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 |
|---|---|---|---|---|---|---|
coord_freen (X : n.-tuple vT) (i j : 'I_n) :
free X -> coord X j (X`_i) = (i == j)%:R.
Proof.
rewrite unlock free_b2mx => /row_freeP[Ct CtK]; rewrite -row_b2mx.
rewrite -row_mul -[pinvmx _]mulmx1 -CtK (mulmxA (b2mx X)) (mulmxA _ _ Ct).
by rewrite mulmxKpV // CtK !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_free | |
coord_sum_freen (X : n.-tuple vT) k j :
free X -> coord X j (\sum_(i < n) k i *: X`_i) = k j.
Proof.
move=> Xfree; rewrite linear_sum (bigD1 j) 1?linearZ //= coord_free // eqxx.
rewrite mulr1 big1 ?addr0 // => i /negPf j'i.
by rewrite linearZ /= coord_free // j'i mulr0.
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_sum_free | |
cat_freeX Y :
free (X ++ Y) = [&& free X, free Y & directv (<<X>> + <<Y>>)].
Proof.
rewrite !free_directv mem_cat directvE /= !big_cat -directvE /= directv_addE /=.
rewrite negb_or -!andbA; do !bool_congr; rewrite -!span_def.
by rewrite (sameP eqP directv_addP).
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 | cat_free | |
catl_freeY X : free (X ++ Y) -> free X.
Proof. by rewrite cat_free => /and3P[]. 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 | catl_free | |
catr_freeX Y : free (X ++ Y) -> free Y.
Proof. by rewrite cat_free => /and3P[]. 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 | catr_free | |
filter_freep X : free X -> free (filter p X).
Proof.
rewrite -(perm_free (etrans (perm_filterC p X _) (perm_refl X))).
exact: catl_free.
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 | filter_free | |
free_consv X : free (v :: X) = (v \notin <<X>>)%VS && free X.
Proof.
rewrite (cat_free [:: v]) seq1_free directvEgeq /= span_seq1 dim_vline.
case: eqP => [-> | _] /=; first by rewrite mem0v.
rewrite andbC ltnNge (geq_leqif (dimv_leqif_sup _)) ?addvSr //.
by rewrite subv_add subvv andbT -memvE.
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_cons | |
freeEn (X : n.-tuple vT) :
free X = [forall i : 'I_n, X`_i \notin <<drop i.+1 X>>%VS].
Proof.
case: X => X /= /eqP <-{n}; rewrite -(big_andE xpredT) /=.
elim: X => [|v X IH_X] /=; first by rewrite nil_free big_ord0.
by rewrite free_cons IH_X big_ord_recl drop0.
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 | freeE | |
freeNEn (X : n.-tuple vT) :
~~ free X = [exists i : 'I_n, X`_i \in <<drop i.+1 X>>%VS].
Proof. by rewrite freeE -negb_exists negbK. 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 | freeNE | |
free_uniqX : free X -> uniq X.
Proof.
elim: X => //= v b IH_X; rewrite free_cons => /andP[X'v /IH_X->].
by rewrite (contra _ X'v) // => /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 | free_uniq | |
free_spanX v (sumX := fun k => \sum_(x <- X) k x *: x) :
free X -> v \in <<X>>%VS ->
{k | v = sumX k & forall k1, v = sumX k1 -> {in X, k1 =1 k}}.
Proof.
rewrite -{2}[X]in_tupleE => freeX /coord_span def_v.
pose k x := oapp (fun i => coord (in_tuple X) i v) 0 (insub (index x X)).
exists k => [|k1 {}def_v _ /(nthP 0)[i ltiX <-]].
rewrite /sumX (big_nth 0) big_mkord def_v; apply: eq_bigr => i _.
by rewrite /k index_uniq ?free_uniq // valK.
rewrite /k /= index_uniq ?free_uniq // insubT //= def_v.
by rewrite /sumX (big_nth 0) big_mkord coord_sum_free.
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_span | |
linear_of_free(rT : lmodType K) X (fX : seq rT) :
{f : {linear vT -> rT} | free X -> size fX = size X -> map f X = fX}.
