :&: component_mx V = 0)%MS. | Proof.
move=> simU simV neUV; apply: contraNeq neUV => ntUV.
apply: (mxsimple_exists _ ntUV) => [|[W simW]].
by rewrite capmx_module ?component_mx_module.
rewrite sub_capmx => /andP[sWU sWV]; apply/component_mx_isoP=> //.
by apply: mx_iso_trans (_ : mx_iso U W) (mx_iso_sym _); apply: component_mx_iso.
Qed. | Lemma | component_mx_disjoint | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"capmx_module",
"component_mx",
"component_mx_iso",
"component_mx_isoP",
"component_mx_module",
"contraNeq",
"mx_iso",
"mx_iso_sym",
"mx_iso_trans",
"mxsimple",
"mxsimple_exists",
"simU",
"sub_capmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socleType | := EnumSocle {
socle_base_enum : seq 'M[F]_n;
_ : forall M, M \in socle_base_enum -> mxsimple M;
_ : forall M, mxsimple M -> has (mxsimple_iso M) socle_base_enum
}. | Record | socleType | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"has",
"mxsimple",
"mxsimple_iso",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_exists : classically socleType. | Proof.
pose V : 'M[F]_n := 0; have: mxsemisimple V by apply: mxsemisimple0.
have: n - \rank V < n.+1 by rewrite mxrank0 subn0.
elim: _.+1 V => // n' IHn' V; rewrite ltnS => le_nV_n' ssimV.
case=> // maxV; apply: (maxV); have [I /= U simU defV _] := ssimV.
exists (codom U) => [M | M simM]; first by case/mapP=> i _ ->.
s... | Lemma | socle_exists | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addsmxSr",
"addsmx_semisimple",
"addsmx_sub",
"apply",
"codom",
"codom_f",
"hasP",
"hom_mxsemisimple_iso",
"leq_trans",
"ltnS",
"ltn_leqif",
"ltn_neqAle",
"ltn_sub2l",
"mapP",
"mulmx1",
"mx_iso_sym",
"mxrank0",
"mxrank_leqif_sup",
"mxsemisimple",
"mxsemisimple0",
"mxsimple_i... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_enum | := map component_mx (socle_base_enum sG0). | Definition | socle_enum | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"component_mx",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_socle M : mxsimple M -> component_mx M \in socle_enum. | Proof.
rewrite /socle_enum; case: sG0 => e0 /= sim_e mem_e simM.
have /hasP[M' e0M' isoMM'] := mem_e M simM; apply/mapP; exists M' => //.
by apply/eqP/component_mx_isoP; [|apply: sim_e | apply/mxsimple_isoP].
Qed. | Lemma | component_socle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx",
"component_mx_isoP",
"e0",
"hasP",
"mapP",
"mxsimple",
"mxsimple_isoP",
"socle_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_sort : predArgType | := PackSocle W of W \in socle_enum. | Inductive | socle_sort | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"socle_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sG | := socle_sort. | Notation | sG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"socle_sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
e0 | := (socle_base_enum sG0). | Notation | e0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_base W | := let: PackSocle W _ := W in e0`_(index W socle_enum). | Definition | socle_base | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"e0",
"index",
"socle_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_val W : 'M[F]_n | := component_mx (socle_base W). | Coercion | socle_val | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"component_mx",
"socle_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_mult (W : sG) | := (\rank W %/ \rank (socle_base W))%N. | Definition | socle_mult | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rank",
"sG",
"socle_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_simple W : mxsimple (socle_base W). | Proof.
case: W => M /=; rewrite /= /socle_enum /=; case: sG0 => e sim_e _ /= e_M.
by apply: sim_e; rewrite mem_nth // -(size_map component_mx) index_mem.
Qed. | Lemma | socle_simple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx",
"index_mem",
"mem_nth",
"mxsimple",
"size_map",
"socle_base",
"socle_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_module (W : sG) | := mxsimple_module (socle_simple W). | Definition | socle_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxsimple_module",
"sG",
"socle_simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_repr W | := submod_repr (socle_module W). | Definition | socle_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"socle_module",
"submod_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz_socle (W : sG) : W != 0 :> 'M_n. | Proof.
have simW := socle_simple W; have [_ nzW _] := simW; apply: contra nzW.
by rewrite -!submx0; apply: submx_trans (component_mx_id simW).
Qed. | Lemma | nz_socle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx_id",
"sG",
"socle_simple",
"submx0",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_mem (W : sG) : (W : 'M_n) \in socle_enum. | Proof. exact: component_socle (socle_simple _). Qed. | Lemma | socle_mem | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"component_socle",
"sG",
"socle_enum",
"socle_simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PackSocleK W e0W : @PackSocle W e0W = W :> 'M_n. | Proof.
rewrite /socle_val /= in e0W *; rewrite -(nth_map _ 0) ?nth_index //.
by rewrite -(size_map component_mx) index_mem.
