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 |
|---|---|---|---|---|---|---|---|---|---|---|
splitting_cyclic_primitive_root_pchar :
cyclic G -> [pchar F]^'.-group G -> group_splitting_field G ->
classically {z : F | #|G|.-primitive_root z}. | Proof.
case/cyclicP=> x defG F'G splitF; case=> // IH.
wlog sG: / irrType G by apply: socle_exists.
have [w prim_w _] := cycle_repr_structure_pchar sG defG F'G splitF.
by apply: IH; exists w.
Qed. | Lemma | splitting_cyclic_primitive_root_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"F'G",
"apply",
"cycle_repr_structure_pchar",
"cyclic",
"cyclicP",
"defG",
"group",
"group_splitting_field",
"irrType",
"pchar",
"sG",
"socle_exists",
"splitF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_Maschke | := (mx_Maschke_pchar) (only parsing). | Notation | mx_Maschke | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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_Maschke_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_regular_submod | := (rsim_regular_submod_pchar) (only parsing). | Notation | rsim_regular_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",
... | [
"rsim_regular_submod_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_mx_sum | := (irr_mx_sum_pchar) (only parsing). | Notation | irr_mx_sum | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_mx_sum_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_sum | := (Wedderburn_sum_pchar) (only parsing). | Notation | Wedderburn_sum | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_sum_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_sum_id | := (Wedderburn_sum_id_pchar) (only parsing). | Notation | Wedderburn_sum_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_sum_id_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_is_id | := (Wedderburn_is_id_pchar) (only parsing). | Notation | Wedderburn_is_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_is_id_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_closed | := (Wedderburn_closed_pchar) (only parsing). | Notation | Wedderburn_closed | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_closed_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_is_ring | := (Wedderburn_is_ring_pchar) (only parsing). | Notation | Wedderburn_is_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",
... | [
"Wedderburn_is_ring_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_min_ideal | := (Wedderburn_min_ideal_pchar) (only parsing). | Notation | Wedderburn_min_ideal | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_min_ideal_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_irr_comp | := (rsim_irr_comp_pchar) (only parsing). | Notation | rsim_irr_comp | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rsim_irr_comp_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp'_op0 | := (irr_comp'_op0_pchar) (only parsing). | Notation | irr_comp'_op0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_comp'_op0_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp_envelop | := (irr_comp_envelop_pchar) (only parsing). | Notation | irr_comp_envelop | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_comp_envelop_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_irr_comp_op | := (ker_irr_comp_op_pchar) (only parsing). | Notation | ker_irr_comp_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",
... | [
"ker_irr_comp_op_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
regular_op_inj | := (regular_op_inj_pchar) (only parsing). | Notation | regular_op_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",
... | [
"regular_op_inj_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_irr_comp | := (rank_irr_comp_pchar) (only parsing). | Notation | rank_irr_comp | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rank_irr_comp_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp_rsim | := (irr_comp_rsim_pchar) (only parsing). | Notation | irr_comp_rsim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_comp_rsim_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_reprK | := (irr_reprK_pchar) (only parsing). | Notation | irr_reprK | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_reprK_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_repr'_op0 | := (irr_repr'_op0_pchar) (only parsing). | Notation | irr_repr'_op0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_repr'_op0_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
op_Wedderburn_id | := (op_Wedderburn_id_pchar) (only parsing). | Notation | op_Wedderburn_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"op_Wedderburn_id_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp_id | := (irr_comp_id_pchar) (only parsing). | Notation | irr_comp_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_comp_id_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_Wedderburn_subring | := (rank_Wedderburn_subring_pchar)
