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 |
|---|---|---|---|---|---|---|---|---|---|---|
"E * F" | := (prodv_aspace E F) : aspace_scope. | Notation | E * F | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"prodv_aspace"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"f @: E" | := (aimg_aspace f E) : aspace_scope. | Notation | f @: E | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"aimg_aspace"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adjoin_degree_aimg : adjoin_degree (f @: K) (f x) = adjoin_degree K x. | Proof.
rewrite !adjoin_degreeE -aimg_adjoin.
by rewrite !limg_dim_eq ?(eqP (AHom_lker0 f)) ?capv0.
Qed. | Lemma | adjoin_degree_aimg | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"AHom_lker0",
"adjoin_degree",
"adjoin_degreeE",
"aimg_adjoin",
"capv0",
"limg_dim_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_minPoly : map_poly f (minPoly K x) = minPoly (f @: K) (f x). | Proof.
set fp := minPoly (f @: K) (f x).
have [p Kp Dp]: exists2 p, p \is a polyOver K & map_poly f p = fp.
have Kfp: fp \is a polyOver (f @: K)%VS by apply: minPolyOver.
exists (map_poly f^-1%VF fp).
apply/polyOver_poly=> j _; have /memv_imgP[y Ky ->] := polyOverP Kfp j.
by rewrite lker0_lfunK ?AHom_lker0.... | Lemma | map_minPoly | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"AHom_lker0",
"allP",
"apply",
"dvdp_map",
"eqp_map",
"eqp_monic",
"eqp_sym",
"fmorph_root",
"fp",
"limg_lfunVK",
"lker0_lfunK",
"map_poly",
"map_poly_comp",
"map_poly_id",
"memv_img",
"memv_imgP",
"memvf",
"minPoly",
"minPolyOver",
"minPoly_dvdp",
"minPoly_irr",
"monic_map... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver & {vspace L} : Type | := L. | Definition | fieldOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
K_F | := (subvs_of F). | Notation | K_F | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"subvs_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
L_F | := (fieldOver F). | Notation | L_F | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"fieldOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_scale (a : K_F) (u : L_F) : L_F | := vsval a * u. | Definition | fieldOver_scale | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"K_F",
"L_F",
"vsval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_scaleA a b u : a *F: (b *F: u) = (a * b) *F: u. | Proof. exact: mulrA. Qed. | Fact | fieldOver_scaleA | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_scale1 u : 1 *F: u = u. | Proof. by rewrite /(1 *F: u) /= algid1 mul1r. Qed. | Fact | fieldOver_scale1 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"algid1",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_scaleDr a u v : a *F: (u + v) = a *F: u + a *F: v. | Proof. exact: mulrDr. Qed. | Fact | fieldOver_scaleDr | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_scaleDl v a b : (a + b) *F: v = a *F: v + b *F: v. | Proof. exact: mulrDl. Qed. | Fact | fieldOver_scaleDl | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_scaleE a (u : L) : a *: (u : L_F) = vsval a * u. | Proof. by []. Qed. | Lemma | fieldOver_scaleE | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"L_F",
"vsval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_scaleAl a u v : a *F: (u * v) = (a *F: u) * v. | Proof. exact: mulrA. Qed. | Fact | fieldOver_scaleAl | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_vectMixin : Lmodule_hasFinDim K_F L_F. | Proof.
have [bL [_ nz_bL] [defL dxSbL]] := field_module_semisimple (subvf (F * _)).
do [set n := \dim_F {:L} in bL nz_bL *; set SbL := (\sum_i _)%VS] in defL dxSbL.
have in_bL i (a : K_F) : val a * (bL`_i : L_F) \in (F * <[bL`_i]>)%VS.
by rewrite memv_mul ?(valP a) ?memv_line.
have nz_bLi (i : 'I_n): bL`_i != 0 by re... | Fact | fieldOver_vectMixin | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"Build",
"K_F",
"L_F",
"apply",
"big_split",
"can2_linear",
"directv_sum_unique",
"eq_bigr",
"field_module_semisimple",
"fin_all_exists",
"forall_inP",
"linear",
"memPn",
"memt_nth",
"memv_cosetP",
"memv_line",
"memv_mul",
"memv_sumP",
"memvf",
"mulIf",
"mxE",
"r2v",
"r2v... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trivial_fieldOver : (1%VS : {vspace L_F}) =i F. | Proof.
move=> x; apply/vlineP/idP=> [[{}x ->] | Fx].
by rewrite fieldOver_scaleE mulr1 (valP x).
by exists (vsproj F x); rewrite fieldOver_scaleE mulr1 vsprojK.
