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 |
|---|---|---|---|---|---|---|
mulr_absz(x : R) i : x *+ `|i| = x *~ `|i|.
Proof. by rewrite -abszE. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | mulr_absz | |
natr_abszi : `|i|%:R = `|i|%:~R :> R.
Proof. by rewrite -abszE. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | natr_absz | |
Posz: nat >-> int. | Coercion | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | Posz | |
distnCm1 m2 : `|m1 - m2| = `|m2 - m1|.
Proof. by rewrite -opprB abszN. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distnC | |
distnDld n1 n2 : `|d + n1 - (d + n2)| = `|n1 - n2|.
Proof. by rewrite !PoszD opprD addrCA -addrA addKr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distnDl | |
distnDrd n1 n2 : `|n1 + d - (n2 + d)| = `|n1 - n2|.
Proof. by rewrite -!(addnC d) distnDl. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distnDr | |
distnErn1 n2 : n1 <= n2 -> `|n1 - n2| = n2 - n1.
Proof. by move/subnK=> {1}<-; rewrite distnC PoszD addrK absz_nat. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distnEr | |
distnEln1 n2 : n2 <= n1 -> `|n1 - n2| = n1 - n2.
Proof. by move/distnEr <-; rewrite distnC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distnEl | |
distn0n : `|n - 0| = n.
Proof. by rewrite subr0 absz_nat. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distn0 | |
dist0nn : `|0 - n| = n.
Proof. by rewrite distnC distn0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | dist0n | |
distnnm : `|m - m| = 0.
Proof. by rewrite subrr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distnn | |
distn_eq0n1 n2 : (`|n1 - n2| == 0) = (n1 == n2).
Proof. by rewrite absz_eq0 subr_eq0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distn_eq0 | |
distnSn : `|n - n.+1| = 1.
Proof. exact: distnDr n 0 1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distnS | |
distSnn : `|n.+1 - n| = 1.
Proof. exact: distnDr n 1 0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distSn | |
distn_eq1n1 n2 :
(`|n1 - n2| == 1) = (if n1 < n2 then n1.+1 == n2 else n1 == n2.+1).
Proof.
case: ltnP => [lt_n12 | le_n21].
by rewrite eq_sym -(eqn_add2r n1) distnEr ?subnK // ltnW.
by rewrite -(eqn_add2r n2) distnEl ?subnK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | distn_eq1 | |
leqD_distm1 m2 m3 : `|m1 - m3| <= `|m1 - m2| + `|m2 - m3|.
Proof. by rewrite -lez_nat PoszD !abszE ler_distD. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | leqD_dist | |
leqifD_distzm1 m2 m3 :
`|m1 - m3| <= `|m1 - m2| + `|m2 - m3|
?= iff (m1 <= m2 <= m3)%R || (m3 <= m2 <= m1)%R.
Proof.
apply/leqifP; rewrite -ltz_nat -eqz_nat PoszD !abszE; apply/leifP.
wlog le_m31 : m1 m3 / (m3 <= m1)%R.
move=> IH; case/orP: (le_total m1 m3) => /IH //.
by rewrite (addrC `|_|)%R orbC !(distrC m1) !(distrC m3).
rewrite ger0_norm ?subr_ge0 // orb_idl => [|/andP[le_m12 le_m23]]; last first.
by have /eqP->: m2 == m3; rewrite ?lexx // eq_le le_m23 (le_trans le_m31).
rewrite -{1}(subrK m2 m1) -(addrA _ m2) -subr_ge0 andbC -[X in X && _]subr_ge0.
by apply: leifD; apply/real_leif_norm/num_real.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | leqifD_distz | |
leqifD_distn1 n2 n3 :
`|n1 - n3| <= `|n1 - n2| + `|n2 - n3|
?= iff (n1 <= n2 <= n3) || (n3 <= n2 <= n1).
Proof. exact: leqifD_distz. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | leqifD_dist | |
sqrn_distn1 n2 : `|n1 - n2| ^ 2 + 2 * (n1 * n2) = n1 ^ 2 + n2 ^ 2.
Proof.
wlog le_n21: n1 n2 / n2 <= n1.
move=> IH; case/orP: (leq_total n2 n1) => /IH //.
by rewrite (addnC (n2 ^ 2)) (mulnC n2) distnC.
by rewrite distnEl ?sqrnB ?subnK ?nat_Cauchy.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | sqrn_dist | |
intr_normm : `|m|%:~R = `|m%:~R : R|.
Proof. by rewrite {2}[m]intEsign rmorphMsign normrMsign abszE normr_nat. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | intr_norm | |
normrMzm (x : R) : `|x *~ m| = `|x| *~ `|m|.
