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 |
|---|---|---|---|---|---|---|---|---|---|---|
kquo_repr | := @MxRepresentation _ _ _ kquo_mx (quo_mx_repr _ _). | Canonical | kquo_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",
... | [
"kquo_mx",
"quo_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kquo_repr_coset x :
x \in G -> kquo_repr (coset (rker rG) x) = rG x. | Proof. exact: quo_repr_coset. Qed. | Lemma | kquo_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",
"kquo_repr",
"quo_repr_coset",
"rG",
"rker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kquo_mx_faithful : mx_faithful kquo_repr. | Proof. by rewrite /mx_faithful rker_quo trivg_quotient. Qed. | Lemma | kquo_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",
... | [
"kquo_repr",
"mx_faithful",
"rker_quo",
"trivg_quotient"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcard | := #|G|. | Definition | gcard | 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 | |
nG | := gcard. | Notation | nG | 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",
... | [
"gcard"
] | hides the projections to set | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gring_index (x : gT) | := enum_rank_in (group1 G) x. | Definition | gring_index | 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_rank_in",
"gT",
"group1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_valK : cancel enum_val gring_index. | Proof. exact: enum_valK_in. Qed. | Lemma | gring_valK | 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",
"enum_valK_in",
"gring_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_indexK : {in G, cancel gring_index enum_val}. | Proof. exact: enum_rankK_in. Qed. | Lemma | gring_indexK | 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_rankK_in",
"enum_val",
"gring_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
regular_mx x : 'M[R]_nG | :=
\matrix_i delta_mx 0 (gring_index (enum_val i * x)). | Definition | regular_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",
... | [
"delta_mx",
"enum_val",
"gring_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
regular_mx_repr : mx_repr G regular_mx. | Proof.
split=> [|x y Gx Gy]; apply/row_matrixP=> i; rewrite !rowK.
by rewrite mulg1 row1 gring_valK.
by rewrite row_mul rowK -rowE rowK mulgA gring_indexK // groupM ?enum_valP.
Qed. | Lemma | regular_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",
... | [
"apply",
"enum_valP",
"gring_indexK",
"gring_valK",
"groupM",
"mulg1",
"mulgA",
"mx_repr",
"regular_mx",
"row1",
"rowE",
"rowK",
"row_matrixP",
"row_mul",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
regular_repr | := MxRepresentation regular_mx_repr. | Canonical | regular_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",
... | [
"regular_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aG | := regular_repr. | Notation | aG | 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",
... | [
"regular_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_ring | := enveloping_algebra_mx aG. | Definition | group_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",
... | [
"aG",
"enveloping_algebra_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R_G | := group_ring. | Notation | R_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",
... | [
"group_ring"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_row : 'M[R]_nG -> 'rV_nG | := row (gring_index 1). | Definition | gring_row | 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",
... | [
"gring_index",
"row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_row_mul A B : gring_row (A *m B) = gring_row A *m B. | Proof. exact: row_mul. Qed. | Lemma | gring_row_mul | 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",
... | [
"gring_row",
"row_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_proj x | := row (gring_index x) \o trmx \o gring_row. | Definition | gring_proj | 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",
... | [
"gring_index",
"gring_row",
"row",
"trmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_projE : {in G &, forall x y, gring_proj x (aG y) = (x == y)%:R}. | Proof.
move=> x y Gx Gy; rewrite /gring_proj /= /gring_row rowK gring_indexK //=.
rewrite mul1g trmx_delta rowE mul_delta_mx_cond [delta_mx 0 0]mx11_scalar !mxE.
by rewrite /= -(inj_eq (can_inj gring_valK)) !gring_indexK.
Qed. | Lemma | gring_projE | 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",
... | [
"aG",
"delta_mx",
"gring_indexK",
"gring_proj",
"gring_row",
"gring_valK",
"inj_eq",
"mul1g",
"mul_delta_mx_cond",
"mx11_scalar",
"mxE",
"rowE",
"rowK",
"trmx_delta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
regular_mx_faithful : mx_faithful aG. | Proof.
apply/subsetP=> x /setIdP[Gx].
rewrite mul1mx inE => /eqP/(congr1 (gring_proj 1%g)).
rewrite -(repr_mx1 aG) !gring_projE ?group1 // eqxx eq_sym.
by case: (x == _) => // /eqP; rewrite eq_sym oner_eq0.