Qed. | Lemma | trivial_fieldOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"L_F",
"apply",
"fieldOver_scaleE",
"mulr1",
"valP",
"vlineP",
"vsproj",
"vsprojK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vspaceOver V | := <<vbasis V : seq L_F>>%VS. | Definition | vspaceOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"L_F",
"seq",
"vbasis"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_vspaceOver V : vspaceOver V =i (F * V)%VS. | Proof.
move=> y; apply/idP/idP; last rewrite unlock; move/coord_span->.
rewrite (@memv_suml F0 L) // => i _.
by rewrite memv_mul ?subvsP // vbasis_mem ?memt_nth.
rewrite memv_suml // => ij _; rewrite -tnth_nth; set x := tnth _ ij.
have/allpairsP[[u z] /= [Fu Vz {x}->]]: x \in _ := mem_tnth ij _.
by rewrite scalerAl... | Lemma | mem_vspaceOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0",
"allpairsP",
"apply",
"coord_span",
"last",
"mem_tnth",
"memt_nth",
"memvZ",
"memv_mul",
"memv_span",
"memv_suml",
"scalerAl",
"subvsP",
"tnth",
"tnth_nth",
"vbasis_mem",
"vspaceOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_aspaceOver E : (F <= E)%VS -> vspaceOver E =i E. | Proof.
by move=> sFE y; rewrite mem_vspaceOver field_module_eq ?sup_field_module.
Qed. | Lemma | mem_aspaceOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"field_module_eq",
"mem_vspaceOver",
"sup_field_module",
"vspaceOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aspaceOver_suproof E : is_aspace (vspaceOver E). | Proof.
rewrite /is_aspace has_algid1; first by rewrite mem_vspaceOver (@mem1v _ L).
by apply/prodvP=> u v; rewrite !mem_vspaceOver; apply: memvM.
Qed. | Fact | aspaceOver_suproof | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"has_algid1",
"is_aspace",
"mem1v",
"mem_vspaceOver",
"memvM",
"prodvP",
"vspaceOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aspaceOver E | := ASpace (aspaceOver_suproof E). | Canonical | aspaceOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"aspaceOver_suproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_vspaceOver M : (F * M <= M)%VS -> \dim (vspaceOver M) = \dim_F M. | Proof.
move=> modM; have [] := field_module_semisimple modM.
set n := \dim_F M => b [Mb nz_b] [defM dx_b].
suff: basis_of (vspaceOver M) b by apply: size_basis.
apply/andP; split.
rewrite eqEsubv; apply/andP; split; apply/span_subvP=> u.
by rewrite mem_vspaceOver field_module_eq // => /Mb.
move/(@vbasis_mem _ _... | Lemma | dim_vspaceOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"L_F",
"apply",
"basis_of",
"dim",
"directv_sum_independent",
"eqEsubv",
"field_module_eq",
"field_module_semisimple",
"freeP",
"memPn",
"mem_vspaceOver",
"memt_nth",
"memvZ",
"memv_cosetP",
"memv_line",
"memv_mul",
"memv_span",
"memv_sumP",
"memv_suml",
"predU1P",
"scaler_eq... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_aspaceOver E : (F <= E)%VS -> \dim (vspaceOver E) = \dim_F E. | Proof. by rewrite -sup_field_module; apply: dim_vspaceOver. Qed. | Lemma | dim_aspaceOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"dim",
"dim_vspaceOver",
"sup_field_module",
"vspaceOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vspaceOverP V_F :
{V | [/\ V_F = vspaceOver V, (F * V <= V)%VS & V_F =i V]}. | Proof.
pose V := (F * <<vbasis V_F : seq L>>)%VS.
have idV: (F * V)%VS = V by rewrite prodvA prodv_id.
suffices defVF: V_F = vspaceOver V.
by exists V; split=> [||u]; rewrite ?defVF ?mem_vspaceOver ?idV.
apply/vspaceP=> v; rewrite mem_vspaceOver idV.
