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 |
|---|---|---|---|---|---|---|
addvSrU V : (V <= U + V)%VS.
Proof. by rewrite /subV vs2mxD addsmxSr. 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
|
addvSr
| |
addvC: commutative addV.
Proof. by move=> U V; apply/vs2mxP; rewrite !vs2mxD addsmxC submx_refl. 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
|
addvC
| |
addvA: associative addV.
Proof. by move=> U V W; apply/vs2mxP; rewrite !vs2mxD addsmxA submx_refl. 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
|
addvA
| |
addv_idPl{U V}: reflect (U + V = U)%VS (V <= U)%VS.
Proof. by rewrite /subV (sameP addsmx_idPl eqmxP) -vs2mxD; apply: vs2mxP. 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_idPl
| |
addv_idPr{U V} : reflect (U + V = V)%VS (U <= V)%VS.
Proof. by rewrite addvC; apply: addv_idPl. 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_idPr
| |
addvv: idempotent_op addV.
Proof. by move=> U; apply/addv_idPl. 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
|
addvv
| |
add0v: left_id 0%VS addV.
Proof. by move=> U; apply/addv_idPr/sub0v. 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
|
add0v
| |
addv0: right_id 0%VS addV.
Proof. by move=> U; apply/addv_idPl/sub0v. 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
|
addv0
| |
sumfv: left_zero fullv addV.
Proof. by move=> U; apply/addv_idPl/subvf. 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
|
sumfv
| |
addvf: right_zero fullv addV.
Proof. by move=> U; apply/addv_idPr/subvf. Qed.
HB.instance Definition _ := Monoid.isComLaw.Build {vspace vT} 0%VS addv
addvA addvC add0v.
|
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
|
addvf
| |
memv_addu v U V : u \in U -> v \in V -> u + v \in (U + V)%VS.
Proof. by rewrite !memvK genmxE linearD; apply: addmx_sub_adds. 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_add
| |
memv_addP{w U V} :
reflect (exists2 u, u \in U & exists2 v, v \in V & w = u + v)
(w \in U + V)%VS.
Proof.
apply: (iffP idP) => [|[u Uu [v Vv ->]]]; last exact: memv_add.
rewrite memvK genmxE => /sub_addsmxP[r /(canRL v2rK)->].
rewrite linearD /=; set u := r2v _; set v := r2v _.
by exists u; last exists v; rewrite // mem_r2v submxMl.
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_addP
| |
sumv_supi0 P U Vs :
P i0 -> (U <= Vs i0)%VS -> (U <= \sum_(i | P i) Vs i)%VS.
Proof. by move=> Pi0 /subv_trans-> //; rewrite (bigD1 i0) ?addvSl. Qed.
Arguments sumv_sup i0 [P U Vs].
|
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_sup
| |
subv_sumP{P Us V} :
reflect (forall i, P i -> Us i <= V)%VS (\sum_(i | P i) Us i <= V)%VS.
Proof.
apply: (iffP idP) => [sUV i Pi | sUV].
by apply: subv_trans sUV; apply: sumv_sup Pi _.
by elim/big_rec: _ => [|i W Pi sWV]; rewrite ?sub0v // subv_add sUV.
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
|
subv_sumP
| |
memv_sumrP vs (Us : I -> {vspace vT}) :
(forall i, P i -> vs i \in Us i) ->
\sum_(i | P i) vs i \in (\sum_(i | P i) Us i)%VS.
Proof. by move=> Uv; apply/rpred_sum=> i Pi; apply/(sumv_sup i Pi)/Uv. 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_sumr
| |
memv_sumP{P} {Us : I -> {vspace vT}} {v} :
reflect (exists2 vs, forall i, P i -> vs i \in Us i
& v = \sum_(i | P i) vs i)
(v \in \sum_(i | P i) Us i)%VS.
Proof.
apply: (iffP idP) => [|[vs Uv ->]]; last exact: memv_sumr.
rewrite memvK vs2mx_sum => /sub_sumsmxP[r /(canRL v2rK)->].
pose f i := r2v (r i *m vs2mx (Us i)); rewrite linear_sum /=.
by exists f => //= i _; rewrite mem_r2v submxMl.
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_sumP
| |
subv_capU V W : (U <= V :&: W)%VS = (U <= V)%VS && (U <= W)%VS.
Proof. by rewrite /subV vs2mxI sub_capmx. 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
|
subv_cap
| |
capvSU1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 :&: V1 <= U2 :&: V2)%VS.
