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 |
|---|---|---|---|---|---|---|---|---|---|---|
mxval_centg x : centgmx rG (mxval x). | Proof.
rewrite [mxval _]horner_rVpoly -memmx_cent_envelop vec_mxK {x}mulmx_sub //.
apply/row_subP=> k; rewrite rowK memmx_cent_envelop; apply/centgmxP => g Gg /=.
by rewrite !mulmxE commrX // /GRing.comm -mulmxE (centgmxP cGA).
Qed. | Lemma | mxval_centg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Gg",
"apply",
"centgmx",
"centgmxP",
"comm",
"commrX",
"horner_rVpoly",
"memmx_cent_envelop",
"mulmxE",
"mulmx_sub",
"mxval",
"rG",
"rowK",
"row_subP",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mulVr x : x != 0 -> genV x * x = 1. | Proof.
rewrite -(inj_eq mxval_inj) mxval0.
move/(mx_Schur irrG (mxval_centg x)) => u_x.
by apply: mxval_inj; rewrite mxvalM mxval_genV mxval1 mulVmx.
Qed. | Lemma | gen_mulVr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"genV",
"inj_eq",
"irrG",
"mulVmx",
"mx_Schur",
"mxval0",
"mxval1",
"mxvalM",
"mxval_centg",
"mxval_genV",
"mxval_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_invr0 : genV 0 = 0. | Proof. by apply: mxval_inj; rewrite mxval_genV !mxval0 -{2}invr0. Qed. | Lemma | gen_invr0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"genV",
"invr0",
"mxval0",
"mxval_genV",
"mxval_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxvalV : {morph mxval : x / x^-1 >-> invmx x}. | Proof. exact: mxval_genV. Qed. | Lemma | mxvalV | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"mxval",
"mxval_genV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_is_zmod_morphism : zmod_morphism gen. | Proof. by move=> x y; apply: mxval_inj; rewrite genK !rmorphB /= !genK. Qed. | Lemma | gen_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",
... | [
"apply",
"gen",
"genK",
"mxval_inj",
"rmorphB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_is_additive | := gen_is_zmod_morphism. | Definition | gen_is_additive | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_is_monoid_morphism : monoid_morphism gen. | Proof. by split=> // x y; apply: mxval_inj; rewrite genK !rmorphM /= !genK. Qed. | Lemma | gen_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",
"gen",
"genK",
"monoid_morphism",
"mxval_inj",
"rmorphM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_is_multiplicative | :=
(fun g => (g.2,g.1)) gen_is_monoid_morphism. | Definition | gen_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",
... | [
"gen_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
groot | := inFA (poly_rV ('X %% pA)). | Definition | groot | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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"
] | cases we want to use the construction solely for that purpose). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mxval_groot : mxval groot = A. | Proof.
rewrite /mxval [pval _]poly_rV_K ?size_mod_mxminpoly // -horner_mxK.
by rewrite mx_inv_hornerK ?horner_mx_mem // horner_mx_X.
Qed. | Lemma | mxval_groot | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"groot",
"horner_mxK",
"horner_mx_X",
"horner_mx_mem",
"mx_inv_hornerK",
"mxval",
"poly_rV_K",
"pval",
"size_mod_mxminpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxval_grootXn k : mxval (groot ^+ k) = A ^+ k. | Proof. by rewrite rmorphXn /= mxval_groot. Qed. | Lemma | mxval_grootXn | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"groot",
"mxval",
"mxval_groot",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mxminpoly_groot : (map_poly gen pA).[groot] = 0. | Proof. (* The [_ groot] prevents divergence of simpl. *)
apply: mxval_inj; rewrite -horner_map [_ groot]/= mxval_groot mxval0.
rewrite -(mx_root_minpoly A); congr ((_ : {poly _}).[A]).
by apply/polyP=> i; rewrite 3!coef_map; apply: genK.
Qed. | Lemma | map_mxminpoly_groot | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"coef_map",
"gen",
"genK",
"groot",
"horner_map",
"map_poly",
"mx_root_minpoly",
"mxval0",
"mxval_groot",
"mxval_inj",
"pA",
"poly",
"polyP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
non_linear_gen_reducible : d > 1 -> mxnonsimple (map_repr gen rG) 1%:M. | Proof.
rewrite ltnNge mxminpoly_linear_is_scalar => Anscal.
pose Af := map_mx gen A; exists (kermx (Af - groot%:M)).
rewrite submx1 kermx_centg_module /=.
apply/centgmxP=> z Gz; rewrite mulmxBl [RHS]mulmxBr [in RHS]scalar_mxC.
by rewrite -!map_mxM 1?(centgmxP cGA).