Proof.
pose f u := \sum_i coord (in_tuple X) i u *: fX`_i.
have lin_f: linear f.
move=> k u v; rewrite scaler_sumr -big_split; apply: eq_bigr => i _.
by rewrite /= scalerA -scalerDl linearP.
pose flM := GRing.isLinear.Build _ _ _ _ f lin_f.
pose fL : {linear _ -> _} := HB.pack f flM.
exists fL => freeX eq_szX.
apply/esym/(@eq_from_nth _ 0); rewrite ?size_map eq_szX // => i ltiX.
rewrite (nth_map 0) //= /f (bigD1 (Ordinal ltiX)) //=.
rewrite big1 => [|j /negbTE neqji]; rewrite (coord_free (Ordinal _)) //.
by rewrite eqxx scale1r addr0.
by rewrite eq_sym neqji 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 | linear_of_free | |
span_basisU X : basis_of U X -> <<X>>%VS = U.
Proof. by case/andP=> /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 | span_basis | |
basis_freeU X : basis_of U X -> free X.
Proof. by case/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 | basis_free | |
coord_basisU n (X : n.-tuple vT) v :
basis_of U X -> v \in U -> v = \sum_i coord X i v *: X`_i.
Proof. by move/span_basis <-; apply: coord_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 | coord_basis | |
nil_basis: basis_of 0 (Nil vT).
Proof. by rewrite /basis_of span_nil eqxx nil_free. 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_basis | |
seq1_basisv : v != 0 -> basis_of <[v]> [:: v].
Proof. by move=> nz_v; rewrite /basis_of span_seq1 // eqxx seq1_free. 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_basis | |
basis_not0x U X : basis_of U X -> x \in X -> x != 0.
Proof. by move/basis_free/free_not0; 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 | basis_not0 | |
basis_memx U X : basis_of U X -> x \in X -> x \in U.
Proof. by move/span_basis=> <- /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 | basis_mem | |
cat_basisU V X Y :
directv (U + V) -> basis_of U X -> basis_of V Y -> basis_of (U + V) (X ++ Y).
Proof.
move=> dxUV /andP[/eqP defU freeX] /andP[/eqP defV freeY].
by rewrite /basis_of span_cat cat_free defU defV // eqxx freeX freeY.
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 | cat_basis | |
size_basisU n (X : n.-tuple vT) : basis_of U X -> \dim U = n.
Proof. by case/andP=> /eqP <- /eqnP->; apply: size_tuple. 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 | size_basis | |
basisEdimX U : basis_of U X = (U <= <<X>>)%VS && (size X <= \dim U).
Proof.
apply/andP/idP=> [[defU /eqnP <-]| ]; first by rewrite -eqEdim eq_sym.
case/andP=> sUX leXU; have leXX := dim_span X.
rewrite /free eq_sym eqEdim sUX eqn_leq !(leq_trans leXX) //.
by rewrite (leq_trans leXU) ?dimvS.
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 | basisEdim | |
basisEfreeX U :
basis_of U X = [&& free X, (<<X>> <= U)%VS & \dim U <= size X].
Proof.
by rewrite andbC; apply: andb_id2r => freeX; rewrite eqEdim (eqnP freeX).
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 | basisEfree | |
perm_basisX Y U : perm_eq X Y -> basis_of U X = basis_of U Y.
Proof.
move=> eqXY; congr ((_ == _) && _); last exact: perm_free.
exact/eq_span/perm_mem.
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_basis | |
vbasisPU : basis_of U (vbasis U).
Proof.
rewrite /basis_of free_b2mx span_b2mx (sameP eqP (vs2mxP _ _)) !genmxE.
have ->: b2mx (vbasis U) = row_base (vs2mx U).
by apply/row_matrixP=> i; rewrite unlock rowK tnth_mktuple r2vK.
by rewrite row_base_free !eq_row_base 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 | vbasisP | |
vbasis_memv U : v \in (vbasis U) -> v \in U.
Proof. exact: basis_mem (vbasisP _). 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 | vbasis_mem | |
coord_vbasisv U :
v \in U -> v = \sum_(i < \dim U) coord (vbasis U) i v *: (vbasis U)`_i.
Proof. exact: coord_basis (vbasisP 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 | coord_vbasis | |
span_bigcat:
(<<\big[cat/[::]]_(i | P i) Xs i>> = \sum_(i | P i) <<Xs i>>)%VS.
Proof. by rewrite (big_morph _ span_cat span_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_bigcat | |
bigcat_free:
directv (\sum_(i | P i) <<Xs i>>) ->
(forall i, P i -> free (Xs i)) -> free (\big[cat/[::]]_(i | P i) Xs i).