Proof. by rewrite /subV !vs2mxI; apply: capmxS. 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
|
capvS
| |
capvSlU V : (U :&: V <= U)%VS.
Proof. by rewrite /subV vs2mxI capmxSl. 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
|
capvSl
| |
capvSrU V : (U :&: V <= V)%VS.
Proof. by rewrite /subV vs2mxI capmxSr. 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
|
capvSr
| |
capvC: commutative capV.
Proof. by move=> U V; apply/vs2mxP; rewrite !vs2mxI capmxC submx_refl. 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
|
capvC
| |
capvA: associative capV.
Proof. by move=> U V W; apply/vs2mxP; rewrite !vs2mxI capmxA submx_refl. 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
|
capvA
| |
capv_idPl{U V} : reflect (U :&: V = U)%VS (U <= V)%VS.
Proof. by rewrite /subV(sameP capmx_idPl eqmxP) -vs2mxI; apply: vs2mxP. 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
|
capv_idPl
| |
capv_idPr{U V} : reflect (U :&: V = V)%VS (V <= U)%VS.
Proof. by rewrite capvC; apply: capv_idPl. 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
|
capv_idPr
| |
capvv: idempotent_op capV.
Proof. by move=> U; apply/capv_idPl. 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
|
capvv
| |
cap0v: left_zero 0%VS capV.
Proof. by move=> U; apply/capv_idPl/sub0v. 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
|
cap0v
| |
capv0: right_zero 0%VS capV.
Proof. by move=> U; apply/capv_idPr/sub0v. 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
|
capv0
| |
capfv: left_id fullv capV.
Proof. by move=> U; apply/capv_idPr/subvf. 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
|
capfv
| |
capvf: right_id fullv capV.
Proof. by move=> U; apply/capv_idPl/subvf. Qed.
HB.instance Definition _ := Monoid.isComLaw.Build {vspace vT} fullv capv
capvA capvC capfv.
|
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
|
capvf
| |
memv_capw U V : (w \in U :&: V)%VS = (w \in U) && (w \in V).
Proof. by rewrite !memvE subv_cap. 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_cap
| |
memv_capP{w U V} : reflect (w \in U /\ w \in V) (w \in U :&: V)%VS.
Proof. by rewrite memv_cap; apply: andP. 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_capP
| |
vspace_modlU V W : (U <= W -> U + (V :&: W) = (U + V) :&: W)%VS.
Proof.
by move=> sUV; apply/vs2mxP; rewrite !(vs2mxD, vs2mxI); apply/eqmxP/matrix_modl.
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
|
vspace_modl
| |
vspace_modrU V W : (W <= U -> (U :&: V) + W = U :&: (V + W))%VS.
Proof. by rewrite -!(addvC W) !(capvC U); apply: vspace_modl. 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
|
vspace_modr
| |
bigcapv_infi0 P Us V :
P i0 -> (Us i0 <= V -> \bigcap_(i | P i) Us i <= V)%VS.
Proof. by move=> Pi0; apply: subv_trans; rewrite (bigD1 i0) ?capvSl. 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
|
bigcapv_inf
| |
subv_bigcapP{P U Vs} :
reflect (forall i, P i -> U <= Vs i)%VS (U <= \bigcap_(i | P i) Vs i)%VS.
Proof.
apply: (iffP idP) => [sUV i Pi | sUV].
by rewrite (subv_trans sUV) ?(bigcapv_inf Pi).
by elim/big_rec: _ => [|i W Pi]; rewrite ?subvf // subv_cap sUV.
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
|
subv_bigcapP
| |
addv_complfU : (U + U^C)%VS = fullv.
Proof.
apply/vs2mxP; rewrite vs2mxD -gen_vs2mx -genmx_adds !genmxE submx1 sub1mx.
exact: addsmx_compl_full.
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_complf
| |
capv_complU : (U :&: U^C = 0)%VS.
Proof.
apply/val_inj; rewrite [val]/= vs2mx0 vs2mxI -gen_vs2mx -genmx_cap.
by rewrite capmx_compl genmx0.
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
|
capv_compl
| |
diffvSlU V : (U :\: V <= U)%VS.
Proof. by rewrite /subV genmxE diffmxSl. 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
|
diffvSl
| |
capv_diffU V : ((U :\: V) :&: V = 0)%VS.
Proof.
apply/val_inj; rewrite [val]/= vs2mx0 vs2mxI -(gen_vs2mx V) -genmx_cap.
by rewrite capmx_diff genmx0.