rewrite andbC mxrank_ker -subn_gt0 mxrank1 subKn ?... | Lemma | non_linear_gen_reducible | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addn2",
"apply",
"centgmxP",
"dvdp_leq",
"factor_theorem",
"gen",
"groot",
"horner_mx",
"horner_mx_C",
"horner_mx_X",
"is_scalar_mxP",
"kermx",
"kermx_centg_module",
"lt0n",
"ltnNge",
"ltnn",
"map_mx",
"map_mxM",
"map_mx_is_scalar",
"map_mxminpoly_groot",
"map_repr",
"moni... | yields a (reducible) tensored representation. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
subbase nA (B : 'rV_nA) : 'M_(nA * d, n) | :=
\matrix_ik mxvec (\matrix_(i, k) (row (B 0 i) (A ^+ k))) 0 ik. | Definition | subbase | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxvec",
"nA",
"row"
] | we have to use explicit bijections to convert between the two views. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gen_dim_ex_proof : exists nA, [exists B : 'rV_nA, row_free (subbase B)]. | Proof. by exists 0; apply/existsP; exists 0; rewrite /row_free unlock. Qed. | Lemma | gen_dim_ex_proof | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"existsP",
"nA",
"row_free",
"subbase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_dim_ub_proof nA :
[exists B : 'rV_nA, row_free (subbase B)] -> (nA <= n)%N. | Proof.
case/existsP=> B /eqnP def_nAd.
by rewrite (leq_trans _ (rank_leq_col (subbase B))) // def_nAd leq_pmulr.
Qed. | Lemma | gen_dim_ub_proof | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"eqnP",
"existsP",
"leq_pmulr",
"leq_trans",
"nA",
"rank_leq_col",
"row_free",
"subbase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_dim | := ex_maxn gen_dim_ex_proof gen_dim_ub_proof. | Definition | gen_dim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"ex_maxn",
"gen_dim_ex_proof",
"gen_dim_ub_proof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nA | := gen_dim. | Notation | nA | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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_dim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_base : 'rV_nA | := odflt 0 [pick B | row_free (subbase B)]. | Definition | gen_base | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"pick",
"row_free",
"subbase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
base | := subbase gen_base. | Definition | base | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gen_base",
"subbase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
base_free : row_free base. | Proof.
rewrite /base /gen_base /nA; case: pickP => //; case: ex_maxnP => nA_max.
by case/existsP=> B Bfree _ no_free; rewrite no_free in Bfree.
Qed. | Lemma | base_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",
... | [
"base",
"ex_maxnP",
"existsP",
"gen_base",
"nA",
"pickP",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
base_full : row_full base. | Proof.
rewrite /row_full (eqnP base_free) /nA; case: ex_maxnP => nA.
case/existsP=> /= B /eqnP Bfree nA_max; rewrite -Bfree eqn_leq rank_leq_col.
rewrite -{1}(mxrank1 F n) mxrankS //; apply/row_subP=> j; set u := row _ _.
move/implyP: {nA_max}(nA_max nA.+1); rewrite ltnn implybF.
apply: contraR => nBj; apply/existsP.
e... | Lemma | base_full | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"addsmx_sub",
"apply",
"base",
"base_free",
"const_mx",
"divff",
"eqVneq",
"eq_row_sub",
"eqnP",
"eqn_leq",
"eqxx",
"ex_maxnP",
"existsP",
"groot",
"horner_rVpoly",
"inFA",
"last",
"lin1_mx",
"linearZ",
"linear_sum",
"lshift",
"ltnn",
"mul0mx",
"mul1mx",
"mulS... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_dim_factor : (nA * d)%N = n. | Proof. by rewrite -(eqnP base_free) (eqnP base_full). Qed. | Lemma | gen_dim_factor | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"base_free",
"base_full",
"eqnP",
"nA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_dim_gt0 : nA > 0. | Proof. by case: posnP gen_dim_factor => // ->. Qed. | Lemma | gen_dim_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",
... | [
"gen_dim_factor",
"nA",
"posnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_gen (W : 'M[F]_(m, n)) : 'M[FA]_(m, nA) | :=
\matrix_(i, j) inFA (row j (vec_mx (row i W *m pinvmx base))). | Definition | in_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",
... | [
"FA",
"base",
"inFA",
"nA",
"pinvmx",
"row",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_gen (W : 'M[FA]_(m, nA)) : 'M[F]_(m, n) | :=
\matrix_i (mxvec (\matrix_j val (W i j)) *m base). | Definition | val_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",
... | [
"FA",
"base",
"mxvec",
"nA",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_genK : cancel in_gen val_gen. | Proof.
move=> W; apply/row_matrixP=> i; rewrite rowK; set w := row i W.
have b_w: (w <= base)%MS by rewrite submx_full ?base_full.
rewrite -{b_w}(mulmxKpV b_w); congr (_ *m _).
by apply/rowP; case/mxvec_indexP=> j k; rewrite mxvecE !mxE.
Qed. | Lemma | in_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",
... | [
"apply",
"base",
"base_full",
"in_gen",
"mulmxKpV",
"mxE",
"mxvecE",
"mxvec_indexP",
"row",
"rowK",
"rowP",
"row_matrixP",
"submx_full",
"val_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_genK : cancel val_gen in_gen. | Proof.
move=> W; apply/matrixP=> i j; apply: val_inj; rewrite mxE /= rowK.
case/row_freeP: base_free => B' BB'; rewrite -[_ *m _]mulmx1 -BB' mulmxA.
by rewrite mulmxKpV ?submxMl // -mulmxA BB' mulmx1 mxvecK rowK.
