fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
lker0Pf : reflect (injective f) (lker f == 0%VS).
Proof.
rewrite -subv0; apply: (iffP subvP) => [injf u v eq_fuv | injf u].
apply/eqP; rewrite -subr_eq0 -memv0 injf //.
by rewrite memv_ker linearB /= eq_fuv subrr.
by rewrite memv_ker memv0 -(inj_eq injf) linear0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0P
| |
limg_ker0f U V : lker f == 0%VS -> (f @: U <= f @: V)%VS = (U <= V)%VS.
Proof.
move/lker0P=> injf; apply/idP/idP=> [/subvP sfUV | ]; last exact: limgS.
by apply/subvP=> u Uu; have /memv_imgP[v Vv /injf->] := sfUV _ (memv_img f Uu).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
limg_ker0
| |
eq_limg_ker0f U V : lker f == 0%VS -> (f @: U == f @: V)%VS = (U == V).
Proof. by move=> injf; rewrite !eqEsubv !limg_ker0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
eq_limg_ker0
| |
lker0_lfunKf : lker f == 0%VS -> cancel f f^-1%VF.
Proof.
by move/lker0P=> injf u; apply: injf; rewrite limg_lfunVK ?memv_img ?memvf.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_lfunK
| |
lker0_compVff : lker f == 0%VS -> (f^-1 \o f = \1)%VF.
Proof. by move/lker0_lfunK=> fK; apply/lfunP=> u; rewrite !lfunE /= fK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_compVf
| |
lker0_img_capf U V : lker f == 0%VS ->
(f @: (U :&: V) = f @: U :&: f @: V)%VS.
Proof.
move=> kf0; apply/eqP; rewrite eqEsubv limg_cap/=; apply/subvP => x.
rewrite memv_cap => /andP[/memv_imgP[u uU ->]] /memv_imgP[v vV].
by move=> /(lker0P _ kf0) eq_uv; rewrite memv_img// memv_cap uU eq_uv vV.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_img_cap
| |
fixedSpacef : {vspace vT} := lker (f - \1%VF).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
fixedSpace
| |
fixedSpacePf a : reflect (f a = a) (a \in fixedSpace f).
Proof.
by rewrite memv_ker add_lfunE opp_lfunE id_lfunE subr_eq0; apply: eqP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
fixedSpaceP
| |
fixedSpacesPf U : reflect {in U, f =1 id} (U <= fixedSpace f)%VS.
Proof. by apply: (iffP subvP) => cUf x /cUf/fixedSpaceP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
fixedSpacesP
| |
fixedSpace_limgf U : (U <= fixedSpace f -> f @: U = U)%VS.
Proof.
move/fixedSpacesP=> cUf; apply/vspaceP=> x.
by apply/memv_imgP/idP=> [[{}x Ux ->] | Ux]; last exists x; rewrite ?cUf.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
fixedSpace_limg
| |
fixedSpace_id: fixedSpace \1 = {:vT}%VS.
Proof.
by apply/vspaceP=> x; rewrite memvf; apply/fixedSpaceP; rewrite lfunE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
fixedSpace_id
| |
lker0_limgf: limg f = fullv.
Proof.
by apply/eqP; rewrite eqEdim subvf limg_dim_eq //= (eqP kerf0) capv0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_limgf
| |
lker0_lfunVK: cancel f^-1%VF f.
Proof. by move=> u; rewrite limg_lfunVK // lker0_limgf memvf. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_lfunVK
| |
lker0_compfV: (f \o f^-1 = \1)%VF.
Proof. by apply/lfunP=> u; rewrite !lfunE /= lker0_lfunVK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_compfV
| |
lker0_compVKfaT g : (f \o (f^-1 \o g))%VF = g :> 'Hom(aT, vT).
Proof. by rewrite comp_lfunA lker0_compfV comp_lfun1l. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_compVKf
| |
lker0_compKfaT g : (f^-1 \o (f \o g))%VF = g :> 'Hom(aT, vT).