Proof.
rewrite /free directvE /= span_bigcat => /directvP-> /= freeXs.
rewrite (big_morph _ (@size_cat _) (erefl _)) /=.
by apply/eqP/eq_bigr=> i /freeXs/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 | bigcat_free | |
bigcat_basisUs (U := (\sum_(i | P i) Us i)%VS) :
directv U -> (forall i, P i -> basis_of (Us i) (Xs i)) ->
basis_of U (\big[cat/[::]]_(i | P i) Xs i).
Proof.
move=> dxU XsUs; rewrite /basis_of span_bigcat.
have defUs i: P i -> span (Xs i) = Us i by case/XsUs/andP=> /eqP.
rewrite (eq_bigr _ defUs) eqxx bigcat_free // => [|_ /XsUs/andP[]//].
apply/directvP; rewrite /= (eq_bigr _ defUs) (directvP dxU) /=.
by apply/eq_bigr=> i /defUs->.
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 | bigcat_basis | |
directvS := (directv_def (Phantom _ S%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 | directv | |
fun_of_lfun_defaT rT (f : 'Hom(aT, rT)) :=
r2v \o mulmxr (f2mx f) \o v2r. | 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 | fun_of_lfun_def | |
fun_of_lfun:= locked_with lfun_key fun_of_lfun_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 | fun_of_lfun | |
fun_of_lfun_unlockable:= [unlockable fun fun_of_lfun]. | 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 | fun_of_lfun_unlockable | |
linfun_defaT rT (f : aT -> rT) :=
Hom (lin1_mx (v2r \o f \o r2v)). | 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 | linfun_def | |
linfun:= locked_with lfun_key linfun_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 | linfun | |
linfun_unlockable:= [unlockable fun linfun]. | 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 | linfun_unlockable | |
id_lfunvT := @linfun vT vT idfun. | 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 | id_lfun | |
comp_lfunaT vT rT (f : 'Hom(vT, rT)) (g : 'Hom(aT, vT)) :=
linfun (fun_of_lfun f \o fun_of_lfun g). | 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 | comp_lfun | |
fun_of_lfun: hom >-> Funclass. | 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 | fun_of_lfun | |
inv_lfunaT rT (f : 'Hom(aT, rT)) := Hom (pinvmx (f2mx f)). | 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 | inv_lfun | |
lkeraT rT (f : 'Hom(aT, rT)) := mx2vs (kermx (f2mx f)).
Fact lfun_img_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 | lker | |
lfun_img_defaT rT f (U : {vspace aT}) : {vspace rT} :=
mx2vs (vs2mx U *m f2mx f). | 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_img_def | |
lfun_img:= locked_with lfun_img_key lfun_img_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 | lfun_img | |
lfun_img_unlockable:= [unlockable fun lfun_img]. | 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 | lfun_img_unlockable | |
lfun_preimaT rT (f : 'Hom(aT, rT)) W :=
(lfun_img (inv_lfun f) (W :&: lfun_img f fullv) + lker f)%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 | lfun_preim | |
limgf := (lfun_img f fullv). | 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 | limg | |
Definition_ := [Choice of 'Hom(aT, rT) by <:].
Fact lfun_is_semilinear f : semilinear f.
Proof. by rewrite unlock; apply: semilinearP. Qed.
HB.instance Definition _ (f : hom aT rT) := GRing.isSemilinear.Build R aT rT _ f
(lfun_is_semilinear f). | 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 | |
lfunE(ff : {linear aT -> rT}) : linfun ff =1 ff.
Proof. by move=> v; rewrite 2!unlock /= mul_rV_lin1 /= !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 | lfunE | |
fun_of_lfunK: cancel (@fun_of_lfun R aT rT) linfun.
Proof.
move=> f; apply/val_inj/row_matrixP=> i.
by rewrite 2!unlock /= !rowE mul_rV_lin1 /= !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 | fun_of_lfunK | |
lfunPf g : f =1 g <-> f = g.
Proof.
split=> [eq_fg | -> //]; rewrite -[f]fun_of_lfunK -[g]fun_of_lfunK unlock.
by apply/val_inj/row_matrixP=> i; rewrite !rowE !mul_rV_lin1 /= eq_fg.
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 | lfunP | |
zero_lfun: 'Hom(aT, rT) := linfun \0. | 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 | zero_lfun | |
add_lfunf g := linfun (f \+ g).