Qed. | Lemma | val_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",
... | [
"apply",
"base_free",
"in_gen",
"matrixP",
"mulmx1",
"mulmxA",
"mulmxKpV",
"mxE",
"mxvecK",
"rowK",
"row_freeP",
"submxMl",
"val_gen",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_gen0 : in_gen 0 = 0. | Proof. by apply/matrixP=> i j; rewrite !mxE !(mul0mx, linear0). Qed. | Lemma | in_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",
... | [
"apply",
"in_gen",
"linear0",
"matrixP",
"mul0mx",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_gen0 : val_gen 0 = 0. | Proof. by apply: (canLR in_genK); rewrite in_gen0. Qed. | Lemma | val_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",
... | [
"apply",
"in_gen0",
"in_genK",
"val_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_genD : {morph in_gen : U V / U + V}. | Proof.
by move=> U V; apply/matrixP=> i j; rewrite !mxE 4!(mulmxDl, linearD).
Qed. | Lemma | in_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",
... | [
"apply",
"in_gen",
"linearD",
"matrixP",
"mulmxDl",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_genD : {morph val_gen : U V / U + V}. | Proof. by move=> U V; apply: (canLR in_genK); rewrite in_genD !val_genK. Qed. | Lemma | val_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",
... | [
"apply",
"in_genD",
"in_genK",
"val_gen",
"val_genK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_genN : {morph in_gen : W / - W}. | Proof. by move=> W; apply/esym/addr0_eq; rewrite -in_genD subrr in_gen0. Qed. | Lemma | in_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",
... | [
"addr0_eq",
"apply",
"in_gen",
"in_gen0",
"in_genD",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_genN : {morph val_gen : W / - W}. | Proof. by move=> W; apply: (canLR in_genK); rewrite in_genN val_genK. Qed. | Lemma | val_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",
... | [
"apply",
"in_genK",
"in_genN",
"val_gen",
"val_genK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_gen_sum | := big_morph in_gen in_genD in_gen0. | Definition | in_gen_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",
"in_gen",
"in_gen0",
"in_genD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_gen_sum | := big_morph val_gen val_genD val_gen0. | Definition | val_gen_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",
"val_gen",
"val_gen0",
"val_genD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_genZ a : {morph in_gen : W / a *: W >-> gen a *: W}. | Proof.
move=> W; apply/matrixP=> i j; apply: mxval_inj.
rewrite !mxE mxvalM genK ![mxval _]horner_rVpoly /=.
by rewrite mul_scalar_mx !(I, scalemxAl, linearZ).
Qed. | Lemma | in_genZ | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"gen",
"genK",
"horner_rVpoly",
"in_gen",
"linearZ",
"matrixP",
"mul_scalar_mx",
"mxE",
"mxval",
"mxvalM",
"mxval_inj",
"scalemxAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_gen_rV (w : 'rV_nA) :
val_gen w = mxvec (\matrix_j val (w 0 j)) *m base. | Proof. by apply/rowP=> j /[1!mxE]. Qed. | Lemma | val_gen_rV | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"base",
"mxE",
"mxvec",
"rowP",
"val",
"val_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_gen_row W (i : 'I_m) : val_gen (row i W) = row i (val_gen W). | Proof.
rewrite val_gen_rV rowK; congr (mxvec _ *m _).
by apply/matrixP=> j k /[!mxE].
Qed. | Lemma | val_gen_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",
... | [
"apply",
"matrixP",
"mxE",
"mxvec",
"row",
"rowK",
"val_gen",
"val_gen_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_gen_row W (i : 'I_m) : in_gen (row i W) = row i (in_gen W). | Proof. by apply: (canLR val_genK); rewrite val_gen_row in_genK. Qed. | Lemma | in_gen_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",
... | [
"apply",
"in_gen",
"in_genK",
"row",
"val_genK",
"val_gen_row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_gen_sum_mxval W (i : 'I_m) :
row i (val_gen W) = \sum_j row (gen_base 0 j) (mxval (W i j)). | Proof.
rewrite -val_gen_row [row i W]row_sum_delta val_gen_sum.
apply: eq_bigr => /= j _ /[1!mxE]; move: {W i}(W i j) => x.
have ->: x = \sum_k gen (val x 0 k) * inFA (delta_mx 0 k).
case: x => u; apply: mxval_inj; rewrite {1}[u]row_sum_delta.
rewrite mxval_sum [mxval _]horner_rVpoly mulmx_suml linear_sum /=.
app... | Lemma | row_gen_sum_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",
... | [
"apply",
"base",
"delta_mx",
"eq_bigr",
"gen",
"genK",
"gen_base",
"horner_rVpoly",
"inFA",
"in_genK",
"in_genZ",
"last",
"linearZ",
"linear_sum",
"mul_delta_mx_cond",
"mul_scalar_mx",
"mulmx_suml",
"mulr_natr",
"mulrb",
"mxE",
"mxval",
"mxvalM",
"mxval_inj",
"mxval_sum... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_genZ x : {morph @val_gen m : W / x *: W >-> W *m mxval x}. | Proof.
move=> W; apply/row_matrixP=> i; rewrite row_mul !row_gen_sum_mxval.
by rewrite mulmx_suml; apply: eq_bigr => j _; rewrite mxE mulrC mxvalM row_mul.