Proof. by rewrite comp_lfunA lker0_compVf ?comp_lfun1l. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_compKf
| |
lker0_compfKrT h : ((h \o f) \o f^-1)%VF = h :> 'Hom(vT, rT).
Proof. by rewrite -comp_lfunA lker0_compfV comp_lfun1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_compfK
| |
lker0_compfVKrT h : ((h \o f^-1) \o f)%VF = h :> 'Hom(vT, rT).
Proof. by rewrite -comp_lfunA lker0_compVf ?comp_lfun1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker0_compfVK
| |
lim1gU : (\1 @: U)%VS = U.
Proof.
have /andP[/eqP <- _] := vbasisP U; rewrite limg_span map_id_in // => u _.
by rewrite lfunE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lim1g
| |
limg_compf g U : ((g \o f) @: U = g @: (f @: U))%VS.
Proof.
have /andP[/eqP <- _] := vbasisP U; rewrite !limg_span; congr (span _).
by rewrite -map_comp; apply/eq_map => u; rewrite lfunE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
limg_comp
| |
lpreim_cap_limgf W : (f @^-1: (W :&: limg f))%VS = (f @^-1: W)%VS.
Proof. by rewrite /lfun_preim -capvA capvv. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lpreim_cap_limg
| |
lpreim0f : (f @^-1: 0)%VS = lker f.
Proof. by rewrite /lfun_preim cap0v limg0 add0v. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lpreim0
| |
lpreimSf V W : (V <= W)%VS-> (f @^-1: V <= f @^-1: W)%VS.
Proof. by move=> sVW; rewrite addvS // limgS // capvS. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lpreimS
| |
lpreimKf W : (W <= limg f)%VS -> (f @: (f @^-1: W))%VS = W.
Proof.
move=> sWf; rewrite limgD (capv_idPl sWf) // -limg_comp.
have /eqP->: (f @: lker f == 0)%VS by rewrite -lkerE.
have /andP[/eqP defW _] := vbasisP W; rewrite addv0 -defW limg_span.
rewrite map_id_in // => x Xx; rewrite lfunE /= limg_lfunVK //.
by apply: span_subvP Xx; rewrite defW.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lpreimK
| |
memv_preimf u W : (f u \in W) = (u \in f @^-1: W)%VS.
Proof.
apply/idP/idP=> [Wfu | /(memv_img f)]; last first.
by rewrite -lpreim_cap_limg lpreimK ?capvSr // => /memv_capP[].
rewrite -[u](addNKr (f^-1%VF (f u))) memv_add ?memv_img //.
by rewrite memv_cap Wfu memv_img ?memvf.
by rewrite memv_ker addrC linearB /= subr_eq0 limg_lfunVK ?memv_img ?memvf.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
memv_preim
| |
lfun_comp_nzRingMixin:= GRing.Zmodule_isNzRing.Build 'End(vT)
comp_lfunA comp_lfun1l comp_lfun1r comp_lfunDl comp_lfunDr lfun1_neq0.
#[deprecated(since="mathcomp 2.4.0",
note="Use lfun_comp_nzRingMixin instead.")]
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_comp_nzRingMixin
| |
lfun_comp_ringMixin:= (lfun_comp_nzRingMixin) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_comp_ringMixin
| |
lfun_comp_nzRingType: nzRingType :=
HB.pack 'End(vT) lfun_comp_nzRingMixin.
#[deprecated(since="mathcomp 2.4.0",
note="Use lfun_comp_nzRingType instead.")]
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_comp_nzRingType
| |
lfun_comp_ringType:= (lfun_comp_nzRingType) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_comp_ringType
| |
lfun_nzRingType: nzRingType := lfun_comp_nzRingType^c.
#[deprecated(since="mathcomp 2.4.0",
note="Use lfun_nzRingType instead.")]