Qed. | Lemma | regular_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",
... | [
"aG",
"apply",
"eq_sym",
"eqxx",
"gring_proj",
"gring_projE",
"group1",
"inE",
"mul1mx",
"mx_faithful",
"oner_eq0",
"repr_mx1",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_mx | := vec_mx \o mulmxr (enveloping_algebra_mx rG). | Definition | gring_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",
... | [
"enveloping_algebra_mx",
"mulmxr",
"rG",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_mxJ a x :
x \in G -> gring_mx (a *m aG x) = gring_mx a *m rG x. | Proof.
move=> Gx; rewrite /gring_mx /= ![a *m _]mulmx_sum_row.
rewrite !(mulmx_suml, linear_sum); apply: eq_bigr => i _.
rewrite linearZ -!scalemxAl linearZ /=; congr (_ *: _) => {a}.
rewrite !rowK /= !mxvecK -rowE rowK mxvecK.
by rewrite gring_indexK ?groupM ?repr_mxM ?enum_valP.
Qed. | Lemma | gring_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",
... | [
"aG",
"apply",
"enum_valP",
"eq_bigr",
"gring_indexK",
"gring_mx",
"groupM",
"linearZ",
"linear_sum",
"mulmx_sum_row",
"mulmx_suml",
"mxvecK",
"rG",
"repr_mxM",
"rowE",
"rowK",
"scalemxAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_mxK : cancel (gring_mx aG) gring_row. | Proof.
move=> a; rewrite /gring_mx /= mulmx_sum_row !linear_sum /= [RHS]row_sum_delta.
apply: eq_bigr => i _; rewrite 2!linearZ /= /gring_row !(rowK, mxvecK).
by rewrite gring_indexK // mul1g gring_valK.
Qed. | Lemma | gring_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",
... | [
"aG",
"apply",
"eq_bigr",
"gring_indexK",
"gring_mx",
"gring_row",
"gring_valK",
"linearZ",
"linear_sum",
"mul1g",
"mulmx_sum_row",
"mxvecK",
"rowK",
"row_sum_delta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_op | := gring_mx rG \o gring_row. | Definition | gring_op | 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",
... | [
"gring_mx",
"gring_row",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_opE a : gring_op a = gring_mx rG (gring_row a). | Proof. by []. Qed. | Lemma | gring_opE | 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",
... | [
"gring_mx",
"gring_op",
"gring_row",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_opG x : x \in G -> gring_op (aG x) = rG x. | Proof.
move=> Gx; rewrite gring_opE /gring_row rowK gring_indexK // mul1g.
by rewrite /gring_mx /= -rowE rowK mxvecK gring_indexK.
Qed. | Lemma | gring_opG | 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",
... | [
"aG",
"gring_indexK",
"gring_mx",
"gring_op",
"gring_opE",
"gring_row",
"mul1g",
"mxvecK",
"rG",
"rowE",
"rowK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_op1 : gring_op 1%:M = 1%:M. | Proof. by rewrite -(repr_mx1 aG) gring_opG ?repr_mx1. Qed. | Lemma | gring_op1 | 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",
... | [
"aG",
"gring_op",
"gring_opG",
"repr_mx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_opJ A b :
gring_op (A *m gring_mx aG b) = gring_op A *m gring_mx rG b. | Proof.
rewrite /gring_mx /= ![b *m _]mulmx_sum_row !linear_sum.
apply: eq_bigr => i _; rewrite !linearZ /= !rowK !mxvecK.
by rewrite gring_opE gring_row_mul gring_mxJ ?enum_valP.
Qed. | Lemma | gring_opJ | 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",
... | [
"aG",
"apply",
"enum_valP",
"eq_bigr",
"gring_mx",
"gring_mxJ",
"gring_op",
"gring_opE",
"gring_row_mul",
"linearZ",
"linear_sum",
"mulmx_sum_row",
"mxvecK",
"rG",
"rowK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_op_mx b : gring_op (gring_mx aG b) = gring_mx rG b. | Proof. by rewrite -[_ b]mul1mx gring_opJ gring_op1 mul1mx. Qed. | Lemma | gring_op_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",
... | [
"aG",
"gring_mx",
"gring_op",
"gring_op1",
"gring_opJ",
"mul1mx",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_mxA a b :
gring_mx rG (a *m gring_mx aG b) = gring_mx rG a *m gring_mx rG b. | Proof.
by rewrite -(gring_op_mx a) -gring_opJ gring_opE gring_row_mul gring_mxK.