Qed. | Lemma | val_genZ | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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_bigr",
"mulmx_suml",
"mulrC",
"mxE",
"mxval",
"mxvalM",
"row_gen_sum_mxval",
"row_matrixP",
"row_mul",
"val_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_in_gen m1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(U <= V -> in_gen U <= in_gen V)%MS. | Proof.
move=> sUV; apply/row_subP=> i; rewrite -in_gen_row.
case/submxP: (row_subP sUV i) => u ->{i}.
rewrite mulmx_sum_row in_gen_sum summx_sub // => j _.
by rewrite in_genZ in_gen_row scalemx_sub ?row_sub.
Qed. | Lemma | submx_in_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",
... | [
"apply",
"in_gen",
"in_genZ",
"in_gen_row",
"in_gen_sum",
"mulmx_sum_row",
"row_sub",
"row_subP",
"scalemx_sub",
"submxP",
"summx_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_in_gen_eq m1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, n)) :
(V *m A <= V -> (in_gen U <= in_gen V) = (U <= V))%MS. | Proof.
move=> sVA_V; apply/idP/idP=> siUV; last exact: submx_in_gen.
apply/row_subP=> i; rewrite -[row i U]in_genK in_gen_row.
case/submxP: (row_subP siUV i) => u ->{i U siUV}.
rewrite mulmx_sum_row val_gen_sum summx_sub // => j _.
rewrite val_genZ val_gen_row in_genK rowE -mulmxA mulmx_sub //.
rewrite [mxval _]horner_... | Lemma | submx_in_gen_eq | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"exprSr",
"horner_poly",
"in_gen",
"in_genK",
"in_gen_row",
"last",
"mul_mx_scalar",
"mulmx1",
"mulmxA",
"mulmx_sub",
"mulmx_sum_row",
"mulmx_sumr",
"mxval",
"row",
"rowE",
"row_subP",
"scalemxAl",
"scalemx_sub",
"submxMr",
"submxP",
"submx_in_gen",
"submx_trans"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mx g | := \matrix_i in_gen (row (gen_base 0 i) (rG g)). | Definition | gen_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",
... | [
"gen_base",
"in_gen",
"rG",
"row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_genJmx m :
{in G, forall g, {morph @val_gen m : W / W *m gen_mx g >-> W *m rG g}}. | Proof.
move=> g Gg /= W; apply/row_matrixP=> i; rewrite -val_gen_row !row_mul.
rewrite mulmx_sum_row val_gen_sum row_gen_sum_mxval mulmx_suml.
apply: eq_bigr => /= j _; rewrite val_genZ rowK in_genK mxE -!row_mul.
by rewrite (centgmxP (mxval_centg _)).
Qed. | Let | val_genJmx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Gg",
"apply",
"centgmxP",
"eq_bigr",
"gen_mx",
"in_genK",
"mulmx_sum_row",
"mulmx_suml",
"mxE",
"mxval_centg",
"rG",
"rowK",
"row_gen_sum_mxval",
"row_matrixP",
"row_mul",
"val_gen",
"val_genZ",
"val_gen_row",
"val_gen_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mx_repr : mx_repr G gen_mx. | Proof.
split=> [|g h Gg Gh]; apply: (can_inj val_genK).
by rewrite -[gen_mx 1]mul1mx val_genJmx // repr_mx1 mulmx1.
rewrite {1}[val_gen]lock -[gen_mx g]mul1mx !val_genJmx // -mulmxA -repr_mxM //.
by rewrite -val_genJmx ?groupM ?mul1mx -?lock.
Qed. | Lemma | gen_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",
... | [
"Gg",
"apply",
"gen_mx",
"groupM",
"mul1mx",
"mulmx1",
"mulmxA",
"mx_repr",
"repr_mx1",
"repr_mxM",
"split",
"val_gen",
"val_genJmx",
"val_genK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_repr | := MxRepresentation gen_mx_repr. | Canonical | gen_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",
... | [
"gen_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGA | := gen_repr. | Notation | rGA | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_genJ m :
{in G, forall g, {morph @val_gen m : W / W *m rGA g >-> W *m rG g}}. | Proof. exact: val_genJmx. Qed. | Lemma | val_genJ | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"rGA",
"val_gen",
"val_genJmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_genJ m :
{in G, forall g, {morph @in_gen m : v / v *m rG g >-> v *m rGA g}}. | Proof.
by move=> g Gg /= v; apply: (canLR val_genK); rewrite val_genJ ?in_genK.
Qed. | Lemma | in_genJ | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Gg",
"apply",
"in_gen",
"in_genK",
"rG",
"rGA",
"val_genJ",
"val_genK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_gen (H : {set gT}) :
H \subset G -> (rfix_mx rGA H :=: in_gen (rfix_mx rG H))%MS. | Proof.
move/subsetP=> sHG; apply/eqmxP/andP; split; last first.
by apply/rfix_mxP=> g Hg; rewrite -in_genJ ?sHG ?rfix_mx_id.
rewrite -[rfix_mx rGA H]val_genK; apply: submx_in_gen.
by apply/rfix_mxP=> g Hg; rewrite -val_genJ ?rfix_mx_id ?sHG.