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_nzRingType
| |
lfun_ringType:= (lfun_nzRingType) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_ringType
| |
lfun_lalgMixin:= GRing.Lmodule_isLalgebra.Build R lfun_nzRingType
(fun k x y => comp_lfunZr k y x).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_lalgMixin
| |
lfun_lalgType: lalgType R :=
HB.pack 'End(vT) lfun_nzRingType lfun_lalgMixin.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_lalgType
| |
lfun_algMixin:= GRing.Lalgebra_isAlgebra.Build R lfun_lalgType
(fun k x y => comp_lfunZl k y x).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_algMixin
| |
lfun_algType: algType R :=
HB.pack 'End(vT) lfun_lalgType lfun_algMixin.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lfun_algType
| |
daddv_piU V := Hom (proj_mx (vs2mx U) (vs2mx V)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
daddv_pi
| |
projvU := daddv_pi U U^C.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
projv
| |
addv_pi1U V := daddv_pi (U :\: V) V.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
addv_pi1
| |
addv_pi2U V := daddv_pi V (U :\: V).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
addv_pi2
| |
memv_piU V w : (daddv_pi U V) w \in U.
Proof. by rewrite unlock memvE /subsetv genmxE /= r2vK proj_mx_sub. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
memv_pi
| |
memv_projU w : projv U w \in U. Proof. exact: memv_pi. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
memv_proj
| |
memv_pi1U V w : (addv_pi1 U V) w \in U.
Proof. by rewrite (subvP (diffvSl U V)) ?memv_pi. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
memv_pi1
| |
memv_pi2U V w : (addv_pi2 U V) w \in V. Proof. exact: memv_pi. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
memv_pi2
| |
daddv_pi_idU V u : (U :&: V = 0)%VS -> u \in U -> daddv_pi U V u = u.
Proof.
move/eqP; rewrite -dimv_eq0 memvE /subsetv /dimv !genmxE mxrank_eq0 => /eqP.
by move=> dxUV Uu; rewrite unlock /= proj_mx_id ?v2rK.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
daddv_pi_id
| |
daddv_pi_projU V w (pi := daddv_pi U V) :
(U :&: V = 0)%VS -> pi (pi w) = pi w.
Proof. by move/daddv_pi_id=> -> //; apply: memv_pi. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
daddv_pi_proj
| |
daddv_pi_addU V w :
(U :&: V = 0)%VS -> (w \in U + V)%VS -> daddv_pi U V w + daddv_pi V U w = w.
Proof.
move/eqP; rewrite -dimv_eq0 memvE /subsetv /dimv !genmxE mxrank_eq0 => /eqP.
by move=> dxUW UVw; rewrite unlock /= -linearD /= add_proj_mx ?v2rK.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
daddv_pi_add
| |
projv_idU u : u \in U -> projv U u = u.
Proof. exact: daddv_pi_id (capv_compl _). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
projv_id
| |
projv_projU w : projv U (projv U w) = projv U w.
Proof. exact: daddv_pi_proj (capv_compl _). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
projv_proj
| |
memv_projCU w : w - projv U w \in (U^C)%VS.
Proof.
rewrite -{1}[w](daddv_pi_add (capv_compl U)) ?addv_complf ?memvf //.
by rewrite addrC addKr memv_pi.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
memv_projC
| |
limg_projU : limg (projv U) = U.
Proof.
apply/vspaceP=> u; apply/memv_imgP/idP=> [[u1 _ ->] | ]; first exact: memv_proj.
by exists (projv U u); rewrite ?projv_id ?memv_img ?memvf.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
limg_proj
| |
lker_projU : lker (projv U) = (U^C)%VS.
Proof.
apply/eqP; rewrite eqEdim andbC; apply/andP; split.
by rewrite dimv_compl -(limg_ker_dim (projv U) fullv) limg_proj addnK capfv.
by apply/subvP=> v; rewrite memv_ker -{2}[v]subr0 => /eqP <-; apply: memv_projC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
lker_proj
| |
addv_pi1_projU V w (pi1 := addv_pi1 U V) : pi1 (pi1 w) = pi1 w.