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
|
capv_diff
| |
addv_diff_capU V : (U :\: V + U :&: V)%VS = U.
Proof.
apply/vs2mxP; rewrite vs2mxD -genmx_adds !genmxE.
exact/eqmxP/addsmx_diff_cap_eq.
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_diff_cap
| |
addv_diffU V : (U :\: V + V = U + V)%VS.
Proof. by rewrite -{2}(addv_diff_cap U V) -addvA (addv_idPr (capvSr U V)). 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_diff
| |
dimv0: \dim (0%VS : {vspace vT}) = 0.
Proof. by rewrite /dimv vs2mx0 mxrank0. 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
|
dimv0
| |
dimv_eq0U : (\dim U == 0) = (U == 0%VS).
Proof. by rewrite /dimv /= mxrank_eq0 [in RHS]/eq_op /= linear0 genmx0. 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
|
dimv_eq0
| |
dimvf: \dim {:vT} = dim vT.
Proof. by rewrite /dimv vs2mxF mxrank1. 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
|
dimvf
| |
dim_vlinev : \dim <[v]> = (v != 0).
Proof. by rewrite /dimv mxrank_gen rank_rV (can2_eq v2rK r2vK) 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
|
dim_vline
| |
dimvSU V : (U <= V)%VS -> \dim U <= \dim V.
Proof. exact: mxrankS. 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
|
dimvS
| |
dimv_leqif_supU V : (U <= V)%VS -> \dim U <= \dim V ?= iff (V <= U)%VS.
Proof. exact: mxrank_leqif_sup. 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
|
dimv_leqif_sup
| |
dimv_leqif_eqU V : (U <= V)%VS -> \dim U <= \dim V ?= iff (U == V).
Proof. by rewrite eqEsubv; apply: mxrank_leqif_eq. 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
|
dimv_leqif_eq
| |
eqEdimU V : (U == V) = (U <= V)%VS && (\dim V <= \dim U).
Proof. by apply/idP/andP=> [/eqP | [/dimv_leqif_eq/geq_leqif]] ->. 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
|
eqEdim
| |
dimv_complU : \dim U^C = (\dim {:vT} - \dim U)%N.
Proof. by rewrite dimvf /dimv mxrank_gen mxrank_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
|
dimv_compl
| |
dimv_cap_complU V : (\dim (U :&: V) + \dim (U :\: V))%N = \dim U.
Proof. by rewrite /dimv !mxrank_gen mxrank_cap_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
|
dimv_cap_compl
| |
dimv_sum_capU V : (\dim (U + V) + \dim (U :&: V) = \dim U + \dim V)%N.
Proof. by rewrite /dimv !mxrank_gen mxrank_sum_cap. 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
|
dimv_sum_cap
| |
dimv_disjoint_sumU V :
(U :&: V = 0)%VS -> \dim (U + V) = (\dim U + \dim V)%N.
Proof. by move=> dxUV; rewrite -dimv_sum_cap dxUV dimv0 addn0. 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
|
dimv_disjoint_sum
| |
dimv_add_leqifU V :
\dim (U + V) <= \dim U + \dim V ?= iff (U :&: V <= 0)%VS.
Proof.
by rewrite /dimv /subV !mxrank_gen vs2mx0 genmxE; apply: mxrank_adds_leqif.
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
|
dimv_add_leqif
| |
diffv_eq0U V : (U :\: V == 0)%VS = (U <= V)%VS.
Proof.
rewrite -dimv_eq0 -(eqn_add2l (\dim (U :&: V))) addn0 dimv_cap_compl eq_sym.
by rewrite (dimv_leqif_eq (capvSl _ _)) (sameP capv_idPl 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
|
diffv_eq0
| |
dimv_leq_sumI r (P : pred I) (Us : I -> {vspace vT}) :
\dim (\sum_(i <- r | P i) Us i) <= \sum_(i <- r | P i) \dim (Us i).
Proof.
elim/big_rec2: _ => [|i d vs _ le_vs_d]; first by rewrite dim_vline eqxx.
by apply: (leq_trans (dimv_add_leqif _ _)); rewrite leq_add2l.
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
|
dimv_leq_sum
| |
addv_expr:= Sumv {
addv_val :> wrapped {vspace vT};
addv_dim : wrapped nat;
_ : mxsum_spec (vs2mx (unwrap addv_val)) (unwrap addv_dim)
}.
|
Structure
|
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_expr
| |
vs2mx_sum_expr_subproof(S : addv_expr) :
mxsum_spec (vs2mx (unwrap S)) (unwrap (addv_dim S)).