Qed. | Lemma | rfix_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",
... | [
"apply",
"eqmxP",
"gT",
"in_gen",
"in_genJ",
"last",
"rG",
"rGA",
"rfix_mx",
"rfix_mxP",
"rfix_mx_id",
"sHG",
"split",
"submx_in_gen",
"subsetP",
"val_genJ",
"val_genK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowval_gen m U | :=
<<\matrix_ik
mxvec (\matrix_(i < m, k < d) (row i (val_gen U) *m A ^+ k)) 0 ik>>%MS. | Definition | rowval_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",
... | [
"mxvec",
"row",
"val_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_rowval_gen m1 m2 (U : 'M_(m1, n)) (V : 'M_(m2, nA)) :
(U <= rowval_gen V)%MS = (in_gen U <= V)%MS. | Proof.
rewrite genmxE; apply/idP/idP=> sUV.
apply: submx_trans (submx_in_gen sUV) _.
apply/row_subP; case/mxvec_indexP=> i k; rewrite -in_gen_row rowK mxvecE mxE.
rewrite -mxval_grootXn -val_gen_row -val_genZ val_genK scalemx_sub //.
exact: row_sub.
rewrite -[U]in_genK; case/submxP: sUV => u ->{U}.
apply/row_su... | Lemma | submx_rowval_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",
... | [
"Ad",
"apply",
"eq_row_sub",
"genmxE",
"horner_rVpoly",
"i0",
"in_gen",
"in_genK",
"in_gen_row",
"linearZ",
"linear_sum",
"mulmx_sum_row",
"mxE",
"mxval",
"mxval_grootXn",
"mxvecE",
"mxvecK",
"mxvec_index",
"mxvec_indexP",
"nA",
"row",
"rowK",
"row_mul",
"row_sub",
"r... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowval_genK m (U : 'M_(m, nA)) : (in_gen (rowval_gen U) :=: U)%MS. | Proof.
apply/eqmxP; rewrite -submx_rowval_gen submx_refl /=.
by rewrite -{1}[U]val_genK submx_in_gen // submx_rowval_gen val_genK.
Qed. | Lemma | rowval_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",
... | [
"apply",
"eqmxP",
"in_gen",
"nA",
"rowval_gen",
"submx_in_gen",
"submx_refl",
"submx_rowval_gen",
"val_genK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowval_gen_stable m (U : 'M_(m, nA)) :
(rowval_gen U *m A <= rowval_gen U)%MS. | Proof.
rewrite -[A]mxval_groot -{1}[_ U]in_genK -val_genZ.
by rewrite submx_rowval_gen val_genK scalemx_sub // rowval_genK.
Qed. | Lemma | rowval_gen_stable | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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_genK",
"mxval_groot",
"nA",
"rowval_gen",
"rowval_genK",
"scalemx_sub",
"submx_rowval_gen",
"val_genK",
"val_genZ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_in_gen m (U : 'M_(m, n)) : rstab rGA (in_gen U) = rstab rG U. | Proof.
apply/setP=> x /[!inE]; case Gx: (x \in G) => //=.
by rewrite -in_genJ // (inj_eq (can_inj in_genK)).
Qed. | Lemma | rstab_in_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",
... | [
"apply",
"inE",
"in_gen",
"in_genJ",
"in_genK",
"inj_eq",
"rG",
"rGA",
"rstab",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_in_gen m (U : 'M_(m, n)) :
rstabs rG U \subset rstabs rGA (in_gen U). | Proof.
by apply/subsetP=> x /[!inE] /andP[Gx nUx]; rewrite -in_genJ Gx // submx_in_gen.
Qed. | Lemma | rstabs_in_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",
... | [
"apply",
"inE",
"in_gen",
"in_genJ",
"rG",
"rGA",
"rstabs",
"submx_in_gen",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_rowval_gen m (U : 'M_(m, nA)) :
rstabs rG (rowval_gen U) = rstabs rGA U. | Proof.
apply/setP=> x /[!inE]; case Gx: (x \in G) => //=.
by rewrite submx_rowval_gen in_genJ // (eqmxMr _ (rowval_genK U)).
Qed. | Lemma | rstabs_rowval_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",
... | [
"apply",
"eqmxMr",
"inE",
"in_genJ",
"nA",
"rG",
"rGA",
"rowval_gen",
"rowval_genK",
"rstabs",
"setP",
"submx_rowval_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_rowval_gen m (U : 'M_(m, nA)) :
mxmodule rG (rowval_gen U) = mxmodule rGA U. | Proof. by rewrite /mxmodule rstabs_rowval_gen. Qed. | Lemma | mxmodule_rowval_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",
... | [
"mxmodule",
"nA",
"rG",
"rGA",
"rowval_gen",
"rstabs_rowval_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mx_irr : mx_irreducible rGA. | Proof.
apply/mx_irrP; split=> [|U Umod nzU]; first exact: gen_dim_gt0.
rewrite -sub1mx -rowval_genK -submx_rowval_gen submx_full //.
case/mx_irrP: irrG => _; apply; first by rewrite mxmodule_rowval_gen.
rewrite -(inj_eq (can_inj in_genK)) in_gen0.
by rewrite -mxrank_eq0 rowval_genK mxrank_eq0.
