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 |
|---|---|---|---|---|---|---|---|---|---|---|
(splitF : group_splitting_field F S) (F'S : [pchar F]^'.-group S). | Hypotheses | splitF | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"group",
"group_splitting_field",
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
p_pr | := extraspecial_prime pS esS. | Let | p_pr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"extraspecial_prime",
"pS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p_gt0 | := prime_gt0 p_pr. | Let | p_gt0 | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"p_pr",
"prime_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p_gt1 | := prime_gt1 p_pr. | Let | p_gt1 | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"p_pr",
"prime_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oZp | := card_center_extraspecial pS esS. | Let | oZp | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"card_center_extraspecial",
"pS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modIp' (i : 'I_p.-1) : (i.+1 %% p = i.+1)%N. | Proof. by case: i => i; rewrite /= -ltnS prednK //; apply: modn_small. Qed. | Let | modIp' | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"ltnS",
"modn_small",
"prednK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extraspecial_repr_structure_pchar (sS : irrType F S) :
[/\ #|linear_irr sS| = (p ^ n.*2)%N,
exists iphi : 'I_p.-1 -> sS, let phi i := irr_repr (iphi i) in
[/\ injective iphi,
codom iphi =i ~: linear_irr sS,
forall i, mx_faithful (phi i),
forall z, z \in 'Z(S)^# ->
... | Proof.
have [[defPhiS defS'] prZ] := esS; set linS := linear_irr sS.
have nb_lin: #|linS| = (p ^ n.*2)%N.
rewrite card_linear_irr // -divgS ?der_sub //=.
by rewrite oSpn defS' oZp expnS mulKn.
have nb_irr: #|sS| = (p ^ n.*2 + p.-1)%N.
pose Zcl := classes S ::&: 'Z(S).
have cardZcl: #|Zcl| = p.
transitivity ... | Theorem | extraspecial_repr_structure_pchar | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"Aut",
"Aut_aut",
"Aut_prime_cyclic",
"Euclid_dvdM",
"TI_center_nil",
"Zp_cast",
"Zp_unitm",
"addKn",
"addSnnS",
"addnC",
"alpha",
"apply",
"autE",
"autm",
"autmE",
"bigID",
"card",
"card_gt0P",
"card_image",
"card_imset",
"card_irr_pchar",
"card_linear_irr",
"card_ord",
... | This is Aschbacher (34.9), parts (1)-(4). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
(simU : mxsimple rS U) (ffulU : rstab rS U == 1%g). | Hypotheses | simU | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mxsimple",
"rstab"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sZS | := center_sub S. | Let | sZS | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"center_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rZ | := subg_repr rS sZS. | Let | rZ | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"sZS",
"subg_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
faithful_repr_extraspecial_pchar :
\rank U = (p ^ n)%N /\
(forall V, mxsimple rS V -> mx_iso rZ U V -> mx_iso rS U V). | Proof.
suffices IH V: mxsimple rS V -> mx_iso rZ U V ->
[&& \rank U == (p ^ n)%N & mxsimple_iso rS U V].
- split=> [|/= V simV isoUV].
by case/andP: (IH U simU (mx_iso_refl _ _)) => /eqP.
by case/andP: (IH V simV isoUV) => _ /(mxsimple_isoP simU).
