phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
vs2mxF : vs2mx {:vT} = 1%:M.	Proof. by rewrite /= genmx1. Qed.	Let	vs2mxF	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "genmx1", "vT", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
row_b2mx n (X : n.-tuple vT) i : row i (b2mx X) = v2r X`_i.	Proof. by rewrite -tnth_nth rowK. Qed.	Let	row_b2mx	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "b2mx", "row", "rowK", "tnth_nth", "tuple", "v2r", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_b2mx n (X : n.-tuple vT) : span X = mx2vs (b2mx X).	Proof. by rewrite unlock tvalK; case: _ / (esym _). Qed.	Let	span_b2mx	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "b2mx", "mx2vs", "span", "tuple", "tvalK", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_b2mx n (X : n.-tuple vT) (rk : 'rV_n) : \sum_i rk 0 i : X`_i = r2v (rk m b2mx X).	Proof. rewrite mulmx_sum_row linear_sum; apply: eq_bigr => i _. by rewrite row_b2mx linearZ /= v2rK. Qed.	Let	mul_b2mx	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "b2mx", "eq_bigr", "linearZ", "linear_sum", "mulmx_sum_row", "r2v", "row_b2mx", "tuple", "v2rK", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lin_b2mx n (X : n.-tuple vT) k : \sum_(i < n) k i : X`_i = r2v (\row_i k i m b2mx X).	Proof. by rewrite -mul_b2mx; apply: eq_bigr => i _; rewrite mxE. Qed.	Let	lin_b2mx	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "b2mx", "eq_bigr", "mul_b2mx", "mxE", "r2v", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
free_b2mx n (X : n.-tuple vT) : free X = row_free (b2mx X).	Proof. by rewrite /free /dimv span_b2mx genmxE size_tuple. Qed.	Let	free_b2mx	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "b2mx", "dimv", "free", "genmxE", "row_free", "size_tuple", "span_b2mx", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvE v U : (v \in U) = (<[v]> <= U)%VS.	Proof. by []. Qed.	Lemma	memvE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[]	end hide	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vlineP v1 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.	Lemma	vlineP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "genmxE", "linearZ", "memvK", "scalemx_sub", "sub_rVP", "v2r_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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.	Fact	memv_submod_closed	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Uu", "addmx_sub", "linear0", "linearP", "memvK", "scalemx_sub", "split", "sub0mx", "submod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem0v U : 0 \in U.	Proof. exact: rpred0. Qed.	Lemma	mem0v	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "rpred0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvN U v : (- v \in U) = (v \in U).	Proof. exact: rpredN. Qed.	Lemma	memvN	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "rpredN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvD U : {in U &, forall u v, u + v \in U}.	Proof. exact: rpredD. Qed.	Lemma	memvD	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "rpredD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvB U : {in U &, forall u v, u - v \in U}.	Proof. exact: rpredB. Qed.	Lemma	memvB	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "rpredB" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvZ U k : {in U, forall v, k *: v \in U}.	Proof. exact: rpredZ. Qed.	Lemma	memvZ	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "rpredZ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_suml I 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	memv_suml	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "rpred_sum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_line u : u \in <[u]>%VS.	Proof. by apply/vlineP; exists 1; rewrite scale1r. Qed.	Lemma	memv_line	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "scale1r", "vlineP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvP U 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	subvP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "genmxE", "memvE", "r2v", "r2vK", "rV_subP", "subsetv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvv U : (U <= U)%VS.	Proof. exact/subvP. Qed.	Lemma	subvv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "subvP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subv_trans : transitive subV.	Proof. by move=> U V W /subvP sUV /subvP sVW; apply/subvP=> u /sUV/sVW. Qed.	Lemma	subv_trans	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "subV", "subvP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subv_anti : antisymmetric subV.	Proof. by move=> U V; apply/vs2mxP. Qed.	Lemma	subv_anti	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "subV", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqEsubv U V : (U == V) = (U <= V <= U)%VS.	Proof. by apply/eqP/idP=> [-> \| /subv_anti//]; rewrite subvv. Qed.	Lemma	eqEsubv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "subv_anti", "subvv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vspaceP U V : U =i V <-> U = V.	