Qed. | Lemma | gen_mx_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"apply",
"gen_dim_gt0",
"in_gen0",
"in_genK",
"inj_eq",
"irrG",
"mx_irrP",
"mx_irreducible",
"mxmodule_rowval_gen",
"mxrank_eq0",
"rGA",
"rowval_genK",
"split",
"sub1mx",
"submx_full",
"submx_rowval_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_gen : rker rGA = rker rG. | Proof.
apply/setP=> g; rewrite !inE !mul1mx; case Gg: (g \in G) => //=.
apply/eqP/eqP=> g1; apply/row_matrixP=> i.
by apply: (can_inj in_genK); rewrite rowE in_genJ //= g1 mulmx1 row1.
by apply: (can_inj val_genK); rewrite rowE val_genJ //= g1 mulmx1 row1.
Qed. | Lemma | rker_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",
... | [
"Gg",
"apply",
"inE",
"in_genJ",
"in_genK",
"mul1mx",
"mulmx1",
"rG",
"rGA",
"rker",
"row1",
"rowE",
"row_matrixP",
"setP",
"val_genJ",
"val_genK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_mx_faithful : mx_faithful rGA = mx_faithful rG. | Proof. by rewrite /mx_faithful rker_gen. Qed. | Lemma | gen_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",
"rGA",
"rker_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(irrG : mx_irreducible 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",
"mx_irreducible",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
inFA | := (Gen irrG cGA). | Notation | 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 | |
d_gt0 : d > 0 | := 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 | |
mxT (u : 'rV_d) | := vec_mx (mulmx_term u (mx_term Ad)). | Let | mxT | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"mulmx_term",
"mx_term",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_mxT e u : eval_mx e (mxT u) = mxval (inFA (eval_mx e u)). | Proof.
by rewrite eval_vec_mx eval_mulmx eval_mx_term [mxval _]horner_rVpoly.
Qed. | Let | eval_mxT | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"eval_mulmx",
"eval_mx",
"eval_mx_term",
"eval_vec_mx",
"horner_rVpoly",
"inFA",
"mxT",
"mxval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Ad'T | := mx_term (pinvmx Ad). | Let | Ad'T | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"mx_term",
"pinvmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulT (u v : 'rV_d) | := mulmx_term (mxvec (mulmx_term (mxT u) (mxT v))) Ad'T. | Let | 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",
... | [
"Ad'T",
"mulmx_term",
"mxT",
"mxvec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_mulT e u v :
eval_mx e (mulT u v) = val (inFA (eval_mx e u) * inFA (eval_mx e v)). | Proof.
rewrite !(eval_mulmx, eval_mxvec) !eval_mxT eval_mx_term.
by apply: (can_inj rVpolyK); rewrite -mxvalM [rVpoly _]horner_rVpolyK.
Qed. | Lemma | eval_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",
... | [
"apply",
"eval_mulmx",
"eval_mx",
"eval_mxT",
"eval_mx_term",
"eval_mxvec",
"horner_rVpolyK",
"inFA",
"mulT",
"mxvalM",
"rVpoly",
"rVpolyK",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_term t | := match t with
| 'X_k => row_var _ d k
| x%:T => mx_term (val (x : FA))
| n1%:R => mx_term (val (n1%:R : FA))%R
| t1 + t2 => \row_i (gen_term t1 0%R i + gen_term t2 0%R i)
| - t1 => \row_i (- gen_term t1 0%R i)
| t1 *+ n1 => mulmx_term (mx_term n1%:R%:M)%R (gen_term t1)
| t1 * t2 => mulT (gen_term t1) (gen_term t2)
| ... | Fixpoint | gen_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",
... | [
"FA",
"iter",
"mulT",
"mulmx_term",
"mx_term",
"row_var",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_env (e : seq FA) | := row_env (map val e). | Definition | gen_env | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"map",
"row_env",
"seq",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_map_rVval (e : seq FA) j : (map val e)`_j = val e`_j. | Proof.
case: (ltnP j (size e)) => [| leej]; first exact: (nth_map 0 0).
by rewrite !nth_default ?size_map.
Qed. | Lemma | nth_map_rVval | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"ltnP",
"map",
"nth_default",
"nth_map",
"seq",
"size",
"size_map",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_nth_map_rVval (e : seq FA) j v :
set_nth 0 (map val e) j v = map val (set_nth 0 e j (inFA v)). | Proof.
apply: (@eq_from_nth _ 0) => [|k _]; first by rewrite !(size_set_nth, size_map).
by rewrite !(nth_map_rVval, nth_set_nth) /= nth_map_rVval [rVval _]fun_if.
Qed. | Lemma | set_nth_map_rVval | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"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",
"apply",
"eq_from_nth",
"inFA",
"map",
"nth_map_rVval",
"nth_set_nth",
"seq",
"set_nth",
"size_map",
"size_set_nth",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_gen_term e t :
GRing.rterm t -> eval_mx (gen_env e) (gen_term t) = val (GRing.eval e t). | Proof.
elim: t => //=.
- by move=> k _; apply/rowP=> i; rewrite !mxE /= nth_row_env nth_map_rVval.
- by move=> x _; rewrite eval_mx_term.
- by move=> x _; rewrite eval_mx_term.
- by move=> t1 + t2 + /andP[rt1 rt2] => <-// <-//; apply/rowP => k /[!mxE].
- by move=> t1 + rt1 => <-//; apply/rowP=> k /[!mxE].