move=> simV isoUV; wlog sS: / irrType F S by apply: socle_exists... | Lemma | faithful_repr_extraspecial_pchar | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"cards1",
"codom",
"eq_prim_root_expr",
"eqmx_rstab",
"eqxx",
"extraspecial_repr_structure_pchar",
"f_iinv",
"i0",
"iinv",
"inE",
"irrType",
"irr_comp",
"irr_mode",
"modIp'",
"mulmxA",
"mulr1n",
"mxE",
"mx_irreducible",
"mx_iso",
"mx_iso_refl",
"mx_rsim",
"mx_rsi... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extraspecial_repr_structure | := (extraspecial_repr_structure_pchar)
(only parsing). | Notation | extraspecial_repr_structure | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"extraspecial_repr_structure_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
faithful_repr_extraspecial | := (faithful_repr_extraspecial_pchar)
(only parsing). | Notation | faithful_repr_extraspecial | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"faithful_repr_extraspecial_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_repr (G : {set gT}) n (r : gT -> 'M[R]_n) | :=
r 1%g = 1%:M /\ {in G &, {morph r : x y / (x * y)%g >-> x *m y}}. | Definition | mx_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",
... | [
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_representation G n | :=
MxRepresentation { repr_mx :> gT -> 'M_n; _ : mx_repr G repr_mx }. | Structure | mx_representation | 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",
... | [
"gT",
"mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mx1 : rG 1 = 1%:M. | Proof. by case: rG => r []. Qed. | Lemma | repr_mx1 | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mxM : {in G &, {morph rG : x y / (x * y)%g >-> x *m y}}. | Proof. by case: rG => r []. Qed. | Lemma | repr_mxM | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mxK m x :
x \in G -> cancel ((@mulmx R m n n)^~ (rG x)) (mulmx^~ (rG x^-1)). | Proof.
by move=> Gx U; rewrite -mulmxA -repr_mxM ?groupV // mulgV repr_mx1 mulmx1.
Qed. | Lemma | repr_mxK | 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",
... | [
"groupV",
"mulgV",
"mulmx",
"mulmx1",
"mulmxA",
"rG",
"repr_mx1",
"repr_mxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mxKV m x :
x \in G -> cancel ((@mulmx R m n n)^~ (rG x^-1)) (mulmx^~ (rG x)). | Proof. by rewrite -groupV -{3}[x]invgK; apply: repr_mxK. Qed. | Lemma | repr_mxKV | 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",
"groupV",
"invgK",
"mulmx",
"rG",
"repr_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mx_unit x : x \in G -> rG x \in unitmx. | Proof. by move=> Gx; case/mulmx1_unit: (repr_mxKV Gx 1%:M). Qed. | Lemma | repr_mx_unit | 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",
... | [
"mulmx1_unit",
"rG",
"repr_mxKV",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mxV : {in G, {morph rG : x / x^-1%g >-> invmx x}}. | Proof.
by move=> x Gx /=; rewrite -[rG x^-1](mulKmx (repr_mx_unit Gx)) mulmxA repr_mxK.
Qed. | Lemma | repr_mxV | 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",
... | [
"invmx",
"mulKmx",
"mulmxA",
"rG",
"repr_mxK",
"repr_mx_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
enveloping_algebra_mx | := \matrix_(i < #|G|) mxvec (rG (enum_val i)). | Definition | enveloping_algebra_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",
... | [
"enum_val",
"mxvec",
"rG"
] | developped the theory of matrix subalgebras for F-algebras. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rstab | := [set x in G | U *m rG x == U]. | Definition | rstab | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_sub : rstab \subset G. | Proof. by apply/subsetP=> x; case/setIdP. Qed. | Lemma | rstab_sub | 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",
"rstab",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_group_set : group_set rstab. | Proof.
apply/group_setP; rewrite inE group1 repr_mx1 mulmx1; split=> //= x y.
case/setIdP=> Gx cUx; case/setIdP=> Gy cUy; rewrite inE repr_mxM ?groupM //.
by rewrite mulmxA (eqP cUx).
Qed. | Lemma | rstab_group_set | 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",
"group1",
"groupM",
"group_set",
"group_setP",
"inE",
"mulmx1",
"mulmxA",
"repr_mx1",
"repr_mxM",
"rstab",
"setIdP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_group | := Group rstab_group_set. | Canonical | rstab_group | 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",
... | [
"rstab_group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent | := [set x in G | f *m rG x == rG x *m f]. | Definition | rcent | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent_sub : rcent \subset G. | Proof. by apply/subsetP=> x; case/setIdP. Qed. | Lemma | rcent_sub | 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",
"rcent",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent_group_set : group_set rcent. | Proof.
apply/group_setP; rewrite inE group1 repr_mx1 mulmx1 mul1mx; split=> //= x y.
case/setIdP=> Gx; move/eqP=> cfx; case/setIdP=> Gy; move/eqP=> cfy.
by rewrite inE repr_mxM ?groupM //= -mulmxA -cfy !mulmxA cfx.