(only parsing). | Notation | rank_Wedderburn_subring | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rank_Wedderburn_subring_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_irr_degree | := (sum_irr_degree_pchar) (only parsing). | Notation | sum_irr_degree | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"sum_irr_degree_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_mx_mult | := (irr_mx_mult_pchar) (only parsing). | Notation | irr_mx_mult | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_mx_mult_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_regular | := (mxtrace_regular_pchar) (only parsing). | Notation | mxtrace_regular | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxtrace_regular_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_irr_comp | := (linear_irr_comp_pchar) (only parsing). | Notation | linear_irr_comp | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"linear_irr_comp_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_subring_center | := (Wedderburn_subring_center_pchar)
(only parsing). | Notation | Wedderburn_subring_center | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_subring_center_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_center | := (Wedderburn_center_pchar) (only parsing). | Notation | Wedderburn_center | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_center_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_irr | := (card_irr_pchar) (only parsing). | Notation | card_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"card_irr_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_repr_structure | := (cycle_repr_structure_pchar) (only parsing). | Notation | cycle_repr_structure | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cycle_repr_structure_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splitting_cyclic_primitive_root | :=
(splitting_cyclic_primitive_root_pchar) (only parsing). | Notation | splitting_cyclic_primitive_root | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"splitting_cyclic_primitive_root_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'Cl" | := (Clifford_action _) : action_scope. | Notation | 'Cl | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''n_' i" | := (irr_degree i) : group_ring_scope. | Notation | ''n_' i | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_degree"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
term | := GRing.term. | Notation | term | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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 | |
mxmodule_form (U : 'M[term F]_n) | :=
\big[And/True]_(x in G) submx_form (mulmx_term U (mx_term (rG x))) U. | Definition | mxmodule_form | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"And",
"True",
"mulmx_term",
"mx_term",
"rG",
"submx_form",
"term"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_form_qf U : qf_form (mxmodule_form U). | Proof.
by rewrite (morphAnd (@qf_form _)) ?big1 //= => x _; rewrite submx_form_qf.
Qed. | Lemma | mxmodule_form_qf | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"big1",
"morphAnd",
"mxmodule_form",
"qf_form",
"submx_form_qf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_mxmodule U e :
qf_eval e (mxmodule_form U) = mxmodule rG (eval_mx e U). | Proof.
rewrite (morphAnd (qf_eval e)) //= big_andE /=.
apply/forallP/mxmoduleP=> Umod x; move/implyP: (Umod x);
by rewrite eval_submx eval_mulmx eval_mx_term.
Qed. | Lemma | eval_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",
... | [
"Umod",
"apply",
"big_andE",
"eval_mulmx",
"eval_mx",
"eval_mx_term",
"eval_submx",
"forallP",
"morphAnd",
"mxmodule",
"mxmoduleP",
"mxmodule_form",
"qf_eval",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxnonsimple_form (U : 'M[term F]_n) | :=
let V := vec_mx (row_var F (n * n) 0) in
let nzV := (~ mxrank_form 0 V)%T in
let properVU := (submx_form V U /\ ~ submx_form U V)%T in
(Exists_row_form (n * n) 0 (mxmodule_form V /\ nzV /\ properVU))%T. | Definition | mxnonsimple_form | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Exists_row_form",
"mxmodule_form",
"mxrank_form",
"row_var",
"submx_form",
"term",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxnonsimple_sat U | :=
GRing.sat (@row_env _ (n * n) [::]) (mxnonsimple_form rG (mx_term U)). | Definition | mxnonsimple_sat | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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_term",
"mxnonsimple_form",
"rG",
"row_env",
"sat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxnonsimpleP U :
U != 0 -> reflect (mxnonsimple rG U) (mxnonsimple_sat U). | Proof.
rewrite /mxnonsimple_sat {1}/mxnonsimple_form; set Vt := vec_mx _ => /= nzU.
pose nsim V := [&& mxmodule rG V, (V <= U)%MS, V != 0 & \rank V < \rank U].
set nsimUt := (_ /\ _)%T; have: qf_form nsimUt.
by rewrite /= mxmodule_form_qf !mxrank_form_qf !submx_form_qf.