Qed. | Lemma | PackSocleK | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"component_mx",
"index_mem",
"nth_index",
"nth_map",
"size_map",
"socle_val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socleP (W W' : sG) : reflect (W = W') (W == W')%MS. | Proof. by rewrite (sameP genmxP eqP) !{1}genmx_component; apply: (W =P _). Qed. | Lemma | socleP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"genmxP",
"genmx_component",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_can_subproof :
cancel (fun W => SeqSub (socle_mem W)) (fun s => PackSocle (valP s)). | Proof. by move=> W /=; apply: val_inj; rewrite /= PackSocleK. Qed. | Fact | socle_can_subproof | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"PackSocleK",
"apply",
"socle_mem",
"valP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_sort : socleType >-> predArgType. | Coercion | socle_sort | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"socleType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
S | := (\sum_(W : sG | P W) socle_val W)%MS. | Notation | S | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"sG",
"socle_val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSocle_module : mxmodule S. | Proof. by rewrite sumsmx_module // => W _; apply: component_mx_module. Qed. | Lemma | subSocle_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx_module",
"mxmodule",
"sumsmx_module"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSocle_semisimple : mxsemisimple S. | Proof.
apply: sumsmx_semisimple => W _; apply: component_mx_semisimple.
exact: socle_simple.
Qed. | Lemma | subSocle_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx_semisimple",
"mxsemisimple",
"socle_simple",
"sumsmx_semisimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ssimS | := subSocle_semisimple. | Notation | ssimS | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"subSocle_semisimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSocle_iso M :
mxsimple M -> (M <= S)%MS -> {W : sG | P W & mx_iso (socle_base W) M}. | Proof.
move=> simM sMS; have [modM nzM _] := simM.
have [V /= modV defMV] := mxsemisimple_reducible ssimS modM sMS.
move/mxdirect_addsP=> dxMV; pose p := proj_mx M V; pose Sp (W : sG) := W *m p.
case: (pickP [pred i | P i & Sp i != 0]) => [/= W | Sp0]; last first.
case/negP: nzM; rewrite -submx0 -[M](proj_mx_id dxMV)... | Lemma | subSocle_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"big1",
"component_mx_module",
"dom_hom_mx",
"hom_component_mx_iso",
"last",
"mx_Schur_onto",
"mx_iso",
"mxdirect_addsP",
"mxsemisimple_reducible",
"mxsimple",
"pickP",
"proj_mx",
"proj_mx_hom",
"proj_mx_id",
"proj_mx_sub",
"sG",
"socle_base",
"socle_simple",
"ssimS",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_subSocle m (M : 'M_(m, n)) :
mxmodule M -> (M :&: S :=: \sum_(W : sG | P W) (M :&: W))%MS. | Proof.
move=> modM; apply/eqmxP/andP; split; last first.
by apply/sumsmx_subP=> W P_W; rewrite capmxS // (sumsmx_sup W).
have modMS: mxmodule (M :&: S)%MS by rewrite capmx_module ?subSocle_module.
have [J /= U simU defMS _] := mxsemisimpleS modMS (capmxSr M S) ssimS.
rewrite -defMS; apply/sumsmx_subP=> j _.
have [sUj... | Lemma | capmx_subSocle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"capmxS",
"capmxSr",
"capmx_module",
"eqmxP",
"last",
"mx_iso_component",
"mxmodule",
"mxsemisimpleS",
"sG",
"simU",
"socle_simple",
"split",
"ssimS",
"subSocle_iso",
"subSocle_module",
"sub_capmx",
"sumsmx_subP",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSocle_direct P : mxdirect (\sum_(W : sG | P W) W). | Proof.
apply/mxdirect_sumsP=> W _; apply/eqP.
rewrite -submx0 capmx_subSocle ?component_mx_module //.
apply/sumsmx_subP=> W' /andP[_ neWW'].
by rewrite capmxC component_mx_disjoint //; apply: socle_simple.
Qed. | Lemma | subSocle_direct | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"capmxC",
"capmx_subSocle",
"component_mx_disjoint",
"component_mx_module",
"mxdirect",
"mxdirect_sumsP",
"sG",
"socle_simple",
"submx0",
"sumsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Socle | := (\sum_(W : sG) W)%MS. | Definition | Socle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simple_Socle M : mxsimple M -> (M <= Socle)%MS. | Proof.
move=> simM; have socM := component_socle sG simM.
by rewrite (sumsmx_sup (PackSocle socM)) // PackSocleK component_mx_id.
Qed. | Lemma | simple_Socle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"PackSocleK",
"Socle",
"component_mx_id",
"component_socle",
"mxsimple",
"sG",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
semisimple_Socle U : mxsemisimple U -> (U <= Socle)%MS. | Proof.
by case=> I M /= simM <- _; apply/sumsmx_subP=> i _; apply: simple_Socle.