do [apply/idP/idP; last rewrite /V unlock] => [/coord_vbasis|/coord... | Lemma | vspaceOverP | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"allpairsP",
"apply",
"coord_span",
"coord_vbasis",
"last",
"mem_tnth",
"mem_vspaceOver",
"memt_nth",
"memvZ",
"memv_mul",
"memv_span",
"memv_suml",
"mulr_sumr",
"prodvA",
"prodv_id",
"scalerAl",
"scalerCA",
"seq",
"split",
"subvsP",
"tnth",
"tnth_nth",
"vbasis",
"vbasi... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aspaceOverP (E_F : {subfield L_F}) :
{E | [/\ E_F = aspaceOver E, (F <= E)%VS & E_F =i E]}. | Proof.
have [V [defEF modV memV]] := vspaceOverP E_F.
have algE: has_algid V && (V * V <= V)%VS.
rewrite has_algid1; first by rewrite -memV mem1v.
by apply/prodvP=> u v; rewrite -!memV; apply: memvM.
by exists (ASpace algE); rewrite -sup_field_module; split; first apply: val_inj.
Qed. | Lemma | aspaceOverP | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"L_F",
"apply",
"aspaceOver",
"has_algid",
"has_algid1",
"mem1v",
"memvM",
"prodvP",
"split",
"sup_field_module",
"val_inj",
"vspaceOverP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseFieldType : Type | := L. | Definition | baseFieldType | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
L0 | := baseFieldType. | Notation | L0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"baseFieldType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseField_scale (a : F0) (u : L0) : L0 | := in_alg F a *: u. | Definition | baseField_scale | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0",
"L0",
"in_alg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseField_scaleA a b u : a *F0: (b *F0: u) = (a * b) *F0: u. | Proof. by rewrite [_ *F0: _]scalerA -rmorphM. Qed. | Fact | baseField_scaleA | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0",
"rmorphM",
"scalerA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseField_scale1 u : 1 *F0: u = u. | Proof. by rewrite /(1 *F0: u) rmorph1 scale1r. Qed. | Fact | baseField_scale1 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0",
"rmorph1",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseField_scaleDr a u v : a *F0: (u + v) = a *F0: u + a *F0: v. | Proof. exact: scalerDr. Qed. | Fact | baseField_scaleDr | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0",
"scalerDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseField_scaleDl v a b : (a + b) *F0: v = a *F0: v + b *F0: v. | Proof. by rewrite -scalerDl -rmorphD. Qed. | Fact | baseField_scaleDl | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0",
"rmorphD",
"scalerDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseField_scaleE a (u : L) : a *: (u : L0) = a%:A *: u. | Proof. by []. Qed. | Lemma | baseField_scaleE | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"L0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseField_scaleAl a (u v : L0) : a *F0: (u * v) = (a *F0: u) * v. | Proof. exact: scalerAl. Qed. | Fact | baseField_scaleAl | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0",
"L0",
"scalerAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n | := \dim {:F}. | Let | n | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"dim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bF : n.-tuple F | := vbasis {:F}. | Let | bF | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"tuple",
"vbasis"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coordF (x : F) | := (coord_vbasis (memvf x)). | Let | coordF | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"coord_vbasis",
"memvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseField_vectMixin : Lmodule_hasFinDim F0 L0. | Proof.
pose bL := vbasis {:L}; set m := \dim {:L} in bL.
pose v2r (x : L0) := mxvec (\matrix_(i, j) coord bF j (coord bL i x)).
have v2r_lin: linear v2r.
move=> a x y; rewrite -linearP; congr mxvec; apply/matrixP=> i j.
by rewrite !mxE linearP /= mulr_algl linearP.
pose r2v r := \sum_(i < m) (\sum_(j < n) vec_mx r ... | Fact | baseField_vectMixin | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0",
"L0",
"apply",
"bF",
"basis_free",
"coord",
"coordF",
"coord_sum_free",
"coord_vbasis",
"dim",
"eq_bigr",
"last",
"linear",
"linearP",
"matrixP",
"memvf",
"mulr_algl",
"mxE",
"mxvec",
"mxvecK",
"r2v",
"v2r",
"v2rK",
"vbasis",
"vbasisP",
"vec_mx",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F0ZEZ a x v : a *: ((x *: v : L) : L0) = (a *: x) *: v. | Proof. by rewrite [a *: _]scalerA -scalerAl mul1r. Qed. | Let | F0ZEZ | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"L0",
"mul1r",
"scalerA",
"scalerAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseVspace_basis V : seq L0 | :=
[seq tnth bF ij.2 *: tnth (vbasis V) ij.1 | ij : 'I_(\dim V) * 'I_n]. | Let | baseVspace_basis | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"L0",
"bF",
"dim",
"seq",
"tnth",
"vbasis"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseVspace V | := <<baseVspace_basis V>>%VS. | Definition | baseVspace | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"baseVspace_basis"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_baseVspace V : baseVspace V =i V. | Proof.