Proof. by rewrite daddv_pi_proj // capv_diff. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
addv_pi1_proj
| |
addv_pi2_idU V v : v \in V -> addv_pi2 U V v = v.
Proof. by apply: daddv_pi_id; rewrite capvC capv_diff. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
addv_pi2_id
| |
addv_pi2_projU V w (pi2 := addv_pi2 U V) : pi2 (pi2 w) = pi2 w.
Proof. by rewrite addv_pi2_id ?memv_pi2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
addv_pi2_proj
| |
addv_pi1_pi2U V w :
w \in (U + V)%VS -> addv_pi1 U V w + addv_pi2 U V w = w.
Proof. by rewrite -addv_diff; exact/daddv_pi_add/capv_diff. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
addv_pi1_pi2
| |
sumV:= (\sum_(i <- r0 | P i) Vs i)%VS.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
sumV
| |
sumv_pi_forV of V = sumV := fun i => sumv_pi_rec i (filter P r0).
Variables (V : {vspace vT}) (defV : V = sumV).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
sumv_pi_for
| |
memv_sum_pii v : sumv_pi_for defV i v \in Vs i.
Proof.
rewrite /sumv_pi_for.
elim: (filter P r0) v => [|j r IHr] v /=; first by rewrite lfunE mem0v.
by case: eqP => [->|_]; rewrite ?lfunE ?memv_pi1 /=.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
memv_sum_pi
| |
sumv_pi_uniq_sumv :
uniq (filter P r0) -> v \in V ->
\sum_(i <- r0 | P i) sumv_pi_for defV i v = v.
Proof.
rewrite /sumv_pi_for defV -!(big_filter r0 P).
elim: (filter P r0) v => [|i r IHr] v /= => [_ | /andP[r'i /IHr{}IHr]].
by rewrite !big_nil memv0 => /eqP.
rewrite !big_cons eqxx => /addv_pi1_pi2; congr (_ + _ = v).
rewrite -[_ v]IHr ?memv_pi2 //; apply: eq_big_seq => j /=.
by case: eqP => [<- /idPn | _]; rewrite ?lfunE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
sumv_pi_uniq_sum
| |
sumv_piV := (sumv_pi_for (erefl V)).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
sumv_pi
| |
sumv_pi_sum(I : finType) (P : pred I) Vs v (V : {vspace vT})
(defV : V = (\sum_(i | P i) Vs i)%VS) :
v \in V -> \sum_(i | P i) sumv_pi_for defV i v = v :> vT.
Proof. by apply: sumv_pi_uniq_sum; have [e _ []] := big_enumP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
sumv_pi_sum
| |
sumv_pi_nat_summ n (P : pred nat) Vs v (V : {vspace vT})
(defV : V = (\sum_(m <= i < n | P i) Vs i)%VS) :
v \in V -> \sum_(m <= i < n | P i) sumv_pi_for defV i v = v :> vT.
Proof. by apply: sumv_pi_uniq_sum; apply/filter_uniq/iota_uniq. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
sumv_pi_nat_sum
| |
subvs_of: predArgType := Subvs u & u \in U.
|
Inductive
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
subvs_of
| |
vsvalw : vT := let: Subvs u _ := w in u.
HB.instance Definition _ := [isSub of subvs_of for vsval].
HB.instance Definition _ := [Choice of subvs_of by <:].
HB.instance Definition _ := [SubChoice_isSubZmodule of subvs_of by <:].
HB.instance Definition _ := [SubZmodule_isSubLmodule of subvs_of by <:].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsval
| |
subvsPw : vsval w \in U. Proof. exact: valP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
subvsP
| |
subvs_inj: injective vsval. Proof. exact: val_inj. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
subvs_inj
| |
congr_subvsu v : u = v -> vsval u = vsval v. Proof. exact: congr1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
congr_subvs
| |
vsval_is_linear: linear vsval. Proof. by []. Qed.