Qed. | Lemma | semisimple_Socle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle",
"apply",
"mxsemisimple",
"simple_Socle",
"sumsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reducible_Socle U :
mxmodule U -> mx_completely_reducible U -> (U <= Socle)%MS. | Proof.
move=> modU redU; apply: (mx_reducible_semisimple modU redU).
exact: semisimple_Socle.
Qed. | Lemma | reducible_Socle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle",
"apply",
"mx_completely_reducible",
"mx_reducible_semisimple",
"mxmodule",
"semisimple_Socle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_Socle : <<Socle>>%MS = Socle. | Proof. by rewrite genmx_sums; apply: eq_bigr => W; rewrite genmx_component. Qed. | Lemma | genmx_Socle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle",
"apply",
"eq_bigr",
"genmx_component",
"genmx_sums"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reducible_Socle1 : mx_completely_reducible 1%:M -> Socle = 1%:M. | Proof.
move=> redG; rewrite -genmx1 -genmx_Socle; apply/genmxP.
by rewrite submx1 reducible_Socle ?mxmodule1.
Qed. | Lemma | reducible_Socle1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle",
"apply",
"genmx1",
"genmxP",
"genmx_Socle",
"mx_completely_reducible",
"mxmodule1",
"reducible_Socle",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Socle_module : mxmodule Socle. | Proof. exact: subSocle_module. Qed. | Lemma | Socle_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle",
"mxmodule",
"subSocle_module"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Socle_semisimple : mxsemisimple Socle. | Proof. exact: subSocle_semisimple. Qed. | Lemma | Socle_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle",
"mxsemisimple",
"subSocle_semisimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Socle_direct : mxdirect Socle. | Proof. exact: subSocle_direct. Qed. | Lemma | Socle_direct | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle",
"mxdirect",
"subSocle_direct"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Socle_iso M : mxsimple M -> {W : sG | mx_iso (socle_base W) M}. | Proof.
by move=> simM; case/subSocle_iso: (simple_Socle simM) => // W _; exists W.
Qed. | Lemma | Socle_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_iso",
"mxsimple",
"sG",
"simple_Socle",
"socle_base",
"subSocle_iso"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_full_dom_hom : row_full (dom_hom_mx f) = centgmx rG f. | Proof.
by rewrite -sub1mx; apply/hom_mxP/centgmxP=> cfG x /cfG; rewrite !mul1mx.
Qed. | Lemma | row_full_dom_hom | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"centgmx",
"centgmxP",
"dom_hom_mx",
"hom_mxP",
"mul1mx",
"rG",
"row_full",
"sub1mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memmx_cent_envelop : (f \in 'C(E_G))%MS = centgmx rG f. | Proof.
apply/cent_rowP/centgmxP=> [cfG x Gx | cfG i].
by have:= cfG (enum_rank_in Gx x); rewrite rowK mxvecK enum_rankK_in.
by rewrite rowK mxvecK /= cfG ?enum_valP.
Qed. | Lemma | memmx_cent_envelop | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"apply",
"cent_rowP",
"centgmx",
"centgmxP",
"enum_rankK_in",
"enum_rank_in",
"enum_valP",
"mxvecK",
"rG",
"rowK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermx_centg_module : centgmx rG f -> mxmodule (kermx f). | Proof.
move/centgmxP=> cGf; apply/mxmoduleP=> x Gx; apply/sub_kermxP.
by rewrite -mulmxA -cGf // mulmxA mulmx_ker mul0mx.
Qed. | Lemma | kermx_centg_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"centgmx",
"centgmxP",
"kermx",
"mul0mx",
"mulmxA",
"mulmx_ker",
"mxmodule",
"mxmoduleP",
"rG",
"sub_kermxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
centgmx_hom m (U : 'M_(m, n)) : centgmx rG f -> (U <= dom_hom_mx f)%MS. | Proof. by rewrite -row_full_dom_hom -sub1mx; apply: submx_trans (submx1 _). Qed. | Lemma | centgmx_hom | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"centgmx",
"dom_hom_mx",
"rG",
"row_full_dom_hom",
"sub1mx",
"submx1",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_irreducible | := mxsimple 1%:M. | Definition | mx_irreducible | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxsimple"
] | that unlike "reducible", "absolutely irreducible" can easily be decided. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mx_irrP :
mx_irreducible <-> n > 0 /\ (forall U, @mxmodule n U -> U != 0 -> row_full U). | Proof.
rewrite /mx_irreducible /mxsimple mxmodule1 -mxrank_eq0 mxrank1 -lt0n.
do [split=> [[_ -> irrG] | [-> irrG]]; split=> // U] => [modU | modU _] nzU.
by rewrite -sub1mx (irrG U) ?submx1.
by rewrite sub1mx irrG.