Qed. | Lemma | rcent_group_set | 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",
"group1",
"groupM",
"group_set",
"group_setP",
"inE",
"mul1mx",
"mulmx1",
"mulmxA",
"rcent",
"repr_mx1",
"repr_mxM",
"setIdP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent_group | := Group rcent_group_set. | Canonical | rcent_group | 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",
... | [
"rcent_group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
centgmx | := G \subset rcent. | Definition | centgmx | 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",
... | [
"rcent"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
centgmxP : reflect (forall x, x \in G -> f *m rG x = rG x *m f) centgmx. | Proof.
by apply: (iffP subsetP) => cGf x Gx; have /[!(inE, Gx)] /eqP := cGf x Gx.
Qed. | Lemma | centgmxP | 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",
"inE",
"rG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker | := rstab 1%:M. | Definition | rker | 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",
... | [
"rstab"
] | Representation kernel, and faithful representations. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rker_group | := Eval hnf in [group of rker]. | Canonical | rker_group | 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",
... | [
"group",
"rker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rkerP x : reflect (x \in G /\ rG x = 1%:M) (x \in rker). | Proof. by apply: (iffP setIdP) => [] [->]; move/eqP; rewrite mul1mx. Qed. | Lemma | rkerP | 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",
"mul1mx",
"rG",
"rker",
"setIdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_norm : G \subset 'N(rker). | Proof.
apply/subsetP=> x Gx; rewrite inE sub_conjg; apply/subsetP=> y.
case/rkerP=> Gy ry1; rewrite mem_conjgV !inE groupJ //=.
by rewrite !repr_mxM ?groupM ?groupV // ry1 !mulmxA mulmx1 repr_mxKV.
Qed. | Lemma | rker_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",
... | [
"apply",
"groupJ",
"groupM",
"groupV",
"inE",
"mem_conjgV",
"mulmx1",
"mulmxA",
"repr_mxKV",
"repr_mxM",
"rker",
"rkerP",
"sub_conjg",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_normal : rker <| G. | Proof. by rewrite /normal rstab_sub rker_norm. Qed. | Lemma | rker_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",
... | [
"normal",
"rker",
"rker_norm",
"rstab_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_faithful | := rker \subset [1]. | Definition | mx_faithful | 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",
... | [
"rker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_faithful_inj : mx_faithful -> {in G &, injective rG}. | Proof.
move=> ffulG x y Gx Gy eq_rGxy; apply/eqP; rewrite eq_mulgV1 -in_set1.
rewrite (subsetP ffulG) // inE groupM ?repr_mxM ?groupV //= eq_rGxy.
by rewrite mulmxA repr_mxK.
Qed. | Lemma | mx_faithful_inj | 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",
"eq_mulgV1",
"groupM",
"groupV",
"inE",
"in_set1",
"mulmxA",
"mx_faithful",
"rG",
"repr_mxK",
"repr_mxM",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_linear : n = 1 -> G^`(1)%g \subset rker. | Proof.
move=> n1; rewrite gen_subG; apply/subsetP=> xy; case/imset2P=> x y Gx Gy ->.
rewrite !inE groupR //= /commg mulgA -invMg repr_mxM ?groupV ?groupM //.
rewrite mulmxA (can2_eq (repr_mxK _) (repr_mxKV _)) ?groupM //.
rewrite !repr_mxV ?repr_mxM ?groupM //; move: (rG x) (rG y).