Qed. | Lemma | gring_mxA | 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",
... | [
"aG",
"gring_mx",
"gring_mxK",
"gring_opE",
"gring_opJ",
"gring_op_mx",
"gring_row_mul",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A ^f" | := (map_mx (GRing.RMorphism.sort f) A) : ring_scope. | Notation | A ^f | 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",
... | [
"map_mx",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_regular_mx x : (regular_mx aR G x)^f = regular_mx rR G x. | Proof. by apply/matrixP=> i j; rewrite !mxE rmorph_nat. Qed. | Lemma | map_regular_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",
"matrixP",
"mxE",
"regular_mx",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_gring_row (A : 'M_#|G|) : (gring_row A)^f = gring_row A^f. | Proof. by rewrite map_row. Qed. | Lemma | map_gring_row | 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",
... | [
"gring_row",
"map_row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_gring_proj x (A : 'M_#|G|) : (gring_proj x A)^f = gring_proj x A^f. | Proof. by rewrite map_row -map_trmx map_gring_row. Qed. | Lemma | map_gring_proj | 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",
... | [
"gring_proj",
"map_gring_row",
"map_row",
"map_trmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_repr_mx (f0 : aR -> rR) rG0 (g : gT) : 'M_n | := map_mx f0 (rG0 g). | Definition | map_repr_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",
... | [
"gT",
"map_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_repr : mx_repr G (map_repr_mx f rG). | Proof.
split=> [|x y Gx Gy]; first by rewrite /map_repr_mx repr_mx1 map_mx1.
by rewrite -map_mxM -repr_mxM.
Qed. | Lemma | map_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",
... | [
"map_mx1",
"map_mxM",
"map_repr_mx",
"mx_repr",
"rG",
"repr_mx1",
"repr_mxM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_repr | := MxRepresentation map_mx_repr. | Canonical | map_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",
... | [
"map_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGf | := map_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",
... | [
"map_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_reprE x : rGf x = (rG x)^f. | Proof. by []. Qed. | Lemma | map_reprE | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_reprJ m (A : 'M_(m, n)) x : (A *m rG x)^f = A^f *m rGf x. | Proof. exact: map_mxM. Qed. | Lemma | map_reprJ | 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",
... | [
"map_mxM",
"rG",
"rGf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_enveloping_algebra_mx :
(enveloping_algebra_mx rG)^f = enveloping_algebra_mx rGf. | Proof. by apply/row_matrixP=> i; rewrite -map_row !rowK map_mxvec. Qed. | Lemma | map_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",
... | [
"apply",
"enveloping_algebra_mx",
"map_mxvec",
"map_row",
"rG",
"rGf",
"rowK",
"row_matrixP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_gring_mx a : (gring_mx rG a)^f = gring_mx rGf a^f. | Proof. by rewrite map_vec_mx map_mxM map_enveloping_algebra_mx. Qed. | Lemma | map_gring_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",
... | [
"gring_mx",
"map_enveloping_algebra_mx",
"map_mxM",
"map_vec_mx",
"rG",
"rGf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_gring_op A : (gring_op rG A)^f = gring_op rGf A^f. | Proof. by rewrite map_gring_mx map_gring_row. Qed. | Lemma | map_gring_op | 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",
... | [
"gring_op",
"map_gring_mx",
"map_gring_row",
"rG",
"rGf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_regular_repr : map_repr (regular_repr aR G) =1 regular_repr rR G. | Proof. exact: map_regular_mx. Qed. | Lemma | map_regular_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",
... | [
"map_regular_mx",
"map_repr",
"regular_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_group_ring : (group_ring aR G)^f = group_ring rR G. | Proof.
rewrite map_enveloping_algebra_mx; apply/row_matrixP=> i.
by rewrite !rowK map_regular_repr.