Fact lfun_addA : associative add_lfun.
Proof. by move=> f g h; apply/lfunP=> v; rewrite !lfunE /= !lfunE addrA. Qed.
Fact lfun_addC : commutative add_lfun.
Proof. by move=> f g; apply/lfunP=> v; rewrite !lfunE /= addrC. Qed.
Fact lfun_add0 : left_id zero_lfun add_lfun.
Proof. by move=> f; apply/lfunP=> v; rewrite lfunE /= lfunE add0r. Qed.
HB.instance Definition _ := GRing.isNmodule.Build 'Hom(aT, rT)
lfun_addA lfun_addC lfun_add0. | 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 | add_lfun | |
zero_lfunEx : (0 : 'Hom(aT, rT)) x = 0. Proof. exact: 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 | zero_lfunE | |
add_lfunEf g x : (f + g) x = f x + g x. Proof. exact: 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 | add_lfunE | |
sum_lfunEI (r : seq I) (P : pred I) (fs : I -> 'Hom(aT, rT)) x :
(\sum_(i <- r | P i) fs i) x = \sum_(i <- r | P i) fs i x.
Proof. by elim/big_rec2: _ => [|i _ f _ <-]; 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 | sum_lfunE | |
opp_lfunf := linfun (-%R \o f). | 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 | opp_lfun | |
lfun_addN: left_inverse zero_lfun opp_lfun add_lfun.
Proof. by move=> f; apply/lfunP=> v; rewrite !lfunE /= lfunE addNr. Qed.
HB.instance Definition _ := GRing.Nmodule_isZmodule.Build 'Hom(aT, rT)
lfun_addN. | 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 | lfun_addN | |
opp_lfunEf x : (- f) x = - f x. Proof. exact: 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 | opp_lfunE | |
scale_lfunk f := linfun (k \*: f).
Local Infix "*:l" := scale_lfun (at level 40).
Fact lfun_scaleA k1 k2 f : k1 *:l (k2 *:l f) = (k1 * k2) *:l f.
Proof. by apply/lfunP=> v; rewrite !lfunE /= lfunE scalerA. Qed.
Fact lfun_scale0 f : 0 *:l f = 0.
Proof. by apply/lfunP=> v; rewrite !lfunE /= scale0r. Qed.
Fact lfun_scale1 f : 1 *:l f = f.
Proof. by apply/lfunP=> v; rewrite lfunE /= scale1r. Qed.
Fact lfun_scaleDr k f1 f2 : k *:l (f1 + f2) = k *:l f1 + k *:l f2.
Proof. by apply/lfunP=> v; rewrite !lfunE /= !lfunE scalerDr. Qed.
Fact lfun_scaleDl f k1 k2 : (k1 + k2) *:l f = k1 *:l f + k2 *:l f.
Proof. by apply/lfunP=> v; rewrite !lfunE /= !lfunE scalerDl. Qed.
HB.instance Definition _ := GRing.Nmodule_isLSemiModule.Build _ 'Hom(aT, rT)
lfun_scaleA lfun_scale0 lfun_scale1 lfun_scaleDr lfun_scaleDl. | 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 | scale_lfun | |
scale_lfunEk f x : (k *: f) x = k *: f x. Proof. exact: lfunE. Qed.
Fact lfun_vect_iso : SemiVector.axiom (dim aT * dim rT) 'Hom(aT, rT).
Proof.
exists (mxvec \o f2mx).
by split => [a f|f g]; rewrite -semilinearP/=;
rewrite -[A in _ = mxvec A]/(f2mx (Hom _)); congr (mxvec (f2mx _));
apply/lfunP=> v; rewrite lfunE/= unlock /= -!semilinearP.
apply: Bijective (Hom \o vec_mx) _ _ => [[A]|A] /=; last exact: vec_mxK.
by rewrite mxvecK.
Qed.
HB.instance Definition _ := LSemiModule_hasFinDim.Build _ 'Hom(aT, rT)
lfun_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 | scale_lfunE | |
Definition_ (R : comNzRingType) (aT rT : vectType R) :=
SemiVector.on 'Hom(aT, rT). | 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 | |
id_lfunEu: \1%VF u = u :> aT. Proof. exact: 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 | id_lfunE | |
comp_lfunEf g u : (f \o g)%VF u = f (g u). Proof. exact: 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 | comp_lfunE | |
comp_lfunAf g h : (f \o (g \o h) = (f \o g) \o h)%VF.