move=> y; apply/idP/idP=> [/coord_span->|/coord_vbasis->]; last first.
apply: memv_suml => i _; rewrite (coordF (coord _ i (y : L))) scaler_suml -/n.
apply: memv_suml => j _; rewrite -/bF -F0ZEZ memvZ ?memv_span // -!tnth_nth.
by apply/imageP; exists (i, j).
(* GG: the F0ZEZ lemma avoids serious performa... | Lemma | mem_baseVspace | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0ZEZ",
"apply",
"bF",
"baseVspace",
"coord",
"coordF",
"coord_span",
"coord_vbasis",
"enum_val",
"imageP",
"last",
"mem_tnth",
"memvZ",
"memv_span",
"memv_suml",
"nth_image",
"scaler_suml",
"tnth_nth",
"vbasis_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_baseVspace V : \dim (baseVspace V) = (\dim V * n)%N. | Proof.
pose bV0 := baseVspace_basis V; set m := \dim V in bV0 *.
suffices /size_basis->: basis_of (baseVspace V) bV0.
by rewrite card_prod !card_ord.
rewrite /basis_of eqxx.
apply/freeP=> s sb0 k; rewrite -(enum_valK k); case/enum_val: k => i j.
have free_baseP := freeP (basis_free (vbasisP _)).
move: j; apply: (free... | Lemma | dim_baseVspace | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F0ZEZ",
"apply",
"baseVspace",
"baseVspace_basis",
"basis_free",
"basis_of",
"card_ord",
"card_prod",
"dim",
"enum_rank",
"enum_rankK",
"enum_val",
"enum_valK",
"enum_val_bij",
"eq_bigr",
"eqxx",
"freeP",
"fullv",
"nth_image",
"pair_bigA",
"reindex",
"scaler_suml",
"size... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseAspace_suproof (E : {subfield L}) : is_aspace (baseVspace E). | Proof.
rewrite /is_aspace has_algid1; first by rewrite mem_baseVspace (mem1v E).
by apply/prodvP=> u v; rewrite !mem_baseVspace; apply: memvM.
Qed. | Fact | baseAspace_suproof | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"baseVspace",
"has_algid1",
"is_aspace",
"mem1v",
"mem_baseVspace",
"memvM",
"prodvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseAspace E | := ASpace (baseAspace_suproof E). | Canonical | baseAspace | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"baseAspace_suproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
refBaseField_unlockable | := Unlockable refBaseField.unlock. | Canonical | refBaseField_unlockable | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F1 | := (refBaseField L). | Notation | F1 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F1unlock | := refBaseField.unlock. | Notation | F1unlock | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
L0 | := (baseFieldType L). | Notation | L0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"baseFieldType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_refBaseField : \dim F1 = n. | Proof. by rewrite F1unlock dim_baseVspace dimv1 mul1n. Qed. | Lemma | dim_refBaseField | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F1",
"F1unlock",
"dim",
"dim_baseVspace",
"dimv1",
"mul1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
baseVspace_module V (V0 := baseVspace V) : (F1 * V0 <= V0)%VS. | Proof.
apply/prodvP=> u v; rewrite F1unlock !mem_baseVspace => /vlineP[x ->] Vv.
by rewrite -(@scalerAl F L) mul1r; apply: memvZ.
Qed. | Lemma | baseVspace_module | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F1",
"F1unlock",
"apply",
"baseVspace",
"mem_baseVspace",
"memvZ",
"mul1r",
"prodvP",
"scalerAl",
"vlineP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_baseField (E : {subfield L}) : (F1 <= baseVspace E)%VS. | Proof. by rewrite -sup_field_module baseVspace_module. Qed. | Lemma | sub_baseField | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F1",
"baseVspace",
"baseVspace_module",
"sup_field_module"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vspaceOver_refBase V : vspaceOver F1 (baseVspace V) =i V. | Proof.
move=> v; rewrite mem_vspaceOver field_module_eq ?baseVspace_module //.
by rewrite mem_baseVspace.