Qed. | Lemma | map_group_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",
"group_ring",
"map_enveloping_algebra_mx",
"map_regular_repr",
"rowK",
"row_matrixP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
E_G | := (enveloping_algebra_mx rG). | Notation | E_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",
... | [
"enveloping_algebra_mx",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_mx_free x : x \in G -> row_free (rG x). | Proof. by move=> Gx; rewrite row_free_unit repr_mx_unit. Qed. | Lemma | repr_mx_free | 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",
"row_free",
"row_free_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs | := [set x in G | U *m rG x <= U]%MS. | Definition | rstabs | 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 | |
rstabs_sub : rstabs \subset G. | Proof. by apply/subsetP=> x /setIdP[]. Qed. | Lemma | rstabs_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",
"rstabs",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_group_set : group_set rstabs. | Proof.
apply/group_setP; rewrite inE group1 repr_mx1 mulmx1.
split=> //= x y /setIdP[Gx nUx] /setIdP[Gy]; rewrite inE repr_mxM ?groupM //.
by apply: submx_trans; rewrite mulmxA submxMr.
Qed. | Lemma | rstabs_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",
"rstabs",
"setIdP",
"split",
"submxMr",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_group | := Group rstabs_group_set. | Canonical | rstabs_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",
... | [
"rstabs_group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_act x m1 (W : 'M_(m1, n)) :
x \in rstab rG U -> (W <= U)%MS -> W *m rG x = W. | Proof. by case/setIdP=> _ /eqP cUx /submxP[w ->]; rewrite -mulmxA cUx. Qed. | Lemma | rstab_act | 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",
"rG",
"rstab",
"setIdP",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_act x m1 (W : 'M_(m1, n)) :
x \in rstabs -> (W <= U)%MS -> (W *m rG x <= U)%MS. | Proof.
by case/setIdP=> [_ nUx] sWU; apply: submx_trans nUx; apply: submxMr.
Qed. | Lemma | rstabs_act | 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",
"rG",
"rstabs",
"setIdP",
"submxMr",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule | := G \subset rstabs. | Definition | mxmodule | 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",
... | [
"rstabs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmoduleP : reflect {in G, forall x, U *m rG x <= U}%MS mxmodule. | Proof.
by apply: (iffP subsetP) => modU x Gx; have:= modU x Gx; rewrite !inE ?Gx.
Qed. | Lemma | mxmoduleP | 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",
"mxmodule",
"rG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabS m1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U <= V)%MS -> rstab rG V \subset rstab rG U. | Proof.
case/submxP=> u ->; apply/subsetP=> x.
by rewrite !inE => /andP[-> /= /eqP cVx]; rewrite -mulmxA cVx.
Qed. | Lemma | rstabS | 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",
"mulmxA",
"rG",
"rstab",
"submxP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_rstab m1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U :=: V)%MS -> rstab rG U = rstab rG V. | Proof. by move=> eqUV; apply/eqP; rewrite eqEsubset !rstabS ?eqUV. Qed. | Lemma | eqmx_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",
... | [
"apply",
"eqEsubset",
"rG",
"rstab",
"rstabS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_rstabs m1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U :=: V)%MS -> rstabs U = rstabs V. | Proof. by move=> eqUV; apply/setP=> x; rewrite !inE eqUV (eqmxMr _ eqUV). Qed. | Lemma | eqmx_rstabs | 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",
"eqmxMr",
"inE",
"rstabs",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_module m1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U :=: V)%MS -> mxmodule U = mxmodule V. | Proof. by move=> eqUV; rewrite /mxmodule (eqmx_rstabs eqUV). Qed. | Lemma | eqmx_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",
... | [
"eqmx_rstabs",
"mxmodule"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule0 m : mxmodule (0 : 'M_(m, n)). | Proof. by apply/mxmoduleP=> x _; rewrite mul0mx. Qed. | Lemma | mxmodule0 | 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",
"mul0mx",
"mxmodule",
"mxmoduleP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule1 : mxmodule 1%:M. | Proof. by apply/mxmoduleP=> x _; rewrite submx1. Qed. | Lemma | mxmodule1 | 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",
"mxmoduleP",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_trans m1 m2 (U : 'M_(m1, n)) (W : 'M_(m2, n)) x :
mxmodule U -> x \in G -> (W <= U -> W *m rG x <= U)%MS. | Proof.
by move=> modU Gx sWU; apply: submx_trans (mxmoduleP modU x Gx); apply: submxMr.