by rewrite n1 => rx ry; rewrite (mx11... | Lemma | rker_linear | 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",
"can2_eq",
"commg",
"gen_subG",
"groupM",
"groupR",
"groupV",
"imset2P",
"inE",
"invMg",
"mulgA",
"mulmxA",
"mx11_scalar",
"rG",
"repr_mxK",
"repr_mxKV",
"repr_mxM",
"repr_mxV",
"rker",
"scalar_mxC",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcenter | := [set g in G | is_scalar_mx (rG g)]. | Definition | rcenter | 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",
... | [
"is_scalar_mx",
"rG"
] | Representation center. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rcenter_group_set : group_set rcenter. | Proof.
apply/group_setP; split=> [|x y].
by rewrite inE group1 repr_mx1 scalar_mx_is_scalar.
move=> /setIdP[Gx /is_scalar_mxP[a defx]] /setIdP[Gy /is_scalar_mxP[b defy]].
by rewrite !inE groupM ?repr_mxM // defx defy -scalar_mxM ?scalar_mx_is_scalar.
Qed. | Fact | rcenter_group_set | 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",
"group1",
"groupM",
"group_set",
"group_setP",
"inE",
"is_scalar_mxP",
"rcenter",
"repr_mx1",
"repr_mxM",
"scalar_mxM",
"scalar_mx_is_scalar",
"setIdP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcenter_group | := Group rcenter_group_set. | Canonical | rcenter_group | 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",
... | [
"rcenter_group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcenter_normal : rcenter <| G. | Proof.
rewrite /normal /rcenter {1}setIdE subsetIl; apply/subsetP=> x Gx /[1!inE].
apply/subsetP=> _ /imsetP[y /setIdP[Gy /is_scalar_mxP[c rGy]] ->].
rewrite inE !repr_mxM ?groupM ?groupV //= mulmxA rGy scalar_mxC repr_mxKV //.
exact: scalar_mx_is_scalar.
Qed. | Lemma | rcenter_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",
... | [
"apply",
"groupM",
"groupV",
"imsetP",
"inE",
"is_scalar_mxP",
"mulmxA",
"normal",
"rcenter",
"repr_mxKV",
"repr_mxM",
"scalar_mxC",
"scalar_mx_is_scalar",
"setIdE",
"setIdP",
"subsetIl",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mxMr : {in G &, {morph rG : x y / (x * y)%g >-> x * y}}. | Proof. exact: repr_mxM. Qed. | Lemma | repr_mxMr | 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",
"repr_mxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mxVr : {in G, {morph rG : x / (x^-1)%g >-> x^-1}}. | Proof. exact: repr_mxV. Qed. | Lemma | repr_mxVr | 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",
"repr_mxV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mx_unitr x : x \in G -> rG x \is a GRing.unit. | Proof. exact: repr_mx_unit. Qed. | Lemma | repr_mx_unitr | 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",
"repr_mx_unit",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mxX m : {in G, {morph rG : x / (x ^+ m)%g >-> x ^+ m}}. | Proof.
elim: m => [|m IHm] x Gx; rewrite /= ?repr_mx1 // expgS exprS -IHm //.
by rewrite repr_mxM ?groupX.
Qed. | Lemma | repr_mxX | 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",
... | [
"expgS",
"exprS",
"groupX",
"rG",
"repr_mx1",
"repr_mxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subg_mx_repr : mx_repr H rG. | Proof.
by split=> [|x y Hx Hy]; rewrite (repr_mx1, repr_mxM) ?(subsetP sHG).