Proof. split=> [eqUV \| -> //]; apply/subv_anti/andP. by split; apply/subvP=> v; rewrite eqUV. Qed.	Lemma	vspaceP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "split", "subvP", "subv_anti" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	subvPn	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Uu", "apply", "last", "memvK", "r2v", "r2vK", "row", "row_sub", "row_subPn", "subvP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub0v U : (0 <= U)%VS.	Proof. exact: mem0v. Qed.	Lemma	sub0v	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "mem0v" ]	Empty space.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subv0 U : (U <= 0)%VS = (U == 0%VS).	Proof. by rewrite eqEsubv sub0v andbT. Qed.	Lemma	subv0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "eqEsubv", "sub0v" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv0 v : (v \in 0%VS) = (v == 0).	Proof. by apply/idP/eqP=> [/vlineP[k ->] \| ->]; rewrite (scaler0, mem0v). Qed.	Lemma	memv0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "mem0v", "scaler0", "vlineP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvf U : (U <= fullv)%VS.	Proof. by rewrite /subsetv vs2mxF submx1. Qed.	Lemma	subvf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "fullv", "submx1", "subsetv", "vs2mxF" ]	Full space	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvf v : v \in fullv.	Proof. exact: subvf. Qed.	Lemma	memvf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "fullv", "subvf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_pick U : vpick U \in U.	Proof. by rewrite mem_r2v nz_row_sub. Qed.	Lemma	memv_pick	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "mem_r2v", "nz_row_sub", "vpick" ]	Picking a non-zero vector in a subspace.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vpick0 U : (vpick U == 0) = (U == 0%VS).	Proof. by rewrite -memv0 mem_r2v -subv0 /subV vs2mx0 !submx0 nz_row_eq0. Qed.	Lemma	vpick0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "mem_r2v", "memv0", "nz_row_eq0", "subV", "submx0", "subv0", "vpick", "vs2mx0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subv_add U V W : (U + V <= W)%VS = (U <= W)%VS && (V <= W)%VS.	Proof. by rewrite /subV vs2mxD addsmx_sub. Qed.	Lemma	subv_add	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addsmx_sub", "subV", "vs2mxD" ]	Sum of subspaces.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addvS U1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 + V1 <= U2 + V2)%VS.	Proof. by rewrite /subV !vs2mxD; apply: addsmxS. Qed.	Lemma	addvS	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addsmxS", "apply", "subV", "vs2mxD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addvSl U V : (U <= U + V)%VS.	Proof. by rewrite /subV vs2mxD addsmxSl. Qed.	Lemma	addvSl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addsmxSl", "subV", "vs2mxD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addvSr U V : (V <= U + V)%VS.	Proof. by rewrite /subV vs2mxD addsmxSr. Qed.	Lemma	addvSr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addsmxSr", "subV", "vs2mxD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addvC : commutative addV.	Proof. by move=> U V; apply/vs2mxP; rewrite !vs2mxD addsmxC submx_refl. Qed.	Lemma	addvC	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addV", "addsmxC", "apply", "submx_refl", "vs2mxD", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addvA : associative addV.	Proof. by move=> U V W; apply/vs2mxP; rewrite !vs2mxD addsmxA submx_refl. Qed.	Lemma	addvA	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addV", "addsmxA", "apply", "submx_refl", "vs2mxD", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	addv_idPl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addsmx_idPl", "apply", "eqmxP", "subV", "vs2mxD", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_idPr {U V} : reflect (U + V = V)%VS (U <= V)%VS.	Proof. by rewrite addvC; apply: addv_idPl. Qed.	Lemma	addv_idPr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addvC", "addv_idPl", "apply" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addvv : idempotent_op addV.	Proof. by move=> U; apply/addv_idPl. Qed.	Lemma	addvv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addV", "addv_idPl", "apply", "idempotent_op" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add0v : left_id 0%VS addV.	Proof. by move=> U; apply/addv_idPr/sub0v. Qed.	Lemma	add0v	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addV", "addv_idPr", "apply", "sub0v" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv0 : right_id 0%VS addV.	Proof. by move=> U; apply/addv_idPl/sub0v. Qed.	Lemma	addv0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addV", "addv_idPl", "apply", "sub0v" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumfv : left_zero fullv addV.	Proof. by move=> U; apply/addv_idPl/subvf. Qed.	