Proof. by rewrite -mulrzl normrM -intr_norm mulrzl. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | normrMz | |
expN1r(i : int) : (-1 : R) ^ i = (-1) ^+ `|i|.
Proof.
case: i => n; first by rewrite exprnP absz_nat.
by rewrite NegzE abszN absz_nat -invr_expz expfV invrN1.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | expN1r | |
coefMrzp n i : (p *~ n)`_i = (p`_i *~ n).
Proof. by case: n => n; rewrite ?NegzE (coefMNn, coefMn). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | coefMrz | |
polyCMzn : {morph (@polyC R) : c / c *~ n}.
Proof. by case: (intP n) => // n' c; rewrite ?mulrNz ?polyCN polyCMn. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | polyCMz | |
hornerMzn p x : (p *~ n).[x] = p.[x] *~ n.
Proof. by case: n => n; rewrite ?NegzE ?mulNzr ?(hornerN, hornerMn). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | hornerMz | |
horner_intn x : (n%:~R : {poly R}).[x] = n%:~R.
Proof. by rewrite hornerMz hornerC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | horner_int | |
derivMzn p : (p *~ n)^`() = p^`() *~ n.
Proof. by case: n => n; rewrite ?NegzE -?pmulrn (derivMn, derivMNn). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | derivMz | |
mulpzp n : p *~ n = n%:~R *: p.
Proof. by rewrite -mul_polyC polyCMz polyC1 mulrzl. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | mulpz | |
rpredMz(M : zmodType) (S : zmodClosed M) m :
{in S, forall u, u *~ m \in S}.
Proof. by case: m => n u Su; rewrite ?rpredN ?rpredMn. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | rpredMz | |
rpred_int(R : pzRingType) (S : subringClosed R) m : m%:~R \in S.
Proof. by rewrite rpredMz ?rpred1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | rpred_int | |
rpredZint(R : pzRingType) (M : lmodType R) (S : zmodClosed M) m :
{in S, forall u, m%:~R *: u \in S}.
Proof. by move=> u Su; rewrite /= scaler_int rpredMz. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | rpredZint | |
rpredXz(R : unitRingType) (S : divClosed R) m :
{in S, forall x, x ^ m \in S}.
Proof. by case: m => n x Sx; rewrite ?rpredV rpredX. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | rpredXz | |
rpredXsign(R : unitRingType) (S : divClosed R) n x :
(x ^ ((-1) ^+ n) \in S) = (x \in S).
Proof. by rewrite -signr_odd; case: (odd n); rewrite ?rpredV. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | rpredXsign | |
semivector_axiom_def(R : nzSemiRingType) n (V : lSemiModType R) := {
v2r : V -> 'rV[R]_n;
v2r_semilinear : semilinear v2r;
v2r_bijective : bijective v2r }.
Arguments semivector_axiom_def [R] n%_N V%_type. | Record | 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 | semivector_axiom_def | |
vector_axiom_def(R : nzRingType) n (V : lmodType R) :=
{v2r : V -> 'rV[R]_n | linear v2r & bijective v2r}.
Arguments vector_axiom_def [R] n%_N V%_type.
HB.mixin Record LSemiModule_hasFinDim (R : nzSemiRingType) (V : Type)
of GRing.LSemiModule R V :=
{ dim : nat;
vector_subdef : semivector_axiom_def dim V }.
#[mathcomp(axiom="semivector_axiom_def"), short(type="semiVectType")]
HB.structure Definition SemiVector (R : nzSemiRingType) :=
{ V of LSemiModule_hasFinDim R V & GRing.LSemiModule R V }.
#[mathcomp(axiom="vector_axiom_def"), short(type="vectType")]
HB.structure Definition Vector (R : nzRingType) :=
{ V of LSemiModule_hasFinDim R V & GRing.Lmodule R V }.
#[deprecated(since="mathcomp 2.2.0", note="Use Vector.axiom 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 | vector_axiom_def | |
vector_axiom:= Vector.axiom.
Arguments dim {R} s.
HB.factory Record Lmodule_hasFinDim (R : nzRingType) (V : Type)
of GRing.Lmodule R V :=
{ dim : nat;
vector_subdef : vector_axiom_def dim V }.
HB.builders Context R V of Lmodule_hasFinDim R 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 | vector_axiom | |
v2r:= sval vector_subdef.
HB.instance Definition _ :=
GRing.isLinear.Build R V 'rV_dim _ v2r (proj1 (proj2_sig vector_subdef)).
HB.instance Definition _ :=
LSemiModule_hasFinDim.Build R V
(Build_semivector_axiom_def
(semilinearP v2r) (proj2 (proj2_sig vector_subdef))).