HB.instance Definition _ := GRing.isSemilinear.Build K subvs_of vT _ vsval
(GRing.semilinear_linear vsval_is_linear).
Fact vsproj_key : unit. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsval_is_linear
| |
vsproj_defu := Subvs (memv_proj U u).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsproj_def
| |
vsproj:= locked_with vsproj_key vsproj_def.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsproj
| |
vsproj_unlockable:= [unlockable fun vsproj].
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsproj_unlockable
| |
vsprojK: {in U, cancel vsproj vsval}.
Proof. by rewrite unlock; apply: projv_id. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsprojK
| |
vsvalK: cancel vsval vsproj.
Proof. by move=> w; apply/val_inj/vsprojK/subvsP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsvalK
| |
vsproj_is_linear: linear vsproj.
Proof. by move=> k w1 w2; apply: val_inj; rewrite unlock /= linearP. Qed.
HB.instance Definition _ := GRing.isSemilinear.Build K vT subvs_of _ vsproj
(GRing.semilinear_linear vsproj_is_linear).
Fact subvs_vect_iso : Vector.axiom (\dim U) subvs_of.
Proof.
exists (fun w => \row_i coord (vbasis U) i (vsval w)).
by move=> k w1 w2; apply/rowP=> i; rewrite !mxE linearP.
exists (fun rw : 'rV_(\dim U) => vsproj (\sum_i rw 0 i *: (vbasis U)`_i)).
move=> w /=; congr (vsproj _ = w): (vsvalK w).
by rewrite {1}(coord_vbasis (subvsP w)); apply: eq_bigr => i _; rewrite mxE.
move=> rw; apply/rowP=> i; rewrite mxE vsprojK.
by rewrite coord_sum_free ?(basis_free (vbasisP U)).
by apply: rpred_sum => j _; rewrite rpredZ ?vbasis_mem ?memt_nth.
Qed.
HB.instance Definition _ := Lmodule_hasFinDim.Build K subvs_of subvs_vect_iso.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsproj_is_linear
| |
SubvsEx (xU : x \in U) : Subvs xU = vsproj x.
Proof. by apply/val_inj; rewrite /= vsprojK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
SubvsE
| |
Definition_ := LSemiModule_hasFinDim.Build _ 'M[R]_(m, n)
matrix_vect_iso.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
Definition
| |
dim_matrix: dim 'M[R]_(m, n) = m * n.
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
dim_matrix
| |
Definition_ (R : nzRingType) (m n : nat) :=
SemiVector.on 'M[R]_(m, n).
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
Definition
| |
Definition_ := LSemiModule_hasFinDim.Build _ R^o regular_vect_iso.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
Definition
| |
Definition_ (R : nzRingType) := SemiVector.on R^o.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
Definition
| |
Definition_ := LSemiModule_hasFinDim.Build _ (vT1 * vT2)%type
pair_vect_iso.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
Definition
| |
Definition_ (R : nzRingType) (vT1 vT2 : vectType R) :=
SemiVector.on (vT1 * vT2)%type.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
Definition
| |
Definition_ := LSemiModule_hasFinDim.Build _ {ffun I -> vT}
ffun_vect_iso.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
Definition
| |
Definition_ (I : finType) (R : nzRingType) (vT : vectType R) :=
SemiVector.on {ffun I -> vT}.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
Definition
| |
vsolve_eqU := finfun (tnth rhs) \in (linfun lhsf @: U)%VS.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsolve_eq
| |
vsolve_eqP(U : {vspace vT}) :
reflect (exists2 u, u \in U & forall i, tnth lhs i u = tnth rhs i)
(vsolve_eq U).