Lemma	sumfv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addV", "addv_idPl", "apply", "fullv", "subvf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addvf : right_zero fullv addV.	Proof. by move=> U; apply/addv_idPr/subvf. Qed.	Lemma	addvf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addV", "addv_idPr", "apply", "fullv", "subvf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_add u 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	memv_add	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addmx_sub_adds", "apply", "genmxE", "linearD", "memvK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	memv_addP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Uu", "apply", "genmxE", "last", "linearD", "mem_r2v", "memvK", "memv_add", "r2v", "sub_addsmxP", "submxMl", "v2rK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumv_sup i0 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.	Lemma	sumv_sup	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Pi0", "addvSl", "bigD1", "i0", "subv_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	subv_sumP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "big_rec", "sub0v", "subv_add", "subv_trans", "sumv_sup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_sumr P 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	memv_sumr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "rpred_sum", "sumv_sup", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	memv_sumP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "last", "linear_sum", "mem_r2v", "memvK", "memv_sumr", "r2v", "sub_sumsmxP", "submxMl", "v2rK", "vT", "vs2mx", "vs2mx_sum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subv_cap U V W : (U <= V :&: W)%VS = (U <= V)%VS && (U <= W)%VS.	Proof. by rewrite /subV vs2mxI sub_capmx. Qed.	Lemma	subv_cap	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "subV", "sub_capmx", "vs2mxI" ]	Intersection	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capvS U1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 :&: V1 <= U2 :&: V2)%VS.	Proof. by rewrite /subV !vs2mxI; apply: capmxS. Qed.	Lemma	capvS	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capmxS", "subV", "vs2mxI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capvSl U V : (U :&: V <= U)%VS.	Proof. by rewrite /subV vs2mxI capmxSl. Qed.	Lemma	capvSl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "capmxSl", "subV", "vs2mxI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capvSr U V : (U :&: V <= V)%VS.	Proof. by rewrite /subV vs2mxI capmxSr. Qed.	Lemma	capvSr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "capmxSr", "subV", "vs2mxI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capvC : commutative capV.	Proof. by move=> U V; apply/vs2mxP; rewrite !vs2mxI capmxC submx_refl. Qed.	Lemma	capvC	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capV", "capmxC", "submx_refl", "vs2mxI", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capvA : associative capV.	Proof. by move=> U V W; apply/vs2mxP; rewrite !vs2mxI capmxA submx_refl. Qed.	Lemma	capvA	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capV", "capmxA", "submx_refl", "vs2mxI", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	capv_idPl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capmx_idPl", "eqmxP", "subV", "vs2mxI", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capv_idPr {U V} : reflect (U :&: V = V)%VS (V <= U)%VS.	Proof. by rewrite capvC; apply: capv_idPl. Qed.	Lemma	capv_idPr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capvC", "capv_idPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capvv : idempotent_op capV.	Proof. by move=> U; apply/capv_idPl. Qed.	Lemma	capvv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capV", "capv_idPl", "idempotent_op" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cap0v : left_zero 0%VS capV.	Proof. by move=> U; apply/capv_idPl/sub0v. Qed.	Lemma	cap0v	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capV", "capv_idPl", "sub0v" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capv0 : right_zero 0%VS capV.	Proof. by move=> U; apply/capv_idPr/sub0v. Qed.	Lemma	capv0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capV", "capv_idPr", "sub0v" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capfv : left_id fullv capV.	Proof. by move=> U; apply/capv_idPr/subvf. Qed.	Lemma	capfv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capV", "capv_idPr", "fullv", "subvf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capvf : right_id fullv capV.	Proof. by move=> U; apply/capv_idPl/subvf. Qed.	Lemma	capvf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capV", "capv_idPl", "fullv", "subvf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_cap w U V : (w \in U :&: V)%VS = (w \in U) && (w \in V).	Proof. by rewrite !memvE subv_cap. Qed.	