Qed. | Lemma | mx_irrP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irrG",
"lt0n",
"mx_irreducible",
"mxmodule",
"mxmodule1",
"mxrank1",
"mxrank_eq0",
"mxsimple",
"row_full",
"split",
"sub1mx",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_Schur :
mx_irreducible -> forall f, centgmx rG f -> f != 0 -> f \in unitmx. | Proof.
move/mx_Schur_onto=> irrG f.
rewrite -row_full_dom_hom -!row_full_unit -!sub1mx => cGf nz.
by rewrite -[f]mul1mx irrG ?submx1 ?mxmodule1 ?mul1mx.
Qed. | Lemma | mx_Schur | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"centgmx",
"irrG",
"mul1mx",
"mx_Schur_onto",
"mx_irreducible",
"mxmodule1",
"rG",
"row_full_dom_hom",
"row_full_unit",
"sub1mx",
"submx1",
"unitmx"
] | Schur's lemma for endomorphisms. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mx_absolutely_irreducible | := (n > 0) && row_full E_G. | Definition | mx_absolutely_irreducible | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_abs_irrP :
reflect (n > 0 /\ exists a_, forall A, A = \sum_(x in G) a_ x A *: rG x)
mx_absolutely_irreducible. | 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.
rewrite /mx_absolutely_irreducible; case: (n > 0); last by right; case.
apply: (iffP row_fullP) => [[E' E'G] | [_ [a_ a_G]]].
split=> //; exists (fun x B => (mxvec B *m E') 0 (enum_rank_i... | Lemma | mx_abs_irrP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"enum_rank_in",
"enum_val",
"enum_valK_in",
"enum_valP",
"enum_val_bij_in",
"eq_big",
"group1",
"last",
"linearZ",
"linear_sum",
"mulmx1",
"mulmxA",
"mxE",
"mx_absolutely_irreducible",
"mxvec",
"mxvecK",
"rG",
"reindex",
"row",
"rowP",
"row_fullP",
"row_matrixP",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_abs_irr_cent_scalar :
mx_absolutely_irreducible -> forall A, centgmx rG A -> is_scalar_mx A. | Proof.
case/mx_abs_irrP=> n_gt0 [a_ a_G] A /centgmxP cGA.
have{cGA a_G} cMA B: A *m B = B *m A.
rewrite {}[B]a_G mulmx_suml mulmx_sumr.
by apply: eq_bigr => x Gx; rewrite -scalemxAl -scalemxAr cGA.
pose i0 := Ordinal n_gt0; apply/is_scalar_mxP; exists (A i0 i0).
apply/matrixP=> i j; move/matrixP/(_ i0 j): (esym (cM... | Lemma | mx_abs_irr_cent_scalar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
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"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"big_ord1",
"centgmx",
"centgmxP",
"delta_mx",
"eq_bigr",
"eq_sym",
"eqxx",
"i0",
"is_scalar_mx",
"is_scalar_mxP",
"matrixP",
"mul_delta_mx",
"mulmxA",
"mulmx_suml",
"mulmx_sumr",
"mulr_natl",
"mxE",
"mx_abs_irrP",
"mx_absolutely_irreducible",
"n_gt0",
"rG",
"row... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_abs_irrW : mx_absolutely_irreducible -> mx_irreducible. | Proof.
case/mx_abs_irrP=> n_gt0 [a_ a_G]; apply/mx_irrP; split=> // U Umod.
case/rowV0Pn=> u Uu; rewrite -mxrank_eq0 -lt0n row_leq_rank -sub1mx.
case/submxP: Uu => v ->{u} /row_freeP[u' vK]; apply/row_subP=> i.
rewrite rowE scalar_mxC -{}vK -2![_ *m _]mulmxA; move: {u' i}(u' *m _) => A.
rewrite mulmx_sub {v}// [A]a_G l... | Lemma | mx_abs_irrW | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
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"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"Uu",
"apply",
"linear_sum",
"lt0n",
"mulmxA",
"mulmx_sub",
"mx_abs_irrP",
"mx_absolutely_irreducible",
"mx_irrP",
"mx_irreducible",
"mxmoduleP",
"mxrank_eq0",
"n_gt0",
"rowE",
"rowV0Pn",
"row_freeP",
"row_leq_rank",
"row_subP",
"scalar_mxC",
"scalemxAr",
"scalemx_s... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_mx_abs_irr : n = 1 -> mx_absolutely_irreducible. | Proof.
move=> n1; rewrite /mx_absolutely_irreducible /row_full eqn_leq rank_leq_col.
rewrite {1 2 3}n1 /= lt0n mxrank_eq0; apply: contraTneq envelop_mx1 => ->.
by rewrite eqmx0 submx0 mxvec_eq0 -mxrank_eq0 mxrank1 n1.