Qed. | Lemma | mxmodule_trans | 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",
"mxmoduleP",
"rG",
"submxMr",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_eigenvector m (U : 'M_(m, n)) :
mxmodule U -> \rank U = 1 ->
{u : 'rV_n & {a | (U :=: u)%MS & {in G, forall x, u *m rG x = a x *: u}}}. | Proof.
move=> modU linU; set u := nz_row U; exists u.
have defU: (U :=: u)%MS.
apply/eqmxP; rewrite andbC -(geq_leqif (mxrank_leqif_eq _)) ?nz_row_sub //.
by rewrite linU lt0n mxrank_eq0 nz_row_eq0 -mxrank_eq0 linU.
pose a x := (u *m rG x *m pinvmx u) 0 0; exists a => // x Gx.
by rewrite -mul_scalar_mx -mx11_scalar... | Lemma | mxmodule_eigenvector | 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",
"defU",
"eqmxP",
"geq_leqif",
"lt0n",
"mul_scalar_mx",
"mulmxKpV",
"mx11_scalar",
"mxmodule",
"mxmodule_trans",
"mxrank_eq0",
"mxrank_leqif_eq",
"nz_row",
"nz_row_eq0",
"nz_row_sub",
"pinvmx",
"rG",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_module m1 m2 U V :
@mxmodule m1 U -> @mxmodule m2 V -> mxmodule (U + V)%MS. | Proof.
move=> modU modV; apply/mxmoduleP=> x Gx.
by rewrite addsmxMr addsmxS ?(mxmoduleP _ x Gx).
Qed. | Lemma | addsmx_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",
... | [
"addsmxMr",
"addsmxS",
"apply",
"mxmodule",
"mxmoduleP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumsmx_module I r (P : pred I) U :
(forall i, P i -> mxmodule (U i)) -> mxmodule (\sum_(i <- r | P i) U i)%MS. | Proof.
by move=> modU; elim/big_ind: _; [apply: mxmodule0 | apply: addsmx_module | ].
Qed. | Lemma | sumsmx_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",
... | [
"addsmx_module",
"apply",
"big_ind",
"mxmodule",
"mxmodule0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_module m1 m2 U V :
@mxmodule m1 U -> @mxmodule m2 V -> mxmodule (U :&: V)%MS. | Proof.
move=> modU modV; apply/mxmoduleP=> x Gx.
by rewrite sub_capmx !mxmodule_trans ?capmxSl ?capmxSr.
Qed. | Lemma | capmx_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",
"capmxSl",
"capmxSr",
"mxmodule",
"mxmoduleP",
"mxmodule_trans",
"sub_capmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcapmx_module I r (P : pred I) U :
(forall i, P i -> mxmodule (U i)) -> mxmodule (\bigcap_(i <- r | P i) U i)%MS. | Proof.
by move=> modU; elim/big_ind: _; [apply: mxmodule1 | apply: capmx_module | ].
Qed. | Lemma | bigcapmx_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",
"big_ind",
"capmx_module",
"mxmodule",
"mxmodule1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submod m : 'M_(m, \rank U) -> 'M_(m, n) | := mulmxr (row_base U). | Definition | val_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",
... | [
"mulmxr",
"rank",
"row_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_submod m : 'M_(m, n) -> 'M_(m, \rank U) | :=
mulmxr (invmx (row_ebase U) *m pid_mx (\rank U)). | Definition | in_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",
... | [
"invmx",
"mulmxr",
"pid_mx",
"rank",
"row_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submodE m W : @val_submod m W = W *m val_submod 1%:M. | Proof. by rewrite mulmxA mulmx1. Qed. | Lemma | val_submodE | 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",
"val_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_submodE m W : @in_submod m W = W *m in_submod 1%:M. | Proof. by rewrite mulmxA mulmx1. Qed. | Lemma | in_submodE | 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",
"mulmx1",
"mulmxA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submod1 : (val_submod 1%:M :=: U)%MS. | Proof. by rewrite /val_submod /= mul1mx; apply: eq_row_base. Qed. | Lemma | val_submod1 | 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_row_base",
"mul1mx",
"val_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submodP m W : (@val_submod m W <= U)%MS. | Proof. by rewrite mulmx_sub ?eq_row_base. Qed. | Lemma | val_submodP | 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",
... | [
"eq_row_base",
"mulmx_sub",
"val_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submodK m : cancel (@val_submod m) (@in_submod m). | Proof.
move=> W; rewrite /in_submod /= -!mulmxA mulKVmx ?row_ebase_unit //.
by rewrite pid_mx_id ?rank_leq_row // pid_mx_1 mulmx1.