Qed. | Lemma | subg_mx_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",
... | [
"mx_repr",
"rG",
"repr_mx1",
"repr_mxM",
"sHG",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subg_repr | := MxRepresentation subg_mx_repr. | Definition | subg_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",
... | [
"subg_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rH | := subg_repr. | 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",
... | [
"subg_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent_subg U : rcent rH U = H :&: rcent rG U. | Proof. by apply/setP=> x; rewrite !inE andbA -in_setI (setIidPl sHG). Qed. | Lemma | rcent_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",
"rcent",
"sHG",
"setIidPl",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_subg : rstab rH U = H :&: rstab rG U. | Proof. by apply/setP=> x; rewrite !inE andbA -in_setI (setIidPl sHG). Qed. | Lemma | rstab_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",
"rstab",
"sHG",
"setIidPl",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_subg : rker rH = H :&: rker rG. | Proof. exact: rstab_subg. Qed. | Lemma | rker_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",
"rker",
"rstab_subg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subg_mx_faithful : mx_faithful rG -> mx_faithful rH. | Proof. by apply: subset_trans; rewrite rker_subg subsetIr. Qed. | Lemma | subg_mx_faithful | 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_faithful",
"rG",
"rH",
"rker_subg",
"subsetIr",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqGH : G | :==: H. | Hypothesis | eqGH | 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 | |
eqg_repr_proof : H \subset G. | Proof. by rewrite (eqP eqGH). Qed. | Lemma | eqg_repr_proof | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqg_repr | := subg_repr eqg_repr_proof. | Definition | eqg_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",
... | [
"eqg_repr_proof",
"subg_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rH | := eqg_repr. | 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",
... | [
"eqg_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent_eqg U : rcent rH U = rcent rG U. | Proof. by rewrite rcent_subg -(eqP eqGH) (setIidPr _) ?rcent_sub. Qed. | Lemma | rcent_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",
"rcent",
"rcent_sub",
"rcent_subg",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_eqg : rstab rH U = rstab rG U. | Proof. by rewrite rstab_subg -(eqP eqGH) (setIidPr _) ?rstab_sub. Qed. | Lemma | rstab_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",
"rstab",
"rstab_sub",
"rstab_subg",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_eqg : rker rH = rker rG. | Proof. exact: rstab_eqg. Qed. | Lemma | rker_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",
"rker",
"rstab_eqg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqg_mx_faithful : mx_faithful rH = mx_faithful rG. | Proof. by rewrite /mx_faithful rker_eqg. Qed. | Lemma | eqg_mx_faithful | 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_faithful",
"rG",
"rH",
"rker_eqg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpre_mx_repr : mx_repr (f @*^-1 G) (rG \o f). | Proof.
split=> [|x y]; first by rewrite /= morph1 repr_mx1.
case/morphpreP=> Dx Gfx; case/morphpreP=> Dy Gfy.
by rewrite /= morphM ?repr_mxM.
Qed. | Lemma | morphpre_mx_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",
... | [
"Dx",
"morph1",
"morphM",
"morphpreP",
"mx_repr",
"rG",
"repr_mx1",
"repr_mxM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpre_repr | := MxRepresentation morphpre_mx_repr. | Canonical | morphpre_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",
... | [
"morphpre_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGf | := morphpre_repr. | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_morphpre : rstab rGf U = f @*^-1 (rstab rG U). | Proof. by apply/setP=> x; rewrite !inE andbA. Qed. | Lemma | rstab_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",
"rstab",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_morphpre : rker rGf = f @*^-1 (rker rG). | Proof. exact: rstab_morphpre. Qed. | Lemma | rker_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",
... | [
"rG",
"rGf",
"rker",
"rstab_morphpre"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_mx & G \subset D | := fun x => rGf (f x). | Definition | morphim_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",
... | [
"rGf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_mxE x : morphim_mx sGD x = rGf (f x). | Proof. by []. Qed. | Lemma | morphim_mxE | 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_mx",
"rGf",
"sGD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sG_f'fG : G \subset f @*^-1 (f @* G). | Proof. by rewrite -sub_morphim_pre. Qed. | Let | sG_f'fG | 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",
... | [
"sub_morphim_pre"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_mx_repr : mx_repr G (morphim_mx sGD). | Proof. exact: subg_mx_repr (morphpre_repr f rGf) sG_f'fG. Qed. | Lemma | morphim_mx_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",
... | [
"morphim_mx",
"morphpre_repr",
"mx_repr",
"rGf",
"sGD",
"sG_f'fG",
"subg_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_repr | := MxRepresentation morphim_mx_repr. | Canonical | morphim_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",
... | [
"morphim_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rG | := morphim_repr. | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_morphim : rstab rG U = G :&: f @*^-1 rstab rGf U. | Proof. by rewrite -rstab_morphpre -(rstab_subg _ sG_f'fG). Qed. | Lemma | rstab_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",
"rstab",
"rstab_morphpre",
"rstab_subg",
"sG_f'fG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_morphim : rker rG = G :&: f @*^-1 (rker rGf). | Proof. exact: rstab_morphim. Qed. | Lemma | rker_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",
"rker",
"rstab_morphim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rconj_mx & B \in unitmx | := fun x => B *m rG x *m invmx B. | Definition | rconj_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",
... | [
"invmx",
"rG",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uB : B \in unitmx. | Hypothesis | uB | 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",
... | [
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rconj_mx_repr : mx_repr G (rconj_mx uB). | Proof.