Qed. | Lemma | linear_mx_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"contraTneq",
"envelop_mx1",
"eqmx0",
"eqn_leq",
"lt0n",
"mx_absolutely_irreducible",
"mxrank1",
"mxrank_eq0",
"mxvec_eq0",
"rank_leq_col",
"row_full",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelian_abs_irr : abelian G -> mx_absolutely_irreducible = (n == 1). | Proof.
move=> cGG; apply/idP/eqP=> [absG|]; last exact: linear_mx_abs_irr.
have [n_gt0 _] := andP absG.
pose M := <<delta_mx 0 (Ordinal n_gt0) : 'rV[F]_n>>%MS.
have rM: \rank M = 1 by rewrite genmxE mxrank_delta.
suffices defM: (M == 1%:M)%MS by rewrite (eqmxP defM) mxrank1 in rM.
case: (mx_abs_irrW absG) => _ _ ->; re... | Lemma | abelian_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"abelian",
"apply",
"cGG",
"centgmxP",
"centsP",
"delta_mx",
"eqmxP",
"genmxE",
"is_scalar_mx",
"is_scalar_mxP",
"last",
"linear_mx_abs_irr",
"mul_mx_scalar",
"mx_abs_irrW",
"mx_abs_irr_cent_scalar",
"mx_absolutely_irreducible",
"mxmoduleP",
"mxrank1",
"mxrank_delta",
"mxrank_e... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
envelop_mx_ring : mxring (enveloping_algebra_mx rG). | Proof.
apply/andP; split; first by apply/mulsmx_subP; apply: envelop_mxM.
apply/mxring_idP; exists 1%:M; split=> *; rewrite ?mulmx1 ?mul1mx //.
by rewrite -mxrank_eq0 mxrank1.
exact: envelop_mx1.
Qed. | Lemma | envelop_mx_ring | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"envelop_mx1",
"envelop_mxM",
"enveloping_algebra_mx",
"mul1mx",
"mulmx1",
"mulsmx_subP",
"mxrank1",
"mxrank_eq0",
"mxring",
"mxring_idP",
"rG",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irrG : mx_irreducible rG. | Hypothesis | irrG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_irreducible",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Hom_G | := 'C(E_G)%MS. | Notation | Hom_G | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_Jacobson_density : ('C(Hom_G) <= E_G)%MS. | Proof.
apply/row_subP=> iB; rewrite -[row iB _]vec_mxK; move defB: (vec_mx _) => B.
have{defB} cBcE: (B \in 'C(Hom_G))%MS by rewrite -defB vec_mxK row_sub.
have rGnP: mx_repr G (fun x => lin_mx (mulmxr (rG x)) : 'A_n).
split=> [|x y Gx Gy]; apply/row_matrixP=> i.
by rewrite !rowE mul_rV_lin repr_mx1 /= !mulmx1 ve... | Lemma | mx_Jacobson_density | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"Hom_G",
"apply",
"cent_mxP",
"centgmx",
"centgmxP",
"cyclic_mx",
"cyclic_mx_id",
"cyclic_mx_module",
"delta_mx",
"enveloping_algebra_mx",
"eq_row_sub",
"eqmxMr",
"eqmxP",
"genmxE",
"intro_mxsemisimple",
"irrG",
"last",
"lin1_mx",
"lin_mx",
"memmx_cent_envelop",
"mul... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cent_mx_scalar_abs_irr : \rank Hom_G <= 1 -> mx_absolutely_irreducible rG. | Proof.
rewrite leqNgt => /(has_non_scalar_mxP (scalar_mx_cent _ _)) scal_cE.
apply/andP; split; first by case/mx_irrP: irrG.
rewrite -sub1mx; apply: submx_trans mx_Jacobson_density.
apply/memmx_subP=> B _; apply/cent_mxP=> A cGA.
case scalA: (is_scalar_mx A); last by case: scal_cE; exists A; rewrite ?scalA.
by case/is_... | Lemma | cent_mx_scalar_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Hom_G",
"apply",
"cent_mxP",
"has_non_scalar_mxP",
"irrG",
"is_scalar_mx",
"is_scalar_mxP",
"last",
"leqNgt",
"memmx_subP",
"mx_Jacobson_density",
"mx_absolutely_irreducible",
"mx_irrP",
"rG",
"rank",
"scalar_mxC",
"scalar_mx_cent",
"split",
"sub1mx",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rH | := (subg_repr rG sHG). | Notation | rH | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rG",
"sHG",
"subg_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_subg : rfix_mx rH = rfix_mx rG. | Proof. by []. Qed. | Lemma | rfix_subg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rG",
"rH",
"rfix_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_subg : rstabs rH U = H :&: rstabs rG U. | Proof. by apply/setP=> x; rewrite !inE andbA -in_setI (setIidPl sHG). Qed. | Lemma | rstabs_subg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"in_setI",
"rG",
"rH",
"rstabs",
"sHG",
"setIidPl",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_subg : mxmodule rG U -> mxmodule rH U. | Proof. by rewrite /mxmodule rstabs_subg subsetI subxx; apply: subset_trans. Qed. | Lemma | mxmodule_subg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mxmodule",
"rG",
"rH",
"rstabs_subg",
"subsetI",
"subset_trans",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_subg M : mxmodule rG M -> mxsimple rH M -> mxsimple rG M. | Proof.
by move=> modM [_ nzM minM]; split=> // U /mxmodule_subg; apply: minM.