HB.end. | 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 | v2r | |
space(K : fieldType) (vT : vectType K) :=
Space (mx : 'M[K]_vT) & <<mx>>%MS == mx. | 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 | space | |
hom(R : nzSemiRingType) (vT wT : semiVectType R) :=
Hom of 'M[R]_(vT, wT). | 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 | hom | |
v2r:= v2r v2r_subproof.
Let v2r_bij : bijective v2r := v2r_bijective v2r_subproof.
Fact r2v_subproof : {r2v | cancel r2v v2r}.
Proof.
have r2vP r: {v | v2r v = r}.
by apply: sig_eqW; have [v _ vK] := v2r_bij; exists (v r).
by exists (fun r => sval (r2vP r)) => r; case: (r2vP r).
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 | v2r | |
r2v:= sval r2v_subproof. | 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 | r2v | |
r2vK: cancel r2v v2r. Proof. exact: svalP r2v_subproof. 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 | r2vK | |
r2v_inj: injective r2v. Proof. exact: can_inj r2vK. 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 | r2v_inj | |
v2rK: cancel v2r r2v. Proof. by have/bij_can_sym:= r2vK; 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 | v2rK | |
v2r_inj: injective v2r. Proof. exact: can_inj v2rK. Qed.
HB.instance Definition _ := GRing.isSemilinear.Build R vT 'rV_vT _ v2r
(v2r_semilinear v2r_subproof).
HB.instance Definition _ := GRing.isSemilinear.Build R 'rV_vT vT _ r2v
(can2_semilinear v2rK r2vK). | 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 | v2r_inj | |
b2mxn (X : n.-tuple vT) := \matrix_i v2r (tnth X 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 | b2mx | |
b2mxKn (X : n.-tuple vT) i : r2v (row i (b2mx X)) = X`_i.
Proof. by rewrite rowK v2rK -tnth_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 | b2mxK | |
vs2mx(U : @space K vT) := let: Space mx _ := U in mx. | 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 | |
gen_vs2mx(U : {vspace vT}) : <<vs2mx U>>%MS = vs2mx U.
Proof. by apply/eqP; rewrite /vs2mx; case: U. Qed.
Fact mx2vs_subproof m (A : 'M[K]_(m, vT)) : <<(<<A>>)>>%MS == <<A>>%MS.
Proof. by rewrite genmx_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 | gen_vs2mx | |
mx2vs{m} A : {vspace vT} := Space (@mx2vs_subproof m A).
HB.instance Definition _ := [isSub of {vspace vT} for vs2mx]. | 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 | mx2vs | |
vs2mxK: cancel vs2mx mx2vs.
Proof. by move=> v; apply: val_inj; rewrite /= gen_vs2mx. 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 | vs2mxK | |
mx2vsKm (M : 'M_(m, vT)) : (vs2mx (mx2vs M) :=: M)%MS.
Proof. exact: genmxE. 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 | mx2vsK | |
f2mx(f : 'Hom(aT, rT)) := let: Hom A := f in A.
HB.instance Definition _ : isSub _ _ 'Hom(aT, rT) := [isNew for f2mx]. | 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 | f2mx | |
Definition_ := [Choice of {vspace vT} by <:]. | 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 | |
dimvU := \rank (vs2mx 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 | dimv | |
subsetvU V := (vs2mx U <= vs2mx V)%MS. | 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 | subsetv | |
vlineu := mx2vs (v2r 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 | vline | |
pred_of_vspace(U : space vT) : {pred vT} :=
fun v => (vs2mx (vline v) <= vs2mx U)%MS. | 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 | pred_of_vspace | |
vspace_predType:= @PredType _ (unkeyed {vspace vT}) pred_of_vspace. | 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 | vspace_predType | |
fullv: {vspace vT} := mx2vs 1%:M. | 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 | fullv | |
addvU V := mx2vs (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 | addv | |
capvU V := mx2vs (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 | capv | |
complvU := mx2vs (vs2mx 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 | complv | |
diffvU V := mx2vs (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 | diffv | |
vpickU := r2v (nz_row (vs2mx U)).
Fact span_key : unit. Proof. by []. 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 | vpick | |
span_expanded_defX := mx2vs (b2mx (in_tuple 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 | span_expanded_def | |
span:= locked_with span_key span_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 | span | |
span_unlockable:= [unlockable fun span]. | 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 | span_unlockable | |
vbasis_defU :=
[tuple r2v (row i (row_base (vs2mx U))) | i < dimv 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 | vbasis_def | |
vbasis:= locked_with span_key vbasis_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 | vbasis | |
vbasis_unlockable:= [unlockable fun vbasis]. | 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 | vbasis_unlockable | |
freeX := dimv (span X) == size 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 | free | |
basis_ofU X := (span X == U) && free 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 | basis_of | |
pred_of_vspace: space >-> pred_sort. | Coercion | 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 | pred_of_vspace | |
memvEv U : (v \in U) = (<[v]> <= U)%VS. 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 | memvE | |
vlinePv1 v2 : reflect (exists k, v1 = k *: v2) (v1 \in <[v2]>)%VS.