split=> [|x y Gx Gy]; rewrite /rconj_mx ?repr_mx1 ?mulmx1 ?mulmxV ?repr_mxM //.
by rewrite !mulmxA mulmxKV.
Qed. | Lemma | rconj_mx_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",
... | [
"mulmx1",
"mulmxA",
"mulmxKV",
"mulmxV",
"mx_repr",
"rconj_mx",
"repr_mx1",
"repr_mxM",
"split",
"uB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rconj_repr | := MxRepresentation rconj_mx_repr. | Canonical | rconj_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",
... | [
"rconj_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGB | := rconj_repr. | Notation | rGB | 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",
... | [
"rconj_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rconj_mxE x : rGB x = B *m rG x *m invmx B. | Proof. by []. Qed. | Lemma | rconj_mxE | 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",
... | [
"invmx",
"rG",
"rGB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rconj_mxJ m (W : 'M_(m, n)) x : W *m rGB x *m B = W *m B *m rG x. | Proof. by rewrite !mulmxA mulmxKV. Qed. | Lemma | rconj_mxJ | 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",
... | [
"mulmxA",
"mulmxKV",
"rG",
"rGB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent_conj A : rcent rGB A = rcent rG (invmx B *m A *m B). | Proof.
apply/setP=> x; rewrite !inE /= rconj_mxE !mulmxA.
rewrite (can2_eq (mulmxKV uB) (mulmxK uB)) -!mulmxA.
by rewrite -(can2_eq (mulKVmx uB) (mulKmx uB)).
Qed. | Lemma | rcent_conj | 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",
"can2_eq",
"inE",
"invmx",
"mulKVmx",
"mulKmx",
"mulmxA",
"mulmxK",
"mulmxKV",
"rG",
"rGB",
"rcent",
"rconj_mxE",
"setP",
"uB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_conj m (U : 'M_(m, n)) : rstab rGB U = rstab rG (U *m B). | Proof.
apply/setP=> x; rewrite !inE /= rconj_mxE !mulmxA.
by rewrite (can2_eq (mulmxKV uB) (mulmxK uB)).
Qed. | Lemma | rstab_conj | 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",
"can2_eq",
"inE",
"mulmxA",
"mulmxK",
"mulmxKV",
"rG",
"rGB",
"rconj_mxE",
"rstab",
"setP",
"uB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_conj : rker rGB = rker rG. | Proof.
apply/setP=> x; rewrite !inE /= mulmxA (can2_eq (mulmxKV uB) (mulmxK uB)).
by rewrite mul1mx -scalar_mxC (inj_eq (can_inj (mulKmx uB))) mul1mx.
Qed. | Lemma | rker_conj | 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",
"can2_eq",
"inE",
"inj_eq",
"mul1mx",
"mulKmx",
"mulmxA",
"mulmxK",
"mulmxKV",
"rG",
"rGB",
"rker",
"scalar_mxC",
"setP",
"uB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_mx_faithful : mx_faithful rGB = mx_faithful rG. | Proof. by rewrite /mx_faithful rker_conj. Qed. | Lemma | conj_mx_faithful | 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_faithful",
"rG",
"rGB",
"rker_conj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_mx (H : {set gT}) & H \subset rker rG & G \subset 'N(H) | :=
fun Hx : coset_of H => rG (repr Hx). | Definition | quo_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",
... | [
"coset_of",
"gT",
"rG",
"repr",
"rker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(krH : H \subset rker rG) (nHG : G \subset 'N(H)). | Hypotheses | krH | 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",
... | [
"nHG",
"rG",
"rker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
nHGs | := subsetP nHG. | Let | nHGs | 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",
... | [
"nHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_mx_coset x : x \in G -> quo_mx krH nHG (coset H x) = rG x. | Proof.