Proof. by case: S. Qed.
|
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
|
vs2mx_sum_expr_subproof
| |
vs2mx_sum_exprS := ProperMxsumExpr (vs2mx_sum_expr_subproof S).
|
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
|
vs2mx_sum_expr
| |
trivial_addvU := @Sumv (Wrap U) (Wrap (\dim U)) (TrivialMxsum _).
|
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
|
trivial_addv
| |
proper_addv_expr:= ProperSumvExpr {
proper_addv_val :> {vspace vT};
proper_addv_dim :> nat;
_ : mxsum_spec (vs2mx proper_addv_val) proper_addv_dim
}.
|
Structure
|
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
|
proper_addv_expr
| |
proper_addvP(S : proper_addv_expr) :=
let: ProperSumvExpr _ _ termS := S return mxsum_spec (vs2mx S) S in termS.
|
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
|
proper_addvP
| |
proper_addv(S : proper_addv_expr) :=
@Sumv (wrap (S : {vspace vT})) (wrap (S : nat)) (proper_addvP S).
|
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
|
proper_addv
| |
binary_addv_expr:= ProperSumvExpr binary_addv_subproof.
|
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
|
binary_addv_expr
| |
nary_addv_expr:= ProperSumvExpr nary_addv_subproof.
|
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
|
nary_addv_expr
| |
directv_defS of phantom {vspace vT} (unwrap (addv_val S)) :=
\dim (unwrap S) == unwrap (addv_dim S).
|
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
|
directv_def
| |
directvE(S : addv_expr) :
directv (unwrap S) = (\dim (unwrap S) == unwrap (addv_dim S)).
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
|
directvE
| |
directvP{S : proper_addv_expr} : reflect (\dim S = S :> nat) (directv S).
Proof. exact: eqnP. 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
|
directvP
| |
directv_trivialU : directv (unwrap (@trivial_addv U)).
Proof. exact: eqxx. 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
|
directv_trivial
| |
dimv_sum_leqif(S : addv_expr) :
\dim (unwrap S) <= unwrap (addv_dim S) ?= iff directv (unwrap S).
Proof.
rewrite directvE; case: S => [[U] [d] /= defUd]; split=> //=.
rewrite /dimv; elim: {1}_ {U}_ d / defUd => // m1 m2 A1 A2 r1 r2 _ leA1 _ leA2.
by apply: leq_trans (leq_add leA1 leA2); rewrite mxrank_adds_leqif.
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
|
dimv_sum_leqif
| |
directvEgeq(S : addv_expr) :
directv (unwrap S) = (\dim (unwrap S) >= unwrap (addv_dim S)).
Proof. by rewrite leq_eqVlt ltnNge eq_sym !dimv_sum_leqif orbF. 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
|
directvEgeq
| |
directv_addE(S1 S2 : addv_expr) :
directv (unwrap S1 + unwrap S2)
= [&& directv (unwrap S1), directv (unwrap S2)
& unwrap S1 :&: unwrap S2 == 0]%VS.
Proof.
by rewrite /directv_def /dimv vs2mxD -mxdirectE mxdirect_addsE -vs2mxI -vs2mx0.
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
|
directv_addE
| |
directv_addP{U V} : reflect (U :&: V = 0)%VS (directv (U + V)).
Proof. by rewrite directv_addE !directv_trivial; 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
|
directv_addP
| |
directv_add_unique{U V} :
reflect (forall u1 u2 v1 v2, u1 \in U -> u2 \in U -> v1 \in V -> v2 \in V ->
(u1 + v1 == u2 + v2) = ((u1, v1) == (u2, v2)))
(directv (U + V)).
Proof.
apply: (iffP directv_addP) => [dxUV u1 u2 v1 v2 Uu1 Uu2 Vv1 Vv2 | dxUV].
apply/idP/idP=> [| /eqP[-> ->] //]; rewrite -subr_eq0 opprD addrACA addr_eq0.
move/eqP=> eq_uv; rewrite xpair_eqE -subr_eq0 eq_uv oppr_eq0 subr_eq0 andbb.
by rewrite -subr_eq0 -memv0 -dxUV memv_cap -memvN -eq_uv !memvB.
apply/eqP; rewrite -subv0; apply/subvP=> v /memv_capP[U1v U2v].
by rewrite memv0 -[v == 0]andbb {1}eq_sym -xpair_eqE -dxUV ?mem0v // addrC.