move/GRing.qf_evalP; set qev := @GRing.qf_eval ... | Lemma | mxnonsimpleP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Exists_rowP",
"apply",
"eval_mx_term",
"eval_mxmodule",
"eval_mxrank",
"eval_row_var",
"eval_submx",
"eval_vec_mx",
"ltn_neqAle",
"muln_gt0",
"mxmodule",
"mxmodule_form_qf",
"mxnonsimple",
"mxnonsimple_form",
"mxnonsimple_sat",
"mxrank_eq0",
"mxrank_form_qf",
"mxrank_leqif_sup",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dec_mxsimple_exists (U : 'M_n) :
mxmodule rG U -> U != 0 -> {V | mxsimple rG V & V <= U}%MS. | Proof.
have [m] := ubnP (\rank U); elim: m U => // m IHm U leUm modU nzU.
have [nsimU | simU] := mxnonsimpleP nzU; last first.
by exists U; first apply/mxsimpleP.
move: (xchooseP nsimU); move: (xchoose _) => W /and4P[modW sWU nzW ltWU].
case: (IHm W) => // [|V simV sVW]; first exact: leq_trans ltWU _.
by exists V; la... | Lemma | dec_mxsimple_exists | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"last",
"leq_trans",
"mxmodule",
"mxnonsimpleP",
"mxsimple",
"mxsimpleP",
"rG",
"rank",
"simU",
"submx_trans",
"ubnP",
"xchoose",
"xchooseP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dec_mx_reducible_semisimple U :
mxmodule rG U -> mx_completely_reducible rG U -> mxsemisimple rG U. | Proof.
have [m] := ubnP (\rank U); elim: m U => // m IHm U leUm modU redU.
have [U0 | nzU] := eqVneq U 0.
have{} U0: (\sum_(i < 0) 0 :=: U)%MS by rewrite big_ord0 U0.
by apply: (intro_mxsemisimple U0); case.
have [V simV sVU] := dec_mxsimple_exists modU nzU; have [modV nzV _] := simV.
have [W modW defVW dxVW] := re... | Lemma | dec_mx_reducible_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"add1n",
"addsmxS",
"addsmxSr",
"apply",
"bigD1",
"big_ord0",
"big_pred0",
"dec_mxsimple_exists",
"defU",
"eqVneq",
"eqmxP",
"eqmx_trans",
"i0",
"intro_mxsemisimple",
"leq_add2r",
"leq_trans",
"lt0n",
"ltnS",
"mx_completely_reducible",
"mx_reducibleS",
"mxdirectP",
"mxmodul... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
DecSocleType : socleType rG. | Proof.
have [n0 | n_gt0] := posnP n.
by exists [::] => // M [_]; rewrite -mxrank_eq0 -leqn0 -n0 rank_leq_row.
have n2_gt0: n ^ 2 > 0 by rewrite muln_gt0 n_gt0.
pose span Ms := (\sum_(M <- Ms) component_mx rG M)%MS.
have: {in [::], forall M, mxsimple rG M} by [].
have [m] := ubnP (n - \rank (span [::])).
elim: m [::] ... | Lemma | DecSocleType | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Exists_rowP",
"Exists_row_form",
"PackSocleK",
"addsmxSr",
"addsmx_sub",
"apply",
"big_cons",
"big_mkord",
"big_nth",
"component_mx",
"component_mx_id",
"component_mx_iso",
"component_mx_module",
"component_socle",
"cyclic_mx",
"cyclic_mx_eq0",
"cyclic_mx_module",
"cyclic_mx_sub",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGf | := (map_repr f rG). | Notation | rGf | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"map_repr",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_rfix_mx H : (rfix_mx rG H)^f = rfix_mx rGf H. | Proof.
rewrite map_kermx //; congr (kermx _); apply: map_lin1_mx => //= v.
rewrite map_mxvec map_mxM; congr (mxvec (_ *m _)); last first.
by apply: map_lin1_mx => //= u; rewrite map_mxM map_vec_mx.
by apply/row_matrixP=> i; rewrite -map_row !rowK map_mxvec map_mxB map_mx1.