Qed. | Lemma | vspaceOver_refBase | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F1",
"baseVspace",
"baseVspace_module",
"field_module_eq",
"mem_baseVspace",
"mem_vspaceOver",
"vspaceOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
module_baseVspace M0 :
(F1 * M0 <= M0)%VS -> {V | M0 = baseVspace V & M0 =i V}. | Proof.
move=> modM0; pose V := <<vbasis (vspaceOver F1 M0) : seq L>>%VS.
suffices memM0: M0 =i V.
by exists V => //; apply/vspaceP=> v; rewrite mem_baseVspace memM0.
move=> v; rewrite -{1}(field_module_eq modM0) -(mem_vspaceOver M0) {}/V.
move: (vspaceOver F1 M0) => M.
apply/idP/idP=> [/coord_vbasis|/coord_span]->; a... | Lemma | module_baseVspace | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F1",
"F1unlock",
"apply",
"baseVspace",
"coord",
"coord_span",
"coord_vbasis",
"fieldOver_scale",
"field_module_eq",
"mem1v",
"mem_baseVspace",
"mem_tnth",
"mem_vspaceOver",
"memt_nth",
"memvZ",
"memv_span",
"memv_suml",
"mul1r",
"scalerAl",
"seq",
"tnth_nth",
"vbasis",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
module_baseAspace (E0 : {subfield L0}) :
(F1 <= E0)%VS -> {E | E0 = baseAspace E & E0 =i E}. | Proof.
rewrite -sup_field_module => /module_baseVspace[E defE0 memE0].
suffices algE: is_aspace E by exists (ASpace algE); first apply: val_inj.
rewrite /is_aspace has_algid1 -?memE0 ?mem1v //.
by apply/prodvP=> u v; rewrite -!memE0; apply: memvM.
Qed. | Lemma | module_baseAspace | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"F1",
"L0",
"apply",
"baseAspace",
"has_algid1",
"is_aspace",
"mem1v",
"memvM",
"module_baseVspace",
"prodvP",
"sup_field_module",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
base_vspaceOver V : baseVspace (vspaceOver F V) =i (F * V)%VS. | Proof. by move=> v; rewrite mem_baseVspace mem_vspaceOver. Qed. | Lemma | base_vspaceOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"baseVspace",
"mem_baseVspace",
"mem_vspaceOver",
"vspaceOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
base_moduleOver V : (F * V <= V)%VS -> baseVspace (vspaceOver F V) =i V. | Proof. by move=> /field_module_eq defV v; rewrite base_vspaceOver defV. Qed. | Lemma | base_moduleOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"baseVspace",
"base_vspaceOver",
"field_module_eq",
"vspaceOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
base_aspaceOver (E : {subfield L}) :
(F <= E)%VS -> baseVspace (vspaceOver F E) =i E. | Proof. by rewrite -sup_field_module; apply: base_moduleOver. Qed. | Lemma | base_aspaceOver | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"baseVspace",
"base_moduleOver",
"sup_field_module",
"vspaceOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^iota" | := (map_poly (GRing.RMorphism.sort iota) p)
(format "p ^iota") : ring_scope. | Notation | p ^iota | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iota",
"map_poly",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
wf_p | := (p != 0) && root p^iota z. | Let | wf_p | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iota",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p0 : {poly F} | := if wf_p then (lead_coef p)^-1 *: p else 'X. | Let | p0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"lead_coef",
"poly",
"wf_p"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
z0 | := if wf_p then z else 0. | Let | z0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"wf_p"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n | := (size p0).-1. | Let | n | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"p0",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p0_mon : p0 \is monic. | Proof.
rewrite /p0; case: ifP => [/andP[nz_p _] | _]; last exact: monicX.
by rewrite monicE lead_coefZ mulVf ?lead_coef_eq0.