Qed. | Lemma | mxsimple_subg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mxmodule",
"mxmodule_subg",
"mxsimple",
"rG",
"rH",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subg_mx_irr : mx_irreducible rH -> mx_irreducible rG. | Proof. by apply: mxsimple_subg; apply: mxmodule1. Qed. | Lemma | subg_mx_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mx_irreducible",
"mxmodule1",
"mxsimple_subg",
"rG",
"rH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subg_mx_abs_irr :
mx_absolutely_irreducible rH -> mx_absolutely_irreducible rG. | Proof.
rewrite /mx_absolutely_irreducible -!sub1mx => /andP[-> /submx_trans-> //].
apply/row_subP=> i; rewrite rowK /= envelop_mx_id //.
by rewrite (subsetP sHG) ?enum_valP.
Qed. | Lemma | subg_mx_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"enum_valP",
"envelop_mx_id",
"mx_absolutely_irreducible",
"rG",
"rH",
"rowK",
"row_subP",
"sHG",
"sub1mx",
"submx_trans",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rH | := (eqg_repr rG eqGH). | Notation | rH | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"eqGH",
"eqg_repr",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_eqg : rfix_mx rH = rfix_mx rG. | Proof. by []. Qed. | Lemma | rfix_eqg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rG",
"rH",
"rfix_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_eqg : rstabs rH U = rstabs rG U. | Proof. by rewrite rstabs_subg -(eqP eqGH) (setIidPr _) ?rstabs_sub. Qed. | Lemma | rstabs_eqg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"eqGH",
"rG",
"rH",
"rstabs",
"rstabs_sub",
"rstabs_subg",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_eqg : mxmodule rH U = mxmodule rG U. | Proof. by rewrite /mxmodule rstabs_eqg -(eqP eqGH). Qed. | Lemma | mxmodule_eqg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"eqGH",
"mxmodule",
"rG",
"rH",
"rstabs_eqg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_eqg M : mxsimple rH M <-> mxsimple rG M. | Proof.
rewrite /mxsimple mxmodule_eqg.
split=> [] [-> -> minM]; split=> // U modU;
by apply: minM; rewrite mxmodule_eqg in modU *.
Qed. | Lemma | mxsimple_eqg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mxmodule_eqg",
"mxsimple",
"rG",
"rH",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqg_mx_irr : mx_irreducible rH <-> mx_irreducible rG. | Proof. exact: mxsimple_eqg. Qed. | Lemma | eqg_mx_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_irreducible",
"mxsimple_eqg",
"rG",
"rH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqg_mx_abs_irr :
mx_absolutely_irreducible rH = mx_absolutely_irreducible rG. | Proof.
by congr (_ && (_ == _)); rewrite /enveloping_algebra_mx /= -(eqP eqGH).
Qed. | Lemma | eqg_mx_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"enveloping_algebra_mx",
"eqGH",
"mx_absolutely_irreducible",
"rG",
"rH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGf | := (morphpre_repr f rG). | Notation | rGf | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"morphpre_repr",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_morphpre : rstabs rGf U = f @*^-1 (rstabs rG U). | Proof. by apply/setP=> x; rewrite !inE andbA. Qed. | Lemma | rstabs_morphpre | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"rG",
"rGf",
"rstabs",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_morphpre : G \subset f @* D -> mxmodule rGf U = mxmodule rG U. | Proof. by move=> sGf; rewrite /mxmodule rstabs_morphpre morphpreSK. Qed. | Lemma | mxmodule_morphpre | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"morphpreSK",
"mxmodule",
"rG",
"rGf",
"rstabs_morphpre"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_morphpre (H : {set aT}) :
H \subset D -> (rfix_mx rGf H :=: rfix_mx rG (f @* H))%MS. | Proof.
move=> sHD; apply/eqmxP/andP; split.
by apply/rfix_mxP=> _ /morphimP[x _ Hx ->]; rewrite rfix_mx_id.
by apply/rfix_mxP=> x Hx; rewrite rfix_mx_id ?mem_morphim ?(subsetP sHD).
Qed. | Lemma | rfix_morphpre | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"aT",
"apply",
"eqmxP",
"mem_morphim",
"morphimP",
"rG",
"rGf",
"rfix_mx",
"rfix_mxP",
"rfix_mx_id",
"sHD",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpre_mx_irr :
G \subset f @* D -> (mx_irreducible rGf <-> mx_irreducible rG). | Proof.
move/mxmodule_morphpre=> modG; split=> /mx_irrP[n_gt0 irrG];
by apply/mx_irrP; split=> // U modU; apply: irrG; rewrite modG in modU *.