Qed. | Lemma | val_submodK | 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",
"mulKVmx",
"mulmx1",
"mulmxA",
"pid_mx_1",
"pid_mx_id",
"rank_leq_row",
"row_ebase_unit",
"val_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submod_inj m : injective (@val_submod m). | Proof. exact: can_inj (@val_submodK m). Qed. | Lemma | val_submod_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",
... | [
"val_submod",
"val_submodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submodS m1 m2 (V : 'M_(m1, \rank U)) (W : 'M_(m2, \rank U)) :
(val_submod V <= val_submod W)%MS = (V <= W)%MS. | Proof.
apply/idP/idP=> sVW; last exact: submxMr.
by rewrite -[V]val_submodK -[W]val_submodK submxMr.
Qed. | Lemma | val_submodS | 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",
"last",
"rank",
"submxMr",
"val_submod",
"val_submodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_submodK m W : (W <= U)%MS -> val_submod (@in_submod m W) = W. | Proof.
case/submxP=> w ->; rewrite /val_submod /= -!mulmxA.
congr (_ *m _); rewrite -{1}[U]mulmx_ebase !mulmxA mulmxK ?row_ebase_unit //.
by rewrite -2!(mulmxA (col_ebase U)) !pid_mx_id ?rank_leq_row // mulmx_ebase.
Qed. | Lemma | in_submodK | 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",
... | [
"col_ebase",
"in_submod",
"mulmxA",
"mulmxK",
"mulmx_ebase",
"pid_mx_id",
"rank_leq_row",
"row_ebase_unit",
"submxP",
"val_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submod_eq0 m W : (@val_submod m W == 0) = (W == 0). | Proof. by rewrite -!submx0 -val_submodS linear0 !(submx0, eqmx0). Qed. | Lemma | val_submod_eq0 | 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",
... | [
"eqmx0",
"linear0",
"submx0",
"val_submod",
"val_submodS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_submod_eq0 m W : (@in_submod m W == 0) = (W <= U^C)%MS. | Proof.
apply/eqP/submxP=> [W_U0 | [w ->{W}]].
exists (W *m invmx (row_ebase U)).
rewrite mulmxA mulmxBr mulmx1 -(pid_mx_id _ _ _ (leqnn _)).
rewrite mulmxA -(mulmxA W) [W *m (_ *m _)]W_U0 mul0mx subr0.
by rewrite mulmxKV ?row_ebase_unit.
rewrite /in_submod /= -!mulmxA mulKVmx ?row_ebase_unit //.
by rewrite mul_... | Lemma | in_submod_eq0 | 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",
"in_submod",
"invmx",
"leqnn",
"mul0mx",
"mulKVmx",
"mul_copid_mx_pid",
"mulmx0",
"mulmx1",
"mulmxA",
"mulmxBr",
"mulmxKV",
"pid_mx_id",
"rank_leq_row",
"row_ebase",
"row_ebase_unit",
"submxP",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_in_submod m (W : 'M_(m, n)) :
(W <= U)%MS -> \rank (in_submod W) = \rank W. | Proof.
by move=> sWU; apply/eqP; rewrite eqn_leq -{3}(in_submodK sWU) !mxrankM_maxl.
Qed. | Lemma | mxrank_in_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",
"eqn_leq",
"in_submod",
"in_submodK",
"mxrankM_maxl",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmod m : _ -> 'M_(m, n) | :=
mulmxr (row_base (cokermx U) *m row_ebase U). | Definition | val_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",
... | [
"cokermx",
"mulmxr",
"row_base",
"row_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_factmod m : 'M_(m, n) -> _ | := mulmxr (col_base (cokermx U)). | Definition | in_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",
... | [
"cokermx",
"col_base",
"mulmxr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmodE m W : @val_factmod m W = W *m val_factmod 1%:M. | Proof. by rewrite mulmxA mulmx1. Qed. | Lemma | val_factmodE | 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",
"val_factmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_factmodE m W : @in_factmod m W = W *m in_factmod 1%:M. | Proof. by rewrite mulmxA mulmx1. Qed. | Lemma | in_factmodE | 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_factmod",
"mulmx1",
"mulmxA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmodP m W : (@val_factmod m W <= U^C)%MS. | Proof.
by rewrite mulmx_sub {m W}// (eqmxMr _ (eq_row_base _)) -mulmxA submxMl.