Proof. by apply/lfunP=> u; do !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 | comp_lfunA | |
comp_lfun1lf : (\1 \o f)%VF = f.
Proof. by apply/lfunP=> u; do !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 | comp_lfun1l | |
comp_lfun1rf : (f \o \1)%VF = f.
Proof. by apply/lfunP=> u; do !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 | comp_lfun1r | |
comp_lfun0lg : (0 \o g)%VF = 0 :> 'Hom(aT, rT).
Proof. by apply/lfunP=> u; do !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 | comp_lfun0l | |
comp_lfun0rf : (f \o 0)%VF = 0 :> 'Hom(aT, rT).
Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite 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 | comp_lfun0r | |
comp_lfunDlf1 f2 g : ((f1 + f2) \o g = (f1 \o g) + (f2 \o g))%VF.
Proof. by apply/lfunP=> u; do !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 | comp_lfunDl | |
comp_lfunDrf g1 g2 : (f \o (g1 + g2) = (f \o g1) + (f \o g2))%VF.
Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearD. 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 | comp_lfunDr | |
comp_lfunNlf g : ((- f) \o g = - (f \o g))%VF.
Proof. by apply/lfunP=> u; do !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 | comp_lfunNl | |
comp_lfunNrf g : (f \o (- g) = - (f \o g))%VF.
Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearN. 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 | comp_lfunNr | |
lfun_simp:=
(comp_lfunE, scale_lfunE, opp_lfunE, add_lfunE, sum_lfunE, lfunE). | 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_simp | |
comp_lfunZlk f g : (k *: (f \o g) = (k *: f) \o g)%VF.
Proof. by apply/lfunP=> u; do !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 | comp_lfunZl | |
comp_lfunZrk f g : (k *: (f \o g) = f \o (k *: g))%VF.
Proof. by apply/lfunP=> u; do !rewrite lfunE /=; rewrite linearZ. 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 | comp_lfunZr | |
limgSf U V : (U <= V)%VS -> (f @: U <= f @: V)%VS.
Proof. by rewrite unlock /subsetv !genmxE; apply: submxMr. 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 | limgS | |
limg_linef v : (f @: <[v]> = <[f v]>)%VS.
Proof.
apply/eqP; rewrite 2!unlock eqEsubv /subsetv /= r2vK !genmxE.
by rewrite !(eqmxMr _ (genmxE _)) 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 | limg_line | |
limg0f : (f @: 0 = 0)%VS. Proof. by rewrite limg_line 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 | limg0 | |
memv_imgf v U : v \in U -> f v \in (f @: U)%VS.
Proof. by move=> Uv; rewrite memvE -limg_line limgS. 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_img | |
memv_imgPf w U :
reflect (exists2 u, u \in U & w = f u) (w \in f @: U)%VS.
Proof.
apply: (iffP idP) => [|[u Uu ->]]; last exact: memv_img.
rewrite 2!unlock memvE /subsetv !genmxE => /submxP[ku Drw].
exists (r2v (ku *m vs2mx U)); last by rewrite /= r2vK -mulmxA -Drw v2rK.
by rewrite memvE /subsetv !genmxE r2vK 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_imgP | |
lim0gU : (0 @: U = 0 :> {vspace rT})%VS.
Proof.
apply/eqP; rewrite -subv0; apply/subvP=> _ /memv_imgP[u _ ->].
by rewrite lfunE 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 | lim0g | |
eq_in_limgV f g : {in V, f =1 g} -> (f @: V = g @: V)%VS.
Proof.
move=> eq_fg; apply/vspaceP=> y.
by apply/memv_imgP/memv_imgP=> [][x Vx ->]; exists x; rewrite ?eq_fg.
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_in_limg | |
limgDf : {morph lfun_img f : U V / U + V}%VS.
Proof.
move=> U V; apply/eqP; rewrite unlock eqEsubv /subsetv /= -genmx_adds.
by rewrite !genmxE !(eqmxMr _ (genmxE _)) !addsmxMr 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 | limgD | |
limg_sumf I r (P : pred I) Us :
(f @: (\sum_(i <- r | P i) Us i) = \sum_(i <- r | P i) f @: Us i)%VS.
Proof. exact: (big_morph _ (limgD f) (limg0 f)). 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_sum | |
limg_capf U V : (f @: (U :&: V) <= f @: U :&: f @: V)%VS.