Qed. | Lemma | map_rfix_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",
"kermx",
"last",
"map_kermx",
"map_lin1_mx",
"map_mx1",
"map_mxB",
"map_mxM",
"map_mxvec",
"map_row",
"map_vec_mx",
"mxvec",
"rG",
"rGf",
"rfix_mx",
"rowK",
"row_matrixP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcent_map A : rcent rGf A^f = rcent rG A. | Proof.
by apply/setP=> x; rewrite !inE -!map_mxM inj_eq //; apply: map_mx_inj.
Qed. | Lemma | rcent_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"inj_eq",
"map_mxM",
"map_mx_inj",
"rG",
"rGf",
"rcent",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_map m (U : 'M_(m, n)) : rstab rGf U^f = rstab rG U. | Proof.
by apply/setP=> x; rewrite !inE -!map_mxM inj_eq //; apply: map_mx_inj.
Qed. | Lemma | rstab_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"inj_eq",
"map_mxM",
"map_mx_inj",
"rG",
"rGf",
"rstab",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_map m (U : 'M_(m, n)) : rstabs rGf U^f = rstabs rG U. | Proof. by apply/setP=> x; rewrite !inE -!map_mxM ?map_submx. Qed. | Lemma | rstabs_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"map_mxM",
"map_submx",
"rG",
"rGf",
"rstabs",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
centgmx_map A : centgmx rGf A^f = centgmx rG A. | Proof. by rewrite /centgmx rcent_map. Qed. | Lemma | centgmx_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"centgmx",
"rG",
"rGf",
"rcent_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_map m (U : 'M_(m, n)) : mxmodule rGf U^f = mxmodule rG U. | Proof. by rewrite /mxmodule rstabs_map. Qed. | Lemma | mxmodule_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"rG",
"rGf",
"rstabs_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_map (U : 'M_n) : mxsimple rGf U^f -> mxsimple rG U. | Proof.
case; rewrite map_mx_eq0 // mxmodule_map // => modU nzU minU.
split=> // V modV sVU nzV; rewrite -(map_submx f).
by rewrite (minU V^f) //= ?mxmodule_map ?map_mx_eq0 // map_submx.
Qed. | Lemma | mxsimple_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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_eq0",
"map_submx",
"mxmodule_map",
"mxsimple",
"rG",
"rGf",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_irr_map : mx_irreducible rGf -> mx_irreducible rG. | Proof. by move=> irrGf; apply: mxsimple_map; rewrite map_mx1. Qed. | Lemma | mx_irr_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"map_mx1",
"mx_irreducible",
"mxsimple_map",
"rG",
"rGf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_map : rker rGf = rker rG. | Proof. by rewrite /rker -rstab_map map_mx1. Qed. | Lemma | rker_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"rG",
"rGf",
"rker",
"rstab_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_faithful : mx_faithful rGf = mx_faithful rG. | Proof. by rewrite /mx_faithful rker_map. Qed. | Lemma | map_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",
"rGf",
"rker_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_abs_irr :
mx_absolutely_irreducible rGf = mx_absolutely_irreducible rG. | Proof.
by rewrite /mx_absolutely_irreducible -map_enveloping_algebra_mx row_full_map.
Qed. | Lemma | map_mx_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"map_enveloping_algebra_mx",
"mx_absolutely_irreducible",
"rG",
"rGf",
"row_full_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_map n1 n2 rG1 rG2 :
@mx_rsim _ _ G n1 rG1 n2 rG2 -> mx_rsim (map_repr f rG1) (map_repr f rG2). | Proof.
case=> g eqn12 inj_g hom_g.
by exists g^f => // [|x Gx]; rewrite ?row_free_map // -!map_mxM ?hom_g.