Qed. | Let | p0_mon | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"last",
"lead_coefZ",
"lead_coef_eq0",
"monic",
"monicE",
"monicX",
"mulVf",
"nz_p",
"p0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz_p0 : p0 != 0. | Proof. by rewrite monic_neq0 // p0_mon. Qed. | Let | nz_p0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"monic_neq0",
"p0",
"p0_mon"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p0z0 : root p0^iota z0. | Proof.
rewrite /p0 /z0; case: ifP => [/andP[_ pz0]|]; last by rewrite map_polyX rootX.
by rewrite map_polyZ rootE hornerZ (rootP pz0) mulr0.
Qed. | Let | p0z0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"hornerZ",
"iota",
"last",
"map_polyX",
"map_polyZ",
"mulr0",
"p0",
"pz0",
"root",
"rootE",
"rootP",
"rootX",
"z0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_gt0: 0 < n. | Proof.
rewrite /n -subn1 subn_gt0 -(size_map_poly iota).
by rewrite (root_size_gt1 _ p0z0) ?map_poly_eq0.
Qed. | Let | n_gt0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iota",
"map_poly_eq0",
"p0z0",
"root_size_gt1",
"size_map_poly",
"subn1",
"subn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
z0Ciota : commr_rmorph iota z0. | Proof. by move=> x; apply: mulrC. Qed. | Let | z0Ciota | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"commr_rmorph",
"iota",
"mulrC",
"z0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iotaPz | := (horner_morph z0Ciota). | Notation | iotaPz | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"horner_morph",
"z0Ciota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iotaFz (x : 'rV[F]_n) | := iotaPz (rVpoly x). | Let | iotaFz | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iotaPz",
"rVpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
equiv_subfext x y | := (iotaFz x == iotaFz y). | Definition | equiv_subfext | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iotaFz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
equiv_subfext_is_equiv : equiv_class_of equiv_subfext. | Proof. by rewrite /equiv_subfext; split=> x // y w /eqP->. Qed. | Fact | equiv_subfext_is_equiv | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"equiv_class_of",
"equiv_subfext",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
equiv_subfext_equiv | := EquivRelPack equiv_subfext_is_equiv. | Canonical | equiv_subfext_equiv | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"equiv_subfext_is_equiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
equiv_subfext_encModRel | := defaultEncModRel equiv_subfext. | Canonical | equiv_subfext_encModRel | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"defaultEncModRel",
"equiv_subfext"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subFExtend | := {eq_quot equiv_subfext}. | Definition | subFExtend | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"equiv_subfext"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inj | := lift_fun1 subFExtend iotaFz. | Definition | subfx_inj | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iotaFz",
"lift_fun1",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfx_inj : {mono \pi : x / iotaFz x >-> subfx_inj x}. | Proof.
unlock subfx_inj => x; apply/eqP; rewrite -/(equiv_subfext _ x).
by rewrite -eqmodE reprK.
Qed. | Fact | pi_subfx_inj | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"eqmodE",
"equiv_subfext",
"iotaFz",
"pi",
"reprK",
"subfx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfx_inj_morph | := PiMono1 pi_subfx_inj. | Canonical | pi_subfx_inj_morph | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"PiMono1",
"pi_subfx_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iotaPz_repr x : iotaPz (rVpoly (repr (\pi_(subFExtend) x))) = iotaFz x. | Proof. by rewrite -/(iotaFz _) -!pi_subfx_inj reprK. Qed. | Let | iotaPz_repr | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iotaFz",
"iotaPz",
"pi_subfx_inj",
"rVpoly",
"repr",
"reprK",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfext0 | := lift_cst subFExtend 0. | Definition | subfext0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"lift_cst",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfext0_morph | := PiConst subfext0. | Canonical | subfext0_morph | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"PiConst",
"subfext0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfext_add | := lift_op2 subFExtend +%R. | Definition | subfext_add | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"lift_op2",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfext_add : {morph \pi : x y / x + y >-> subfext_add x y}. | Proof.
unlock subfext_add => x y /=; apply/eqmodP/eqP.
by rewrite /iotaFz !linearD /= !iotaPz_repr.
Qed. | Fact | pi_subfext_add | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"eqmodP",
"iotaFz",
"iotaPz_repr",
"linearD",
"pi",
"subfext_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfx_add_morph | := PiMorph2 pi_subfext_add. | Canonical | pi_subfx_add_morph | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"PiMorph2",
"pi_subfext_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfext_opp | := lift_op1 subFExtend -%R. | Definition | subfext_opp | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"lift_op1",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfext_opp : {morph \pi : x / - x >-> subfext_opp x}. | Proof.
unlock subfext_opp => y /=; apply/eqmodP/eqP.
by rewrite /iotaFz !linearN /= !iotaPz_repr.