Qed. | Lemma | morphpre_mx_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"irrG",
"modG",
"mx_irrP",
"mx_irreducible",
"mxmodule_morphpre",
"n_gt0",
"rG",
"rGf",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpre_mx_abs_irr :
G \subset f @* D ->
mx_absolutely_irreducible rGf = mx_absolutely_irreducible rG. | Proof.
move=> sGfD; congr (_ && (_ == _)); apply/eqP; rewrite mxrank_leqif_sup //.
apply/row_subP=> i; rewrite rowK (envelop_mx_id rG) //.
by case/morphpreP: (enum_valP i).
apply/row_subP=> i; rewrite rowK.
case/morphimP: (subsetP sGfD _ (enum_valP i)) => x Dx _ def_i.
by rewrite def_i (envelop_mx_id rGf) // !inE D... | Lemma | morphpre_mx_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Dx",
"apply",
"enum_valP",
"envelop_mx_id",
"inE",
"morphimP",
"morphpreP",
"mx_absolutely_irreducible",
"mxrank_leqif_sup",
"rG",
"rGf",
"rowK",
"row_subP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rG | := (morphim_repr rGf sGD). | Notation | rG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"morphim_repr",
"rGf",
"sGD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_morphim : rstabs rG U = G :&: f @*^-1 rstabs rGf U. | Proof. by rewrite -rstabs_morphpre -(rstabs_subg _ sG_f'fG). Qed. | Lemma | rstabs_morphim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rG",
"rGf",
"rstabs",
"rstabs_morphpre",
"rstabs_subg",
"sG_f'fG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_morphim : mxmodule rG U = mxmodule rGf U. | Proof. by rewrite /mxmodule rstabs_morphim subsetI subxx -sub_morphim_pre. Qed. | Lemma | mxmodule_morphim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"rG",
"rGf",
"rstabs_morphim",
"sub_morphim_pre",
"subsetI",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_morphim (H : {set aT}) :
H \subset D -> (rfix_mx rG H :=: rfix_mx rGf (f @* H))%MS. | Proof. exact: rfix_morphpre. Qed. | Lemma | rfix_morphim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"aT",
"rG",
"rGf",
"rfix_morphpre",
"rfix_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_morphim M : mxsimple rG M <-> mxsimple rGf M. | Proof.
rewrite /mxsimple mxmodule_morphim.
split=> [] [-> -> minM]; split=> // U modU;
by apply: minM; rewrite mxmodule_morphim in modU *.
Qed. | Lemma | mxsimple_morphim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mxmodule_morphim",
"mxsimple",
"rG",
"rGf",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_mx_irr : (mx_irreducible rG <-> mx_irreducible rGf). | Proof. exact: mxsimple_morphim. Qed. | Lemma | morphim_mx_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_irreducible",
"mxsimple_morphim",
"rG",
"rGf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_mx_abs_irr :
mx_absolutely_irreducible rG = mx_absolutely_irreducible rGf. | Proof.
have fG_onto: f @* G \subset restrm sGD f @* G.
by rewrite (morphim_restrm sGD) setIid.
rewrite -(morphpre_mx_abs_irr _ fG_onto); congr (_ && (_ == _)).
by rewrite /enveloping_algebra_mx /= morphpre_restrm (setIidPl _).
Qed. | Lemma | morphim_mx_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"enveloping_algebra_mx",
"morphim_restrm",
"morphpre_mx_abs_irr",
"morphpre_restrm",
"mx_absolutely_irreducible",
"rG",
"rGf",
"restrm",
"sGD",
"setIid",
"setIidPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rU | := (submod_repr Umod). | Notation | rU | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"submod_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rU' | := (factmod_repr Umod). | Notation | rU' | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"factmod_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_submod (H : {set gT}) :
H \subset G -> (rfix_mx rU H :=: in_submod U (U :&: rfix_mx rG H))%MS. | Proof.
move=> sHG; apply/eqmxP/andP; split; last first.
apply/rfix_mxP=> x Hx; rewrite -in_submodJ ?capmxSl //.
by rewrite (rfix_mxP H _) ?capmxSr.
rewrite -val_submodS in_submodK ?capmxSl // sub_capmx val_submodP //=.
apply/rfix_mxP=> x Hx.
by rewrite -(val_submodJ Umod) ?(subsetP sHG) ?rfix_mx_id.
Qed. | Lemma | rfix_submod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"apply",
"capmxSl",
"capmxSr",
"eqmxP",
"gT",
"in_submod",
"in_submodJ",
"in_submodK",
"last",
"rG",
"rU",
"rfix_mx",
"rfix_mxP",
"rfix_mx_id",
"sHG",
"split",
"sub_capmx",
"subsetP",
"val_submodJ",
"val_submodP",
"val_submodS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_factmod (H : {set gT}) :
H \subset G -> (in_factmod U (rfix_mx rG H) <= rfix_mx rU' H)%MS. | Proof.
move=> sHG; apply/rfix_mxP=> x Hx.
by rewrite -(in_factmodJ Umod) ?(subsetP sHG) ?rfix_mx_id.