Qed. | Lemma | val_factmodP | 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",
... | [
"eq_row_base",
"eqmxMr",
"mulmxA",
"mulmx_sub",
"submxMl",
"val_factmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmodK m : cancel (@val_factmod m) (@in_factmod m). | Proof.
move=> W /=; rewrite /in_factmod /=; set Uc := cokermx U.
apply: (row_free_inj (row_base_free Uc)); rewrite -mulmxA mulmx_base.
rewrite /val_factmod /= 2!mulmxA -/Uc mulmxK ?row_ebase_unit //.
have /submxP[u ->]: (row_base Uc <= Uc)%MS by rewrite eq_row_base.
by rewrite -!mulmxA copid_mx_id ?rank_leq_row.
Qed. | Lemma | val_factmodK | 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",
"cokermx",
"copid_mx_id",
"eq_row_base",
"in_factmod",
"mulmxA",
"mulmxK",
"mulmx_base",
"rank_leq_row",
"row_base",
"row_base_free",
"row_ebase_unit",
"row_free_inj",
"submxP",
"val_factmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmod_inj m : injective (@val_factmod m). | Proof. exact: can_inj (@val_factmodK m). Qed. | Lemma | val_factmod_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",
... | [
"val_factmod",
"val_factmodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmodS m1 m2 (V : 'M_(m1, _)) (W : 'M_(m2, _)) :
(val_factmod V <= val_factmod W)%MS = (V <= W)%MS. | Proof.
apply/idP/idP=> sVW; last exact: submxMr.
by rewrite -[V]val_factmodK -[W]val_factmodK submxMr.
Qed. | Lemma | val_factmodS | 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",
"last",
"submxMr",
"val_factmod",
"val_factmodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmod_eq0 m W : (@val_factmod m W == 0) = (W == 0). | Proof. by rewrite -!submx0 -val_factmodS linear0 !(submx0, eqmx0). Qed. | Lemma | val_factmod_eq0 | 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",
... | [
"eqmx0",
"linear0",
"submx0",
"val_factmod",
"val_factmodS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_factmod_eq0 m (W : 'M_(m, n)) : (in_factmod W == 0) = (W <= U)%MS. | Proof.
rewrite submxE -!mxrank_eq0 -{2}[_ U]mulmx_base mulmxA.
by rewrite (mxrankMfree _ (row_base_free _)).
Qed. | Lemma | in_factmod_eq0 | 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_factmod",
"mulmxA",
"mulmx_base",
"mxrankMfree",
"mxrank_eq0",
"row_base_free",
"submxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_factmodK m (W : 'M_(m, n)) :
(W <= U^C)%MS -> val_factmod (in_factmod W) = W. | Proof.
case/submxP=> w ->{W}; rewrite /val_factmod /= -2!mulmxA.
congr (_ *m _); rewrite (mulmxA (col_base _)) mulmx_base -2!mulmxA.
by rewrite mulKVmx ?row_ebase_unit // mulmxA copid_mx_id ?rank_leq_row.
Qed. | Lemma | in_factmodK | 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",
... | [
"col_base",
"copid_mx_id",
"in_factmod",
"mulKVmx",
"mulmxA",
"mulmx_base",
"rank_leq_row",
"row_ebase_unit",
"submxP",
"val_factmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_factmod_addsK m (W : 'M_(m, n)) :
(in_factmod (U + W)%MS :=: in_factmod W)%MS. | Proof.
apply: eqmx_trans (addsmxMr _ _ _) _.
by rewrite ((_ *m _ =P 0) _) ?in_factmod_eq0 //; apply: adds0mx.
Qed. | Lemma | in_factmod_addsK | 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",
... | [
"adds0mx",
"addsmxMr",
"apply",
"eqmx_trans",
"in_factmod",
"in_factmod_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_sub_fact_mod m (W : 'M_(m, n)) :
val_submod (in_submod W) + val_factmod (in_factmod W) = W. | Proof.
rewrite /val_submod /val_factmod /= -!mulmxA -mulmxDr.
rewrite addrC ) pid_mx_id //.
rewrite (mulmxA (col_ebase _)) (mulmxA _ _ (row_ebase _)) mulmx_ebase.
rewrite (mulmxA (pid_mx _)) pid_mx_id // mulmxA -mulmxDl -mulmxDr.
by rewrite subrK mulmx1 mulmxA mulmxKV ?row_ebase_unit.