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
|
directv_add_unique
| |
directv_sumP{Us : I -> {vspace vT}} :
reflect (forall i, P i -> Us i :&: (\sum_(j | P j && (j != i)) Us j) = 0)%VS
(directv (\sum_(i | P i) Us i)).
Proof.
rewrite directvE /= /dimv vs2mx_sum -mxdirectE; apply: (equivP mxdirect_sumsP).
by do [split=> dxU i /dxU; rewrite -vs2mx_sum -vs2mxI -vs2mx0] => [/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
|
directv_sumP
| |
directv_sumE{Ss : I -> addv_expr} (xunwrap := unwrap) :
reflect [/\ forall i, P i -> directv (unwrap (Ss i))
& directv (\sum_(i | P i) xunwrap (Ss i))]
(directv (\sum_(i | P i) unwrap (Ss i))).
Proof.
by rewrite !directvE /= /dimv 2!{1}vs2mx_sum -!mxdirectE; apply: mxdirect_sumsE.
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
|
directv_sumE
| |
directv_sum_independent{Us : I -> {vspace vT}} :
reflect (forall us,
(forall i, P i -> us i \in Us i) -> \sum_(i | P i) us i = 0 ->
(forall i, P i -> us i = 0))
(directv (\sum_(i | P i) Us i)).
Proof.
apply: (iffP directv_sumP) => [dxU us Uu u_0 i Pi | dxU i Pi].
apply/eqP; rewrite -memv0 -(dxU i Pi) memv_cap Uu //= -memvN -sub0r -{1}u_0.
by rewrite (bigD1 i) //= [_ - us i]addrC addKr memv_sumr // => j /andP[/Uu].
apply/eqP; rewrite -subv0; apply/subvP=> v.
rewrite memv_cap memv0 => /andP[Uiv /memv_sumP[us Uu Dv]].
have: \sum_(j | P j) [eta us with i |-> - v] j = 0.
rewrite (bigD1 i) //= eqxx {1}Dv addrC -sumrB big1 // => j /andP[_ i'j].
by rewrite (negPf i'j) subrr.
move/dxU/(_ i Pi); rewrite /= eqxx -oppr_eq0 => -> // j Pj.
by have [-> | i'j] := eqVneq; rewrite ?memvN // Uu ?Pj.
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
|
directv_sum_independent
| |
directv_sum_unique{Us : I -> {vspace vT}} :
reflect (forall us vs,
(forall i, P i -> us i \in Us i) ->
(forall i, P i -> vs i \in Us i) ->
(\sum_(i | P i) us i == \sum_(i | P i) vs i)
= [forall (i | P i), us i == vs i])
(directv (\sum_(i | P i) Us i)).
Proof.
apply: (iffP directv_sum_independent) => [dxU us vs Uu Uv | dxU us Uu u_0 i Pi].
apply/idP/forall_inP=> [|eq_uv]; last by apply/eqP/eq_bigr => i /eq_uv/eqP.
rewrite -subr_eq0 -sumrB => /eqP/dxU eq_uv i Pi.
by rewrite -subr_eq0 eq_uv // => j Pj; apply: memvB; move: j Pj.
apply/eqP; have:= esym (dxU us \0 Uu _); rewrite u_0 big1_eq eqxx.
by move/(_ _)/forall_inP=> -> // j _; apply: mem0v.
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
|
directv_sum_unique
| |
memv_spanX v : v \in X -> v \in <<X>>%VS.
Proof.
by case/seq_tnthP=> i {v}->; rewrite unlock memvK genmxE (eq_row_sub i) // rowK.
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_span
| |
memv_span1v : v \in <<[:: v]>>%VS.
Proof. by rewrite memv_span ?mem_head. 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_span1
| |
dim_spanX : \dim <<X>> <= size X.
Proof. by rewrite unlock /dimv genmxE rank_leq_row. 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_span
| |
span_subvP{X U} : reflect {subset X <= U} (<<X>> <= U)%VS.
Proof.
rewrite /subV [@span _ _]unlock genmxE.
apply: (iffP row_subP) => /= [sXU | sXU i].
by move=> _ /seq_tnthP[i ->]; have:= sXU i; rewrite rowK memvK.
by rewrite rowK -memvK sXU ?mem_tnth.
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_subvP
| |
sub_spanX Y : {subset X <= Y} -> (<<X>> <= <<Y>>)%VS.
Proof. by move=> sXY; apply/span_subvP=> v /sXY/memv_span. 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
|
sub_span
| |
eq_spanX Y : X =i Y -> (<<X>> = <<Y>>)%VS.