Qed. | Lemma | mx_rsim_map | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"map_repr",
"mx_rsim",
"row_free_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_section_repr n (rG : mx_representation aF G n) rGf U V
(modU : mxmodule rG U) (modV : mxmodule rG V)
(modUf : mxmodule rGf U^f) (modVf : mxmodule rGf V^f) :
map_repr f rG =1 rGf ->
mx_rsim (map_repr f (section_repr modU modV)) (section_repr modUf modVf). | Proof.
move=> def_rGf; set VU := <<_>>%MS.
pose valUV := val_factmod (val_submod (1%:M : 'M[aF]_(\rank VU))).
have sUV_Uf: (valUV^f <= U^f + V^f)%MS.
rewrite -map_addsmx map_submx; apply: submx_trans (proj_factmodS _ _).
by rewrite val_factmodS val_submod1 genmxE.
exists (in_submod _ (in_factmod U^f valUV^f)) => [|... | Lemma | map_section_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",
"cokermx",
"eqmx0",
"genmxE",
"in_factmod",
"in_factmodE",
"in_factmodJ",
"in_factmod_addsK",
"in_factmod_eq0",
"in_submod",
"in_submodE",
"in_submodJ",
"in_submodK",
"kermx_eq0",
"linear0",
"linearB",
"map_addsmx",
"map_cokermx",
"map_col_base",
"map_mx0",
"map_mx1"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_regular_subseries U i (modU : mx_subseries (regular_repr aF G) U)
(modUf : mx_subseries (regular_repr rF G) [seq M^f | M <- U]) :
mx_rsim (map_repr f (subseries_repr i modU)) (subseries_repr i modUf). | Proof.
set mf := map _ in modUf *; rewrite /subseries_repr.
do 2!move: (mx_subseries_module' _ _) (mx_subseries_module _ _).
have mf_i V: nth 0^f (mf V) i = (V`_i)^f.
case: (ltnP i (size V)) => [ltiV | leVi]; first exact: nth_map.
by rewrite !nth_default ?size_map.
rewrite -(map_mx0 f) mf_i (mf_i (0 :: U)) => modUi... | Lemma | map_regular_subseries | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"ltnP",
"map",
"map_mx0",
"map_regular_repr",
"map_repr",
"map_section_repr",
"mx_rsim",
"mx_subseries",
"mx_subseries_module",
"mx_subseries_module'",
"nth",
"nth_default",
"nth_map",
"regular_repr",
"seq",
"size",
"size_map",
"subseries_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extend_group_splitting_field :
group_splitting_field aF G -> group_splitting_field rF G. | Proof.
move=> splitG n rG irrG.
have modU0: all ((mxmodule (regular_repr aF G)) #|G|) [::] by [].
apply: (mx_Schreier modU0 _) => // [[U [compU lastU _]]]; have [modU _]:= compU.
pose Uf := map (map_mx f) U.
have{lastU} lastUf: (last 0 Uf :=: 1%:M)%MS.
by rewrite -(map_mx0 f) -(map_mx1 f) last_map; apply/map_eqmx.