Proof.
have lhsZ: linear lhsf by move=> a u v; apply/ffunP=> i; rewrite !ffunE linearP.
pose lhslM := GRing.isLinear.Build _ _ _ _ lhsf lhsZ.
pose lhsL : {linear _ -> _} := HB.pack lhsf lhslM.
apply: (iffP memv_imgP) => [] [u Uu sol_u]; exists u => //.
by move=> i; rewrite -[tnth rhs i]ffunE sol_u (lfunE lhsL) ffunE.
by apply/ffunP=> i; rewrite (lfunE lhsL) !ffunE sol_u.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vsolve_eqP
| |
span_lfunP(U : seq uT) (phi psi : 'Hom(uT,vT)) :
{in <<U>>%VS, phi =1 psi} <-> {in U, phi =1 psi}.
Proof.
split=> eq_phi_psi u uU; first by rewrite eq_phi_psi ?memv_span.
rewrite [u](@coord_span _ _ _ (in_tuple U))// !linear_sum/=.
by apply: eq_bigr=> i _; rewrite 2!linearZ/= eq_phi_psi// ?mem_nth.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
span_lfunP
| |
fullv_lfunP(U : seq uT) (phi psi : 'Hom(uT,vT)) : <<U>>%VS = fullv ->
phi = psi <-> {in U, phi =1 psi}.
Proof.
by move=> Uf; split=> [->//|/span_lfunP]; rewrite Uf=> /(_ _ (memvf _))-/lfunP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
fullv_lfunP
| |
rVof(v : vT) := \row_i coord e i v.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
rVof
| |
rVof_linear: linear rVof.
Proof. by move=> x v1 v2; apply/rowP=> i; rewrite !mxE linearP. Qed.
HB.instance Definition _ := GRing.isSemilinear.Build F _ _ _ rVof
(GRing.semilinear_linear rVof_linear).
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
rVof_linear
| |
coord_rVofi v : coord e i v = rVof v 0 i.
Proof. by rewrite !mxE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
coord_rVof
| |
vecof(v : 'rV_n) := \sum_i v 0 i *: e`_i.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vecof
| |
vecof_deltai : vecof (delta_mx 0 i) = e`_i.
Proof.
rewrite /vecof (bigD1 i)//= mxE !eqxx scale1r big1 ?addr0// => j neq_ji.
by rewrite mxE (negPf neq_ji) andbF scale0r.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vecof_delta
| |
vecof_linear: linear vecof.
Proof.
move=> x v1 v2; rewrite linear_sum -big_split/=.
by apply: eq_bigr => i _/=; rewrite !mxE scalerDl scalerA.
Qed.
HB.instance Definition _ := GRing.isSemilinear.Build F _ _ _ vecof
(GRing.semilinear_linear vecof_linear).
Variable e_basis : basis_of {:vT} e.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vecof_linear
| |
rVofK: cancel rVof vecof.
Proof.
move=> v; rewrite [v in RHS](coord_basis e_basis) ?memvf//.
by apply: eq_bigr => i; rewrite !mxE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
rVofK
| |
vecofK: cancel vecof rVof.
Proof.
move=> v; apply/rowP=> i; rewrite !(lfunE, mxE).
by rewrite coord_sum_free ?(basis_free e_basis).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vecofK
| |
rVofE(i : 'I_n) : rVof e`_i = delta_mx 0 i.
Proof.
apply/rowP=> k; rewrite !mxE.
by rewrite eqxx coord_free ?(basis_free e_basis)// eq_sym.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
rVofE
| |
coord_vecofi v : coord e i (vecof v) = v 0 i.
Proof. by rewrite coord_rVof vecofK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
coord_vecof
| |
rVof_eq0v : (rVof v == 0) = (v == 0).
Proof. by rewrite -(inj_eq (can_inj vecofK)) rVofK linear0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
rVof_eq0
| |
vecof_eq0v : (vecof v == 0) = (v == 0).
Proof. by rewrite -(inj_eq (can_inj rVofK)) vecofK linear0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] |
algebra/vector.v
|
vecof_eq0
|
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