Lemma	memv_cap	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "memvE", "subv_cap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	memv_capP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "memv_cap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vspace_modl U 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	vspace_modl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "eqmxP", "matrix_modl", "vs2mxD", "vs2mxI", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vspace_modr U V W : (W <= U -> (U :&: V) + W = U :&: (V + W))%VS.	Proof. by rewrite -!(addvC W) !(capvC U); apply: vspace_modl. Qed.	Lemma	vspace_modr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addvC", "apply", "capvC", "vspace_modl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigcapv_inf i0 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	bigcapv_inf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Pi0", "apply", "bigD1", "capvSl", "i0", "subv_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	subv_bigcapP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "big_rec", "bigcapv_inf", "subv_cap", "subv_trans", "subvf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_complf U : (U + U^C)%VS = fullv.	Proof. apply/vs2mxP; rewrite vs2mxD -gen_vs2mx -genmx_adds !genmxE submx1 sub1mx. exact: addsmx_compl_full. Qed.	Lemma	addv_complf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addsmx_compl_full", "apply", "fullv", "gen_vs2mx", "genmxE", "genmx_adds", "sub1mx", "submx1", "vs2mxD", "vs2mxP" ]	Complement	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capv_compl U : (U :&: U^C = 0)%VS.	Proof. apply/val_inj; rewrite [val]/= vs2mx0 vs2mxI -gen_vs2mx -genmx_cap. by rewrite capmx_compl genmx0. Qed.	Lemma	capv_compl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capmx_compl", "gen_vs2mx", "genmx0", "genmx_cap", "val", "val_inj", "vs2mx0", "vs2mxI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffvSl U V : (U :\: V <= U)%VS.	Proof. by rewrite /subV genmxE diffmxSl. Qed.	Lemma	diffvSl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "diffmxSl", "genmxE", "subV" ]	Difference	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capv_diff U 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	capv_diff	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "capmx_diff", "gen_vs2mx", "genmx0", "genmx_cap", "val", "val_inj", "vs2mx0", "vs2mxI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_diff_cap U V : (U :\: V + U :&: V)%VS = U.	Proof. apply/vs2mxP; rewrite vs2mxD -genmx_adds !genmxE. exact/eqmxP/addsmx_diff_cap_eq. Qed.	Lemma	addv_diff_cap	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addsmx_diff_cap_eq", "apply", "eqmxP", "genmxE", "genmx_adds", "vs2mxD", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_diff U V : (U :\: V + V = U + V)%VS.	Proof. by rewrite -{2}(addv_diff_cap U V) -addvA (addv_idPr (capvSr U V)). Qed.	Lemma	addv_diff	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addvA", "addv_diff_cap", "addv_idPr", "capvSr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv0 : \dim (0%VS : {vspace vT}) = 0.	Proof. by rewrite /dimv vs2mx0 mxrank0. Qed.	Lemma	dimv0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "dimv", "mxrank0", "vT", "vs2mx0" ]	Subspace dimension.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_eq0 U : (\dim U == 0) = (U == 0%VS).	Proof. by rewrite /dimv /= mxrank_eq0 [in RHS]/eq_op /= linear0 genmx0. Qed.	Lemma	dimv_eq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "dimv", "genmx0", "linear0", "mxrank_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimvf : \dim {:vT} = dim vT.	Proof. by rewrite /dimv vs2mxF mxrank1. Qed.	Lemma	dimvf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "dimv", "mxrank1", "vT", "vs2mxF" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_vline v : \dim <[v]> = (v != 0).	Proof. by rewrite /dimv mxrank_gen rank_rV (can2_eq v2rK r2vK) linear0. Qed.	Lemma	dim_vline	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "can2_eq", "dim", "dimv", "linear0", "mxrank_gen", "r2vK", "rank_rV", "v2rK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimvS U V : (U <= V)%VS -> \dim U <= \dim V.	Proof. exact: mxrankS. Qed.	Lemma	dimvS	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "mxrankS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_leqif_sup U V : (U <= V)%VS -> \dim U <= \dim V ?= iff (V <= U)%VS.	Proof. exact: mxrank_leqif_sup. Qed.	Lemma	dimv_leqif_sup	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "mxrank_leqif_sup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_leqif_eq U V : (U <= V)%VS -> \dim U <= \dim V ?= iff (U == V).	Proof. by rewrite eqEsubv; apply: mxrank_leqif_eq. Qed.	Lemma	dimv_leqif_eq	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "dim", "eqEsubv", "mxrank_leqif_eq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqEdim U V : (U == V) = (U <= V)%VS && (\dim V <= \dim U).	Proof. by apply/idP/andP=> [/eqP \| [/dimv_leqif_eq/geq_leqif]] ->. Qed.	Lemma	eqEdim	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "dim", "dimv_leqif_eq", "geq_leqif" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_compl U : \dim U^C = (\dim {:vT} - \dim U)%N.	