Proof.
apply: (iffP idP) => [|[k ->]]; rewrite memvK genmxE ?linearZ ?scalemx_sub //.
by case/sub_rVP=> k; rewrite -linearZ => /v2r_inj->; exists k.
Qed.
Fact memv_submod_closed U : submod_closed U.
Proof.
split=> [|a u v]; rewrite !memvK 1?linear0 1?sub0mx // => Uu Uv.
by rewrite linearP addmx_sub ?scalemx_sub.
Qed.
HB.instance Definition _ (U : {vspace vT}) :=
GRing.isSubmodClosed.Build K vT (pred_of_vspace U) (memv_submod_closed U). | 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 | vlineP | |
mem0vU : 0 \in U. Proof. exact: rpred0. 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 | mem0v | |
memvNU v : (- v \in U) = (v \in U). Proof. exact: rpredN. 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 | memvN | |
memvDU : {in U &, forall u v, u + v \in U}. Proof. exact: rpredD. 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 | memvD | |
memvBU : {in U &, forall u v, u - v \in U}. Proof. exact: rpredB. 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 | memvB | |
memvZU k : {in U, forall v, k *: v \in U}. Proof. exact: rpredZ. 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 | memvZ | |
memv_sumlI r (P : pred I) vs U :
(forall i, P i -> vs i \in U) -> \sum_(i <- r | P i) vs i \in U.
Proof. exact: rpred_sum. 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_suml | |
memv_lineu : u \in <[u]>%VS.
Proof. by apply/vlineP; exists 1; rewrite scale1r. 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_line | |
subvPU V : reflect {subset U <= V} (U <= V)%VS.
Proof.
apply: (iffP rV_subP) => sU12 u.
by rewrite !memvE /subsetv !genmxE => /sU12.
by have:= sU12 (r2v u); rewrite !memvE /subsetv !genmxE r2vK.
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 | subvP | |
subvvU : (U <= U)%VS. Proof. exact/subvP. Qed.
Hint Resolve subvv : core. | 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 | subvv | |
subv_trans: transitive subV.
Proof. by move=> U V W /subvP sUV /subvP sVW; apply/subvP=> u /sUV/sVW. 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_trans | |
subv_anti: antisymmetric subV.
Proof. by move=> U V; 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 | subv_anti | |
eqEsubvU V : (U == V) = (U <= V <= U)%VS.
Proof. by apply/eqP/idP=> [-> | /subv_anti//]; rewrite subvv. 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 | eqEsubv | |
vspacePU V : U =i V <-> U = V.
Proof.
split=> [eqUV | -> //]; apply/subv_anti/andP.
by split; apply/subvP=> v; rewrite eqUV.
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 | vspaceP | |
subvPn{U V} : reflect (exists2 u, u \in U & u \notin V) (~~ (U <= V)%VS).
Proof.
apply: (iffP idP) => [|[u Uu]]; last by apply: contra => /subvP->.
case/row_subPn=> i; set vi := row i _ => V'vi.
by exists (r2v vi); rewrite memvK r2vK ?row_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 | subvPn | |
sub0vU : (0 <= U)%VS.
Proof. exact: 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 | sub0v | |
subv0U : (U <= 0)%VS = (U == 0%VS).
Proof. by rewrite eqEsubv sub0v andbT. 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 | subv0 | |
memv0v : v \in 0%VS = (v == 0).
Proof. by apply/idP/eqP=> [/vlineP[k ->] | ->]; rewrite (scaler0, 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 | memv0 | |
subvfU : (U <= fullv)%VS. Proof. by rewrite /subsetv vs2mxF submx1. 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 | subvf | |
memvfv : v \in fullv. Proof. exact: 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 | memvf | |
memv_pickU : vpick U \in U. Proof. by rewrite mem_r2v nz_row_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_pick | |
vpick0U : (vpick U == 0) = (U == 0%VS).
Proof. by rewrite -memv0 mem_r2v -subv0 /subV vs2mx0 !submx0 nz_row_eq0. 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 | vpick0 | |
subv_addU V W : (U + V <= W)%VS = (U <= W)%VS && (V <= W)%VS.
Proof. by rewrite /subV vs2mxD addsmx_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 | subv_add | |
addvSU1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 + V1 <= U2 + V2)%VS.
Proof. by rewrite /subV !vs2mxD; apply: addsmxS. 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 | addvS | |
addvSlU V : (U <= U + V)%VS.
Proof. by rewrite /subV vs2mxD addsmxSl. 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 | addvSl |
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