- move=> t1 I... | Lemma | eval_gen_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",
... | [
"apply",
"eval",
"eval_mulT",
"eval_mulmx",
"eval_mx",
"eval_mx_term",
"exprS",
"gen_env",
"gen_term",
"mul_scalar_mx",
"mulrS",
"mxE",
"nth_map_rVval",
"nth_row_env",
"rowP",
"rterm",
"scaler_nat",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_form f | := match f with
| Bool b => Bool b
| t1 == t2 => mxrank_form 0 (gen_term (t1 - t2))
| GRing.Unit t1 => mxrank_form 1 (gen_term t1)
| f1 /\ f2 => gen_form f1 /\ gen_form f2
| f1 \/ f2 => gen_form f1 \/ gen_form f2
| f1 ==> f2 => gen_form f1 ==> gen_form f2
| ~ f1 => ~ gen_form f1
| ('exists 'X_k, f1) => Exists_row_form... | Fixpoint | gen_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",
... | [
"Bool",
"Exists_row_form",
"f1",
"f2",
"gen_term",
"mxrank_form"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sat_gen_form e f : GRing.rformula f ->
reflect (GRing.holds e f) (GRing.sat (gen_env e) (gen_form f)). | Proof.
have ExP := Exists_rowP; have set_val := set_nth_map_rVval.
elim: f e => //.
- by move=> b e _; apply: (iffP satP).
- rewrite /gen_form => t1 t2 e rt_t; set t := (_ - _)%T.
have:= GRing.qf_evalP (gen_env e) (mxrank_form_qf 0 (gen_term t)).
rewrite eval_mxrank mxrank_eq0 eval_gen_term // => tP.
by rewrite (... | Lemma | sat_gen_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_rowP",
"apply",
"eval_gen_term",
"eval_mxrank",
"f1",
"f2",
"gen_env",
"gen_form",
"gen_term",
"holds",
"inFA",
"mxrank_eq0",
"mxrank_form_qf",
"qf_evalP",
"rformula",
"sat",
"satP",
"set_nth_map_rVval",
"subr_eq0",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_sat e f | := GRing.sat (gen_env e) (gen_form (GRing.to_rform f)). | Definition | gen_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",
... | [
"gen_env",
"gen_form",
"sat",
"to_rform"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_satP :
GRing.MathCompCompatDecidableField.DecidableField.axiom gen_sat. | Proof.
move=> e f; have [tor rto] := GRing.to_rformP e f.
exact: (iffP (sat_gen_form e (GRing.to_rform_rformula f))).
Qed. | Lemma | gen_satP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"axiom",
"gen_sat",
"sat_gen_form",
"to_rformP",
"to_rform_rformula"
] | FIXME : why this MathCompCompatDecidableField | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
card_gen : #|{:FA}| = (#|F| ^ degree_mxminpoly A)%N. | Proof. by rewrite card_sub card_mx mul1n. Qed. | Lemma | card_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",
... | [
"FA",
"card_mx",
"card_sub",
"degree_mxminpoly",
"mul1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_splitting_field_exists gT (G : {group gT}) F :
classically {Fs : fieldType & {rmorphism F -> Fs}
& group_splitting_field Fs G}. | Proof.
move: F => F0 [] // nosplit; pose nG := #|G|; pose aG F := regular_repr F G.
pose m := nG.+1; pose F := F0; pose U : seq 'M[F]_nG := [::].
suffices: size U + m <= nG by rewrite ltnn.
have: mx_subseries (aG F) U /\ path ltmx 0 U by [].
pose f : {rmorphism F0 -> F} := idfun.
elim: m F U f => [|m IHm] F U f [modU l... | Lemma | group_splitting_field_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",
... | [
"F0",
"aG",
"addn0",
"addnS",
"all_map",
"apply",
"cent_mx_scalar_abs_irr",
"enveloping_algebra_mx",
"eq_all",
"eq_subset_r",
"eq_sym",
"gT",
"gen",
"geq_leqif",
"group",
"group_splitting_field",
"has_non_scalar_mxP",
"inE",
"irrG",
"is_scalar_mx",
"last",
"leq_add2r",
"l... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_closure_field_exists gT F :
classically {Fs : fieldType & {rmorphism F -> Fs}
& group_closure_field Fs gT}. | Proof.
set n := #|{group gT}|.
suffices: classically {Fs : fieldType & {rmorphism F -> Fs}
& forall G : {group gT}, enum_rank G < n -> group_splitting_field Fs G}.
- apply: classic_bind => [[Fs f splitFs]] _ -> //.
by exists Fs => // G; apply: splitFs.
elim: (n) => [|i IHi]; first by move=> _ -> //; exists F => //... | Lemma | group_closure_field_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",
"defG",
"enum_rank",
"enum_rankK",
"enum_val",
"extend_group_splitting_field",
"gT",
"group",
"group_closure_field",
"group_splitting_field",
"group_splitting_field_exists",
"id",
"leqP",
"leq_eqVlt",
"leq_trans",
"ltnS",
"predU1P",
"val",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_closure_closed_field (F : closedFieldType) gT :
group_closure_field F gT. | Proof.
move=> G [|n] rG irrG; first by case/mx_irrP: irrG.
apply: cent_mx_scalar_abs_irr => //; rewrite leqNgt.