ha... | Lemma | extend_group_splitting_field | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"all",
"all_map",
"apply",
"compU",
"eq_all",
"eq_subset_r",
"group_splitting_field",
"inE",
"irrG",
"last",
"last_map",
"leqW",
"ltmxW",
"map",
"map_cons",
"map_eqmx",
"map_mx",
"map_mx0",
"map_mx1",
"map_mx_abs_irr",
"map_regular_repr",
"map_regular_subseries",
"map_sub... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_of {F : fieldType} {gT : finGroupType} {G : {group gT}} {n' : nat}
{rG : mx_representation F G n'.+1} {A : 'M[F]_n'.+1}
(irrG : mx_irreducible rG) (cGA : centgmx rG A) | :=
Gen {rVval : 'rV[F]_(degree_mxminpoly A)}. | Record | gen_of | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"centgmx",
"degree_mxminpoly",
"gT",
"group",
"irrG",
"mx_irreducible",
"mx_representation",
"n'",
"nat",
"rG"
] | of the section. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
d | := (degree_mxminpoly A). | Notation | d | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"degree_mxminpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pA | := (mxminpoly A). | Notation | pA | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxminpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
d_gt0 | := mxminpoly_nonconstant A. | Let | d_gt0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxminpoly_nonconstant"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr | := mx_irreducible. | Notation | irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_irreducible"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(irrG : irr rG) (cGA : centgmx rG A). | Hypotheses | irrG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"centgmx",
"irr",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
FA | := (gen_of irrG cGA). | Notation | FA | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gen_of",
"irrG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inFA | := Gen irrG cGA. | Let | inFA | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irrG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen0 | := inFA 0. | Definition | gen0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inFA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genN (x : FA) | := inFA (- val x). | Definition | genN | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"FA",
"inFA",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genD (x y : FA) | := inFA (val x + val y). | Definition | genD | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"FA",
"inFA",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_addA : associative genD. | Proof. by move=> x y z; apply: val_inj; rewrite /= addrA. Qed. | Lemma | gen_addA | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addrA",
"apply",
"genD",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_addC : commutative genD. | Proof. by move=> x y; apply: val_inj; rewrite /= addrC. Qed. | Lemma | gen_addC | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"apply",
"genD",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_add0r : left_id gen0 genD. | Proof. by move=> x; apply: val_inj; rewrite /= add0r. Qed. | Lemma | gen_add0r | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"add0r",
"apply",
"gen0",
"genD",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_addNr : left_inverse gen0 genN genD. | Proof. by move=> x; apply: val_inj; rewrite /= addNr. Qed. | Lemma | gen_addNr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addNr",
"apply",
"gen0",
"genD",
"genN",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pval (x : FA) | := rVpoly (val x). | Definition | pval | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"FA",
"rVpoly",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval (x : FA) | := horner_mx A (pval x). | Definition | mxval | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"FA",
"horner_mx",
"pval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen (x : F) | := inFA (poly_rV x%:P). | Definition | gen | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inFA",
"poly_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genK x : mxval (gen x) = x%:M. | Proof.
by rewrite /mxval [pval _]poly_rV_K ?horner_mx_C // size_polyC; case: (x != 0).
Qed. | Lemma | genK | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gen",
"horner_mx_C",
"mxval",
"poly_rV_K",
"pval",
"size_polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_inj : injective mxval. | Proof. exact: inj_comp horner_rVpoly_inj val_inj. Qed. | Lemma | mxval_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",
... | [
"horner_rVpoly_inj",
"mxval",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval0 : mxval 0 = 0. | Proof. by rewrite /mxval [pval _]raddf0 rmorph0. Qed. | Lemma | mxval0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxval",
"pval",
"raddf0",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvalN : {morph mxval : x / - x}. | Proof. by move=> x; rewrite /mxval [pval _](@raddfN 'rV[F]_d) rmorphN. Qed. | Lemma | mxvalN | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxval",
"pval",
"raddfN",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvalD : {morph mxval : x y / x + y}. | Proof. by move=> x y; rewrite /mxval [pval _]raddfD rmorphD. Qed. | Lemma | mxvalD | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxval",
"pval",
"raddfD",
"rmorphD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_sum | := big_morph mxval mxvalD mxval0. | Definition | mxval_sum | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"big_morph",
"mxval",
"mxval0",
"mxvalD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen1 | := inFA (poly_rV 1). | Definition | gen1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inFA",
"poly_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genM x y | := inFA (poly_rV (pval x * pval y %% pA)). | Definition | genM | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inFA",
"pA",
"poly_rV",
"pval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genV x | := inFA (poly_rV (mx_inv_horner A (mxval x)^-1)). | Definition | genV | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inFA",
"mx_inv_horner",
"mxval",
"poly_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_gen1 : mxval gen1 = 1%:M. | Proof. by rewrite /mxval [pval _]poly_rV_K ?size_poly1 // horner_mx_C. Qed. | Lemma | mxval_gen1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gen1",
"horner_mx_C",
"mxval",
"poly_rV_K",
"pval",
"size_poly1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_genM : {morph mxval : x y / genM x y >-> x *m y}. | Proof.
move=> x y; rewrite /mxval [pval _]poly_rV_K ?size_mod_mxminpoly //.
by rewrite -horner_mxK mx_inv_hornerK ?horner_mx_mem // rmorphM.