... | Lemma | add_sub_fact_mod | 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",
... | [
"addrC",
"col_ebase",
"in_factmod",
"in_submod",
"mulmx1",
"mulmxA",
"mulmxDl",
"mulmxDr",
"mulmxKV",
"mulmx_ebase",
"pid_mx",
"pid_mx_id",
"row_ebase",
"row_ebase_unit",
"subrK",
"val_factmod",
"val_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_factmodS m (W : 'M_(m, n)) :
(val_factmod (in_factmod W) <= U + W)%MS. | Proof.
by rewrite -{2}[W]add_sub_fact_mod addsmx_addKl ?val_submodP ?addsmxSr.
Qed. | Lemma | proj_factmodS | 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",
... | [
"add_sub_fact_mod",
"addsmxSr",
"addsmx_addKl",
"in_factmod",
"val_factmod",
"val_submodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_factmodsK m (W : 'M_(m, n)) :
(U <= W)%MS -> (U + val_factmod (in_factmod W) :=: W)%MS. | Proof.
move/addsmx_idPr; apply: eqmx_trans (eqmx_sym _).
by rewrite -{1}[W]add_sub_fact_mod; apply: addsmx_addKl; apply: val_submodP.
Qed. | Lemma | in_factmodsK | 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",
... | [
"add_sub_fact_mod",
"addsmx_addKl",
"addsmx_idPr",
"apply",
"eqmx_sym",
"eqmx_trans",
"in_factmod",
"val_factmod",
"val_submodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_in_factmod m (W : 'M_(m, n)) :
(\rank (in_factmod W) + \rank U)%N = \rank (U + W). | Proof.
rewrite -in_factmod_addsK in_factmodE; set fU := in_factmod 1%:M.
suffices <-: ((U + W) :&: kermx fU :=: U)%MS by rewrite mxrank_mul_ker.
apply: eqmx_trans (capmx_idPr (addsmxSl U W)).
apply: cap_eqmx => //; apply/eqmxP/rV_eqP => u.
by rewrite (sameP sub_kermxP eqP) -in_factmodE in_factmod_eq0.
Qed. | Lemma | mxrank_in_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",
... | [
"addsmxSl",
"apply",
"cap_eqmx",
"capmx_idPr",
"eqmxP",
"eqmx_trans",
"in_factmod",
"in_factmodE",
"in_factmod_addsK",
"in_factmod_eq0",
"kermx",
"mxrank_mul_ker",
"rV_eqP",
"rank",
"sub_kermxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submod_mx & mxmodule U | :=
fun x => in_submod (val_submod 1%:M *m rG x). | Definition | submod_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",
... | [
"in_submod",
"mxmodule",
"rG",
"val_submod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factmod_mx & mxmodule U | :=
fun x => in_factmod (val_factmod 1%:M *m rG x). | Definition | factmod_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",
... | [
"in_factmod",
"mxmodule",
"rG",
"val_factmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Umod : mxmodule U. | Hypothesis | Umod | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
in_submodJ m (W : 'M_(m, n)) x :
(W <= U)%MS -> in_submod (W *m rG x) = in_submod W *m submod_mx Umod x. | Proof.
move=> sWU; rewrite mulmxA; congr (in_submod _).
by rewrite mulmxA -val_submodE in_submodK.
Qed. | Lemma | in_submodJ | 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",
"in_submod",
"in_submodK",
"mulmxA",
"rG",
"submod_mx",
"val_submodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_submodJ m (W : 'M_(m, \rank U)) x :
x \in G -> val_submod (W *m submod_mx Umod x) = val_submod W *m rG x. | Proof.
move=> Gx; rewrite 2!(mulmxA W) -val_submodE in_submodK //.
by rewrite mxmodule_trans ?val_submodP.
Qed. | Lemma | val_submodJ | 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",
"in_submodK",
"mulmxA",
"mxmodule_trans",
"rG",
"rank",
"submod_mx",
"val_submod",
"val_submodE",
"val_submodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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