Proof. by rewrite subv_cap !limgS ?capvSl ?capvSr. 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_cap | |
limg_bigcapf I r (P : pred I) Us :
(f @: (\bigcap_(i <- r | P i) Us i) <= \bigcap_(i <- r | P i) f @: Us i)%VS.
Proof.
elim/big_rec2: _ => [|i V U _ sUV]; first exact: subvf.
by rewrite (subv_trans (limg_cap f _ U)) ?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 | limg_bigcap | |
limg_spanf X : (f @: <<X>> = <<map f X>>)%VS.
Proof.
by rewrite !span_def big_map limg_sum; apply: eq_bigr => x _; rewrite limg_line.
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_span | |
subset_limgPf U (r : seq rT) :
{subset r <= (f @: U)%VS} <-> (exists2 a, all (mem U) a & r = map f a).
Proof.
split => [|[{}r /allP/= rE ->] _ /mapP[x xr ->]]; last by rewrite memv_img ?rE.
move=> /(_ _ _)/memv_imgP/sig2_eqW-/(all_sig_cond (0 : aT))[f' f'P].
exists (map f' r); first by apply/allP => _ /mapP [x /f'P[? ?] ->].
by symmetry; rewrite -map_comp; apply: map_id_in => x /f'P[].
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 | subset_limgP | |
lfunPnf g : reflect (exists u, f u != g u) (f != g).
Proof.
apply: (iffP idP) => [f'g|[x]]; last by apply: contraNneq => /lfunP->.
suffices /subvPn[_ /memv_imgP[u _ ->]]: ~~ (limg (f - g) <= 0)%VS.
by rewrite lfunE /= lfunE /= memv0 subr_eq0; exists u.
apply: contra f'g => /subvP fg0; apply/eqP/lfunP=> u; apply/eqP.
by rewrite -subr_eq0 -opp_lfunE -add_lfunE -memv0 fg0 ?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 | lfunPn | |
inv_lfun_deff : (f \o f^-1 \o f)%VF = f.
Proof.
apply/lfunP=> u; do !rewrite lfunE /=; rewrite unlock /= !r2vK.
by rewrite mulmxKpV ?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 | inv_lfun_def | |
limg_lfunVKf : {in limg f, cancel f^-1%VF f}.
Proof. by move=> _ /memv_imgP[u _ ->]; rewrite -!comp_lfunE inv_lfun_def. 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_lfunVK | |
lkerEf U : (U <= lker f)%VS = (f @: U == 0)%VS.
Proof.
rewrite unlock -dimv_eq0 /dimv /subsetv !genmxE mxrank_eq0.
by rewrite (sameP sub_kermxP 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 | lkerE | |
memv_kerf v : (v \in lker f) = (f v == 0).
Proof. by rewrite -memv0 !memvE subv0 lkerE limg_line. 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_ker | |
eqlfunPf g v : reflect (f v = g v) (v \in lker (f - g)).
Proof. by rewrite memv_ker !lfun_simp 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 | eqlfunP | |
eqlfun_inPV f g : reflect {in V, f =1 g} (V <= lker (f - g))%VS.
Proof. by apply: (iffP subvP) => E x /E/eqlfunP. 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 | eqlfun_inP | |
limg_ker_complf U : (f @: (U :\: lker f) = f @: U)%VS.
Proof.
rewrite -{2}(addv_diff_cap U (lker f)) limgD; apply/esym/addv_idPl.
by rewrite (subv_trans _ (sub0v _)) // subv0 -lkerE capvSr.
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_ker_compl | |
limg_ker_dimf U : (\dim (U :&: lker f) + \dim (f @: U) = \dim U)%N.
Proof.
rewrite unlock /dimv /= genmx_cap genmx_id -genmx_cap !genmxE.
by rewrite addnC mxrank_mul_ker.
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_ker_dim | |
limg_dim_eqf U : (U :&: lker f = 0)%VS -> \dim (f @: U) = \dim U.
Proof. by rewrite -(limg_ker_dim f U) => ->; rewrite 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 | limg_dim_eq | |
limg_basis_off U X :
(U :&: lker f = 0)%VS -> basis_of U X -> basis_of (f @: U) (map f X).
Proof.
move=> injUf /andP[/eqP defU /eqnP freeX].
by rewrite /basis_of /free size_map -limg_span -freeX defU limg_dim_eq ?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 | limg_basis_of |
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