Proof.
by move=> eqXY; apply: subv_anti; rewrite !sub_span // => u; rewrite eqXY.
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_span
| |
span_defX : span X = (\sum_(u <- X) <[u]>)%VS.
Proof.
apply/subv_anti/andP; split.
by apply/span_subvP=> v Xv; rewrite (big_rem v) // memvE addvSl.
by rewrite big_tnth; apply/subv_sumP=> i _; rewrite -memvE memv_span ?mem_tnth.
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_def
| |
span_nil: (<<Nil vT>> = 0)%VS.
Proof. by rewrite span_def big_nil. 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_nil
| |
span_seq1v : (<<[:: v]>> = <[v]>)%VS.
Proof. by rewrite span_def big_seq1. 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_seq1
| |
span_consv X : (<<v :: X>> = <[v]> + <<X>>)%VS.
Proof. by rewrite !span_def big_cons. 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_cons
| |
span_catX Y : (<<X ++ Y>> = <<X>> + <<Y>>)%VS.
Proof. by rewrite !span_def big_cat. 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_cat
| |
coord_expanded_defn (X : n.-tuple vT) i v :=
(v2r v *m pinvmx (b2mx X)) 0 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
|
coord_expanded_def
| |
coord:= locked_with span_key coord_expanded_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
|
coord
| |
coord_unlockable:= [unlockable fun coord].
Fact coord_is_scalar n (X : n.-tuple vT) i : scalar (coord X i).
Proof. by move=> k u v; rewrite unlock linearP mulmxDl -scalemxAl !mxE. Qed.
HB.instance Definition _ n Xn i :=
GRing.isSemilinear.Build K vT K _ (coord Xn i)
(GRing.semilinear_linear (@coord_is_scalar n Xn i)).
|
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
|
coord_unlockable
| |
coord_spann (X : n.-tuple vT) v :
v \in span X -> v = \sum_i coord X i v *: X`_i.
Proof.
rewrite memvK span_b2mx genmxE => Xv.
by rewrite unlock_with mul_b2mx mulmxKpV ?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
|
coord_span
| |
coord0i v : coord [tuple 0] i v = 0.
Proof.
rewrite unlock /pinvmx rank_rV; case: negP => [[] | _].
by apply/eqP/rowP=> j; rewrite !mxE (tnth_nth 0) /= linear0 mxE.
by rewrite pid_mx_0 !(mulmx0, mul0mx) 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
|
coord0
| |
nil_free: free (Nil vT).
Proof. by rewrite /free span_nil dimv0. 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
|
nil_free
| |
seq1_freev : free [:: v] = (v != 0).
Proof. by rewrite /free span_seq1 dim_vline; case: (~~ _). 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
|
seq1_free
| |
perm_freeX Y : perm_eq X Y -> free X = free Y.
Proof.
by move=> eqXY; rewrite /free (perm_size eqXY) (eq_span (perm_mem eqXY)).
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
|
perm_free
| |
free_directvX : free X = (0 \notin X) && directv (\sum_(v <- X) <[v]>).
Proof.
have leXi i (v := tnth (in_tuple X) i): true -> \dim <[v]> <= 1 ?= iff (v != 0).
by rewrite -seq1_free -span_seq1 => _; apply/leqif_eq/dim_span.
have [_ /=] := leqif_trans (dimv_sum_leqif _) (leqif_sum leXi).
rewrite sum1_card card_ord !directvE /= /free andbC span_def !(big_tnth _ _ X).
by congr (_ = _ && _); rewrite -has_pred1 -all_predC -big_all big_tnth big_andE.
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
|
free_directv
| |
free_not0v X : free X -> v \in X -> v != 0.
Proof. by rewrite free_directv andbC => /andP[_ /memPn]; apply. 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
|
free_not0
| |
freePn (X : n.-tuple vT) :
reflect (forall k, \sum_(i < n) k i *: X`_i = 0 -> (forall i, k i = 0))
(free X).
Proof.
rewrite free_b2mx; apply: (iffP idP) => [t_free k kt0 i | t_free].
suffices /rowP/(_ i): \row_i k i = 0 by rewrite !mxE.
by apply/(row_free_inj t_free)/r2v_inj; rewrite mul0mx -lin_b2mx kt0 linear0.
rewrite -kermx_eq0; apply/rowV0P=> rk /sub_kermxP kt0.
by apply/rowP=> i; rewrite mxE {}t_free // mul_b2mx kt0 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
|
freeP
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.