Proof. by rewrite dimvf /dimv mxrank_gen mxrank_compl. Qed.	Lemma	dimv_compl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "dimv", "dimvf", "mxrank_compl", "mxrank_gen", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_cap_compl U V : (\dim (U :&: V) + \dim (U :\: V))%N = \dim U.	Proof. by rewrite /dimv !mxrank_gen mxrank_cap_compl. Qed.	Lemma	dimv_cap_compl	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "dimv", "mxrank_cap_compl", "mxrank_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_sum_cap U V : (\dim (U + V) + \dim (U :&: V) = \dim U + \dim V)%N.	Proof. by rewrite /dimv !mxrank_gen mxrank_sum_cap. Qed.	Lemma	dimv_sum_cap	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "dimv", "mxrank_gen", "mxrank_sum_cap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_disjoint_sum U 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	dimv_disjoint_sum	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addn0", "dim", "dimv0", "dimv_sum_cap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_add_leqif U 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	dimv_add_leqif	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "dim", "dimv", "genmxE", "mxrank_adds_leqif", "mxrank_gen", "subV", "vs2mx0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffv_eq0 U 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	diffv_eq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addn0", "capvSl", "capv_idPl", "dim", "dimv_cap_compl", "dimv_eq0", "dimv_leqif_eq", "eq_sym", "eqn_add2l" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_leq_sum I 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	dimv_leq_sum	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "big_rec2", "dim", "dim_vline", "dimv_add_leqif", "eqxx", "leq_add2l", "leq_trans", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_expr	:= Sumv { addv_val :> wrapped {vspace vT}; addv_dim : wrapped nat; _ : mxsum_spec (vs2mx (unwrap addv_val)) (unwrap addv_dim) }.	Structure	addv_expr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "mxsum_spec", "nat", "vT", "vs2mx", "wrapped" ]	nevetheless reuses much of the matrix theory.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mx_sum_expr_subproof (S : addv_expr) : mxsum_spec (vs2mx (unwrap S)) (unwrap (addv_dim S)).	Proof. by case: S. Qed.	Definition	vs2mx_sum_expr_subproof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_expr", "mxsum_spec", "vs2mx" ]	Piggyback on mxalgebra theory.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mx_sum_expr S	:= ProperMxsumExpr (vs2mx_sum_expr_subproof S).	Canonical	vs2mx_sum_expr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vs2mx_sum_expr_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivial_addv U	:= @Sumv (Wrap U) (Wrap (\dim U)) (TrivialMxsum _).	Canonical	trivial_addv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_addv_expr	:= ProperSumvExpr { proper_addv_val :> {vspace vT}; proper_addv_dim :> nat; _ : mxsum_spec (vs2mx proper_addv_val) proper_addv_dim }.	Structure	proper_addv_expr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "mxsum_spec", "nat", "vT", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_addvP (S : proper_addv_expr)	:= let: ProperSumvExpr _ _ termS := S return mxsum_spec (vs2mx S) S in termS.	Definition	proper_addvP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "mxsum_spec", "proper_addv_expr", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_addv (S : proper_addv_expr)	:= @Sumv (wrap (S : {vspace vT})) (wrap (S : nat)) (proper_addvP S).	Canonical	proper_addv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "nat", "proper_addvP", "proper_addv_expr", "vT", "wrap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
binary_addv_subproof : mxsum_spec (vs2mx (unwrap S1 + unwrap S2)) (unwrap (addv_dim S1) + unwrap (addv_dim S2)).	Proof. by rewrite vs2mxD; apply: proper_mxsumP. Qed.	Fact	binary_addv_subproof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "S1", "S2", "apply", "mxsum_spec", "proper_mxsumP", "vs2mx", "vs2mxD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
binary_addv_expr	:= ProperSumvExpr binary_addv_subproof.	Canonical	binary_addv_expr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "binary_addv_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nary_addv_subproof : mxsum_spec (vs2mx (\sum_(i <- r \| P i) unwrap (S_ i))) (\sum_(i <- r \| P i) unwrap (addv_dim (S_ i))).	Proof. by rewrite vs2mx_sum; apply: proper_mxsumP. Qed.	Fact	nary_addv_subproof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "mxsum_spec", "proper_mxsumP", "vs2mx", "vs2mx_sum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nary_addv_expr	:= ProperSumvExpr nary_addv_subproof.	Canonical	nary_addv_expr	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "nary_addv_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mxF : vs2mx {:vT} = 1%:M.