Qed. | Fact | pi_subfext_opp | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"eqmodP",
"iotaFz",
"iotaPz_repr",
"linearN",
"pi",
"subfext_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfext_opp_morph | := PiMorph1 pi_subfext_opp. | Canonical | pi_subfext_opp_morph | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"PiMorph1",
"pi_subfext_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addfxA : associative subfext_add. | Proof. by move=> x y t; rewrite -[x]reprK -[y]reprK -[t]reprK !piE addrA. Qed. | Fact | addfxA | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"addrA",
"piE",
"reprK",
"subfext_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addfxC : commutative subfext_add. | Proof. by move=> x y; rewrite -[x]reprK -[y]reprK !piE addrC. Qed. | Fact | addfxC | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"addrC",
"piE",
"reprK",
"subfext_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add0fx : left_id subfext0 subfext_add. | Proof. by move=> x; rewrite -[x]reprK !piE add0r. Qed. | Fact | add0fx | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"add0r",
"piE",
"reprK",
"subfext0",
"subfext_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addfxN : left_inverse subfext0 subfext_opp subfext_add. | Proof. by move=> x; rewrite -[x]reprK !piE addNr. Qed. | Fact | addfxN | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"addNr",
"piE",
"reprK",
"subfext0",
"subfext_add",
"subfext_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_rV_modp_K q : rVpoly (poly_rV (q %% p0) : 'rV[F]_n) = q %% p0. | Proof. by apply: poly_rV_K; rewrite -ltnS -polySpred // ltn_modp. Qed. | Let | poly_rV_modp_K | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"ltnS",
"ltn_modp",
"p0",
"polySpred",
"poly_rV",
"poly_rV_K",
"rVpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iotaPz_modp q : iotaPz (q %% p0) = iotaPz q. | Proof.
rewrite {2}(divp_eq q p0) rmorphD rmorphM /=.
by rewrite [iotaPz p0](rootP p0z0) mulr0 add0r.
Qed. | Let | iotaPz_modp | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"add0r",
"divp_eq",
"iotaPz",
"mulr0",
"p0",
"p0z0",
"rmorphD",
"rmorphM",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_mul_rep (x y : 'rV[F]_n) : 'rV[F]_n | :=
poly_rV ((rVpoly x) * (rVpoly y) %% p0). | Definition | subfx_mul_rep | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"p0",
"poly_rV",
"rVpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfext_mul | := lift_op2 subFExtend subfx_mul_rep. | Definition | subfext_mul | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"lift_op2",
"subFExtend",
"subfx_mul_rep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfext_mul :
{morph \pi : x y / subfx_mul_rep x y >-> subfext_mul x y}. | Proof.
unlock subfext_mul => x y /=; apply/eqmodP/eqP.
by rewrite /iotaFz !poly_rV_modp_K !iotaPz_modp !rmorphM /= !iotaPz_repr.
Qed. | Fact | pi_subfext_mul | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"eqmodP",
"iotaFz",
"iotaPz_modp",
"iotaPz_repr",
"pi",
"poly_rV_modp_K",
"rmorphM",
"subfext_mul",
"subfx_mul_rep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfext_mul_morph | := PiMorph2 pi_subfext_mul. | Canonical | pi_subfext_mul_morph | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"PiMorph2",
"pi_subfext_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfext1 | := lift_cst subFExtend (poly_rV 1). | Definition | subfext1 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"lift_cst",
"poly_rV",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfext1_morph | := PiConst subfext1. | Canonical | subfext1_morph | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"PiConst",
"subfext1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulfxA : associative (subfext_mul). | Proof.
elim/quotW=> x; elim/quotW=> y; elim/quotW=> w; rewrite !piE /subfx_mul_rep.
by rewrite !poly_rV_modp_K [_ %% p0 * _]mulrC !modp_mul // mulrA mulrC.
Qed. | Fact | mulfxA | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"modp_mul",
"mulrA",
"mulrC",
"p0",
"piE",
"poly_rV_modp_K",
"quotW",
"subfext_mul",
"subfx_mul_rep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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