Qed. | Lemma | mxval_genM | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"genM",
"horner_mxK",
"horner_mx_mem",
"mx_inv_hornerK",
"mxval",
"poly_rV_K",
"pval",
"rmorphM",
"size_mod_mxminpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_genV : {morph mxval : x / genV x >-> invmx x}. | Proof.
move=> x; rewrite /mxval [pval _]poly_rV_K ?size_poly ?mx_inv_hornerK //.
pose m B : 'M[F]_(n * n) := lin_mx (mulmxr B); set B := mxval x.
case uB: (B \is a GRing.unit); last by rewrite invr_out ?uB ?horner_mx_mem.
have defAd: Ad = Ad *m m B *m m B^-1.
apply/row_matrixP=> i.
by rewrite !row_mul mul_rV_lin /=... | Lemma | mxval_genV | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Ad",
"apply",
"genV",
"geq_leqif",
"horner_mx_X",
"horner_mx_mem",
"invmx",
"invr_out",
"last",
"lin_mx",
"mul1mx",
"mul_rV_lin",
"mul_vec_lin",
"mulmxE",
"mulmxK",
"mulmxr",
"mx_inv_hornerK",
"mx_rV_lin",
"mxrankM_maxl",
"mxrank_leqif_sup",
"mxval",
"mxval_gen1",
"poly_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mulA : associative genM. | Proof. by move=> x y z; apply: mxval_inj; rewrite !mxval_genM mulmxA. Qed. | Lemma | gen_mulA | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"genM",
"mulmxA",
"mxval_genM",
"mxval_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mulC : commutative genM. | Proof. by move=> x y; rewrite /genM mulrC. Qed. | Lemma | gen_mulC | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"genM",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mul1r : left_id gen1 genM. | Proof. by move=> x; apply: mxval_inj; rewrite mxval_genM mxval_gen1 mul1mx. Qed. | Lemma | gen_mul1r | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"gen1",
"genM",
"mul1mx",
"mxval_gen1",
"mxval_genM",
"mxval_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mulDr : left_distributive genM +%R. | Proof.
by move=> x y z; apply: mxval_inj; rewrite !(mxvalD, mxval_genM) mulmxDl.
Qed. | Lemma | gen_mulDr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"genM",
"mulmxDl",
"mxvalD",
"mxval_genM",
"mxval_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_ntriv : gen1 != 0. | Proof. by rewrite -(inj_eq mxval_inj) mxval_gen1 mxval0 oner_eq0. Qed. | Lemma | gen_ntriv | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gen1",
"inj_eq",
"mxval0",
"mxval_gen1",
"mxval_inj",
"oner_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval1 : mxval 1 = 1%:M. | Proof. exact: mxval_gen1. Qed. | Lemma | mxval1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxval",
"mxval_gen1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvalM : {morph mxval : x y / x * y >-> x *m y}. | Proof. exact: mxval_genM. Qed. | Lemma | mxvalM | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxval",
"mxval_genM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_is_zmod_morphism : zmod_morphism mxval. | Proof. by move=> x y; rewrite mxvalD mxvalN. Qed. | Lemma | mxval_is_zmod_morphism | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxval",
"mxvalD",
"mxvalN",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_sub | := mxval_is_zmod_morphism. | Definition | mxval_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",
... | [
"mxval_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_is_monoid_morphism : monoid_morphism mxval. | Proof. by split; [apply: mxval1 | apply: mxvalM]. Qed. | Lemma | mxval_is_monoid_morphism | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"monoid_morphism",
"mxval",
"mxval1",
"mxvalM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_is_multiplicative | :=
(fun g => (g.2,g.1)) mxval_is_monoid_morphism. | Definition | mxval_is_multiplicative | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxval_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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