Qed. | Lemma | rfix_factmod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"apply",
"gT",
"in_factmod",
"in_factmodJ",
"rG",
"rU'",
"rfix_mx",
"rfix_mxP",
"rfix_mx_id",
"sHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_submod m (W : 'M_(m, \rank U)) :
rstab rU W = rstab rG (val_submod W). | Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
by rewrite -(inj_eq val_submod_inj) val_submodJ.
Qed. | Lemma | rstab_submod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"inj_eq",
"rG",
"rU",
"rank",
"rstab",
"setP",
"val_submod",
"val_submodJ",
"val_submod_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_submod m (W : 'M_(m, \rank U)) :
rstabs rU W = rstabs rG (val_submod W). | Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
by rewrite -val_submodS val_submodJ.
Qed. | Lemma | rstabs_submod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"rG",
"rU",
"rank",
"rstabs",
"setP",
"val_submod",
"val_submodJ",
"val_submodS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submod_module m (W : 'M_(m, \rank U)) :
mxmodule rG (val_submod W) = mxmodule rU W. | Proof. by rewrite /mxmodule rstabs_submod. Qed. | Lemma | val_submod_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"rG",
"rU",
"rank",
"rstabs_submod",
"val_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_submod_module m (V : 'M_(m, n)) :
(V <= U)%MS -> mxmodule rU (in_submod U V) = mxmodule rG V. | Proof. by move=> sVU; rewrite -val_submod_module in_submodK. Qed. | Lemma | in_submod_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"in_submod",
"in_submodK",
"mxmodule",
"rG",
"rU",
"val_submod_module"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_factmod m (W : 'M_(m, n)) :
rstab rG W \subset rstab rU' (in_factmod U W). | Proof.
by apply/subsetP=> x /setIdP[Gx /eqP cUW]; rewrite inE Gx -in_factmodJ //= cUW.
Qed. | Lemma | rstab_factmod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"in_factmod",
"in_factmodJ",
"rG",
"rU'",
"rstab",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_factmod m (W : 'M_(m, \rank (cokermx U))) :
rstabs rU' W = rstabs rG (U + val_factmod W)%MS. | Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
rewrite addsmxMr addsmx_sub (submx_trans (mxmoduleP Umod x Gx)) ?addsmxSl //.
rewrite -val_factmodS val_factmodJ //= val_factmodS; apply/idP/idP=> nWx.
rewrite (submx_trans (addsmxSr U _)) // -(in_factmodsK (addsmxSl U _)) //.
by rewrite addsmxS // val_factmodS... | Lemma | rstabs_factmod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"addsmxMr",
"addsmxS",
"addsmxSl",
"addsmxSr",
"addsmx_sub",
"apply",
"cokermx",
"inE",
"in_factmodE",
"in_factmod_addsK",
"in_factmodsK",
"mxmoduleP",
"rG",
"rU'",
"rank",
"rstabs",
"setP",
"submxMr",
"submx_trans",
"val_factmod",
"val_factmodJ",
"val_factmodK",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmod_module m (W : 'M_(m, \rank (cokermx U))) :
mxmodule rG (U + val_factmod W)%MS = mxmodule rU' W. | Proof. by rewrite /mxmodule rstabs_factmod. Qed. | Lemma | val_factmod_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cokermx",
"mxmodule",
"rG",
"rU'",
"rank",
"rstabs_factmod",
"val_factmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_factmod_module m (V : 'M_(m, n)) :
mxmodule rU' (in_factmod U V) = mxmodule rG (U + V)%MS. | Proof.
rewrite -(eqmx_module _ (in_factmodsK (addsmxSl U V))).
by rewrite val_factmod_module (eqmx_module _ (in_factmod_addsK _ _)).
Qed. | Lemma | in_factmod_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addsmxSl",
"eqmx_module",
"in_factmod",
"in_factmod_addsK",
"in_factmodsK",
"mxmodule",
"rG",
"rU'",
"val_factmod_module"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_submod : rker rU = rstab rG U. | Proof. by rewrite /rker rstab_submod; apply: eqmx_rstab (val_submod1 U). Qed. | Lemma | rker_submod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmx_rstab",
"rG",
"rU",
"rker",
"rstab",
"rstab_submod",
"val_submod1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_norm : G \subset 'N(rstab rG U). | Proof. by rewrite -rker_submod rker_norm. Qed. | Lemma | rstab_norm | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rG",
"rker_norm",
"rker_submod",
"rstab"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_normal : rstab rG U <| G. | Proof. by rewrite -rker_submod rker_normal. Qed. | Lemma | rstab_normal | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rG",
"rker_normal",
"rker_submod",
"rstab"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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