apply/(has_non_scalar_mxP (scalar_mx_cent _ _)) => [[A cGA nscalA]].
have [a]: exists a, eigenvalue A a.
pose P := mxminpoly A; pose d := degree_mxminpoly A.
have Pd1: P`_d = 1.
by rewrite -(eqP (mxmi... | Lemma | group_closure_closed_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",
... | [
"addmx_sub",
"apply",
"big_ord_recr",
"cent_mx_scalar_abs_irr",
"coefN",
"contraNneq",
"d_gt0",
"degree_mxminpoly",
"eigenvalue",
"eigenvalue_root_min",
"gT",
"group_closure_field",
"has_non_scalar_mxP",
"hornerN",
"horner_coef",
"irrG",
"kermx_eq0",
"lead_coef",
"leqNgt",
"lin... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zchar : {pred 'CF(B)} | :=
[pred phi in 'CF(B, A) | dec_Cint_span (in_tuple S) phi]. | Definition | Zchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dec_Cint_span",
"in_tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun0_zchar : 0 \in Zchar. | Proof.
rewrite inE mem0v; apply/sumboolP; exists 0.
by rewrite big1 // => i _; rewrite ffunE.
Qed. | Lemma | cfun0_zchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Zchar",
"apply",
"big1",
"ffunE",
"inE",
"mem0v"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zchar_zmod : zmod_closed Zchar. | Proof.
split; first exact: cfun0_zchar.
move=> phi xi /andP[Aphi /sumboolP[a Da]] /andP[Axi /sumboolP[b Db]].
rewrite inE rpredB // Da Db -sumrB; apply/sumboolP; exists (a - b).
by apply: eq_bigr => i _; rewrite -mulrzBr !ffunE.
Qed. | Fact | Zchar_zmod | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Da",
"Zchar",
"apply",
"cfun0_zchar",
"eq_bigr",
"ffunE",
"inE",
"mulrzBr",
"rpredB",
"split",
"sumrB",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_zchar a phi : a \in Num.int -> phi \in Zchar -> a *: phi \in Zchar. | Proof. by case/intrP=> m -> Zphi; rewrite scaler_int rpredMz. Qed. | Lemma | scale_zchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Zchar",
"int",
"intrP",
"rpredMz",
"scaler_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Z[' S , A ]" | := (Zchar S A) (format "''Z[' S , A ]") : group_scope. | Notation | ''Z[' S , A ] | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Zchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Z[' S ]" | := 'Z[S, setT] (format "''Z[' S ]") : group_scope. | Notation | ''Z[' S ] | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_split S A phi :
(phi \in 'Z[S, A]) = (phi \in 'Z[S]) && (phi \in 'CF(G, A)). | Proof. by rewrite !inE cfun_onT andbC. Qed. | Lemma | zchar_split | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onT",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcharD1E phi S : (phi \in 'Z[S, G^#]) = (phi \in 'Z[S]) && (phi 1%g == 0). | Proof. by rewrite zchar_split cfunD1E. Qed. | Lemma | zcharD1E | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfunD1E",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcharD1 phi S A :
(phi \in 'Z[S, A^#]) = (phi \in 'Z[S, A]) && (phi 1%g == 0). | Proof. by rewrite zchar_split cfun_onD1 andbA -zchar_split. Qed. | Lemma | zcharD1 | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onD1",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcharW S A : {subset 'Z[S, A] <= 'Z[S]}. | Proof. by move=> phi; rewrite zchar_split => /andP[]. Qed. | Lemma | zcharW | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_on S A : {subset 'Z[S, A] <= 'CF(G, A)}. | Proof. by move=> phi /andP[]. Qed. | Lemma | zchar_on | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_onS A B S : A \subset B -> {subset 'Z[S, A] <= 'Z[S, B]}. | Proof.
move=> sAB phi; rewrite zchar_split (zchar_split _ B) => /andP[->].
exact: cfun_onS.
Qed. | Lemma | zchar_onS | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onS",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_onG S : 'Z[S, G] =i 'Z[S]. | Proof. by move=> phi; rewrite zchar_split cfun_onG andbT. Qed. | Lemma | zchar_onG | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onG",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_vchar_on A : {subset 'Z[irr G, A] <= 'CF(G, A)}. | Proof. exact: zchar_on. Qed. | Lemma | irr_vchar_on | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"irr",
"zchar_on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
support_zchar S A phi : phi \in 'Z[S, A] -> support phi \subset A. | Proof. by move/zchar_on; rewrite cfun_onE. Qed. | Lemma | support_zchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onE",
"support",
"zchar_on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_zchar_on S A phi :
phi \in 'CF(G, A) -> phi \in S -> phi \in 'Z[S, A]. | Proof.
move=> Aphi /(@tnthP _ _ (in_tuple S))[i Dphi]; rewrite inE /= {}Aphi {phi}Dphi.
apply/sumboolP; exists [ffun j => (j == i)%:Z].
rewrite (bigD1 i) //= ffunE eqxx (tnth_nth 0) big1 ?addr0 // => j i'j.
by rewrite ffunE (negPf i'j).
Qed. | Lemma | mem_zchar_on | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"addr0",
"apply",
"big1",
"bigD1",
"eqxx",
"ffunE",
"inE",
"in_tuple",
"tnthP",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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