move=> Gx; rewrite /quo_mx val_coset ?nHGs //; case: repr_rcosetP => z Hz.
by case/rkerP: (subsetP krH z Hz) => Gz rz1; rewrite repr_mxM // rz1 mul1mx.
Qed. | Lemma | quo_mx_coset | 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",
... | [
"coset",
"krH",
"mul1mx",
"nHG",
"nHGs",
"quo_mx",
"rG",
"repr_mxM",
"repr_rcosetP",
"rkerP",
"subsetP",
"val_coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_mx_repr : mx_repr (G / H)%g (quo_mx krH nHG). | Proof.
split=> [|Hx Hy]; first by rewrite /quo_mx repr_coset1 repr_mx1.
case/morphimP=> x Nx Gx ->{Hx}; case/morphimP=> y Ny Gy ->{Hy}.
by rewrite -morphM // !quo_mx_coset ?groupM ?repr_mxM.
Qed. | Lemma | quo_mx_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",
... | [
"groupM",
"krH",
"morphM",
"morphimP",
"mx_repr",
"nHG",
"quo_mx",
"quo_mx_coset",
"repr_coset1",
"repr_mx1",
"repr_mxM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_repr | := MxRepresentation quo_mx_repr. | Canonical | quo_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",
... | [
"quo_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGH | := quo_repr. | Notation | rGH | 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",
... | [
"quo_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_repr_coset x : x \in G -> rGH (coset H x) = rG x. | Proof. exact: quo_mx_coset. Qed. | Lemma | quo_repr_coset | 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",
... | [
"coset",
"quo_mx_coset",
"rG",
"rGH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent_quo A : rcent rGH A = (rcent rG A / H)%g. | Proof.
apply/setP=> Hx /[!inE]; apply/andP/idP=> [[]|]; case/morphimP=> x Nx Gx ->{Hx}.
by rewrite quo_repr_coset // => cAx; rewrite mem_morphim // inE Gx.
by case/setIdP: Gx => Gx cAx; rewrite quo_repr_coset ?mem_morphim.
Qed. | Lemma | rcent_quo | 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",
"mem_morphim",
"morphimP",
"quo_repr_coset",
"rG",
"rGH",
"rcent",
"setIdP",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_quo m (U : 'M_(m, n)) : rstab rGH U = (rstab rG U / H)%g. | Proof.
apply/setP=> Hx /[!inE]; apply/andP/idP=> [[]|]; case/morphimP=> x Nx Gx ->{Hx}.
by rewrite quo_repr_coset // => nUx; rewrite mem_morphim // inE Gx.
by case/setIdP: Gx => Gx nUx; rewrite quo_repr_coset ?mem_morphim.
Qed. | Lemma | rstab_quo | 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",
"mem_morphim",
"morphimP",
"quo_repr_coset",
"rG",
"rGH",
"rstab",
"setIdP",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_quo : rker rGH = (rker rG / H)%g. | Proof. exact: rstab_quo. Qed. | Lemma | rker_quo | 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",
"rGH",
"rker",
"rstab_quo"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kquo_mx | := quo_mx (subxx (rker rG)) (rker_norm rG). | Definition | kquo_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",
... | [
"quo_mx",
"rG",
"rker",
"rker_norm",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kquo_mxE : kquo_mx = quo_mx (subxx (rker rG)) (rker_norm rG). | Proof. by []. Qed. | Lemma | kquo_mxE | 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",
... | [
"kquo_mx",
"quo_mx",
"rG",
"rker",
"rker_norm",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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