Proof. by rewrite /= genmx1. Qed.

Let

vs2mxF

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "genmx1", "vT", "vs2mx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

row_b2mx n (X : n.-tuple vT) i : row i (b2mx X) = v2r X`_i.

Proof. by rewrite -tnth_nth rowK. Qed.

Let

row_b2mx

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "b2mx", "row", "rowK", "tnth_nth", "tuple", "v2r", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_b2mx n (X : n.-tuple vT) : span X = mx2vs (b2mx X).

Proof. by rewrite unlock tvalK; case: _ / (esym _). Qed.

Let

span_b2mx

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "b2mx", "mx2vs", "span", "tuple", "tvalK", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul_b2mx n (X : n.-tuple vT) (rk : 'rV_n) : \sum_i rk 0 i *: X`_i = r2v (rk *m b2mx X).

Proof. rewrite mulmx_sum_row linear_sum; apply: eq_bigr => i _. by rewrite row_b2mx linearZ /= v2rK. Qed.

Let

mul_b2mx

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "b2mx", "eq_bigr", "linearZ", "linear_sum", "mulmx_sum_row", "r2v", "row_b2mx", "tuple", "v2rK", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lin_b2mx n (X : n.-tuple vT) k : \sum_(i < n) k i *: X`_i = r2v (\row_i k i *m b2mx X).

Proof. by rewrite -mul_b2mx; apply: eq_bigr => i _; rewrite mxE. Qed.

Let

lin_b2mx

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "b2mx", "eq_bigr", "mul_b2mx", "mxE", "r2v", "tuple", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

free_b2mx n (X : n.-tuple vT) : free X = row_free (b2mx X).

Proof. by rewrite /free /dimv span_b2mx genmxE size_tuple. Qed.

Let

free_b2mx

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "b2mx", "dimv", "free", "genmxE", "row_free", "size_tuple", "span_b2mx", "tuple", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memvE v U : (v \in U) = (<[v]> <= U)%VS.

Proof. by []. Qed.

Lemma

memvE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[]

end hide

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vlineP v1 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.

Lemma

vlineP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "genmxE", "linearZ", "memvK", "scalemx_sub", "sub_rVP", "v2r_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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.

Fact

memv_submod_closed

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Uu", "addmx_sub", "linear0", "linearP", "memvK", "scalemx_sub", "split", "sub0mx", "submod_closed" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem0v U : 0 \in U.

Proof. exact: rpred0. Qed.

Lemma

mem0v

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "rpred0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memvN U v : (- v \in U) = (v \in U).

Proof. exact: rpredN. Qed.

Lemma

memvN

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "rpredN" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memvD U : {in U &, forall u v, u + v \in U}.

Proof. exact: rpredD. Qed.

Lemma

memvD

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "rpredD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memvB U : {in U &, forall u v, u - v \in U}.

Proof. exact: rpredB. Qed.

Lemma

memvB

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "rpredB" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memvZ U k : {in U, forall v, k *: v \in U}.

Proof. exact: rpredZ. Qed.

Lemma

memvZ

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "rpredZ" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_suml I 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

memv_suml

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "rpred_sum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_line u : u \in <[u]>%VS.

Proof. by apply/vlineP; exists 1; rewrite scale1r. Qed.

Lemma

memv_line

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "scale1r", "vlineP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subvP U 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

subvP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "genmxE", "memvE", "r2v", "r2vK", "rV_subP", "subsetv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subvv U : (U <= U)%VS.

Proof. exact/subvP. Qed.

Lemma

subvv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "subvP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subv_trans : transitive subV.

Proof. by move=> U V W /subvP sUV /subvP sVW; apply/subvP=> u /sUV/sVW. Qed.

Lemma

subv_trans

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "subV", "subvP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subv_anti : antisymmetric subV.

Proof. by move=> U V; apply/vs2mxP. Qed.

Lemma

subv_anti

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "subV", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqEsubv U V : (U == V) = (U <= V <= U)%VS.

Proof. by apply/eqP/idP=> [-> | /subv_anti//]; rewrite subvv. Qed.

Lemma

eqEsubv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "subv_anti", "subvv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vspaceP U V : U =i V <-> U = V.

Proof. split=> [eqUV | -> //]; apply/subv_anti/andP. by split; apply/subvP=> v; rewrite eqUV. Qed.

Lemma

vspaceP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "split", "subvP", "subv_anti" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

subvPn

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Uu", "apply", "last", "memvK", "r2v", "r2vK", "row", "row_sub", "row_subPn", "subvP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub0v U : (0 <= U)%VS.

Proof. exact: mem0v. Qed.

Lemma

sub0v

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "mem0v" ]

Empty space.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subv0 U : (U <= 0)%VS = (U == 0%VS).

Proof. by rewrite eqEsubv sub0v andbT. Qed.

Lemma

subv0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "eqEsubv", "sub0v" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv0 v : (v \in 0%VS) = (v == 0).

Proof. by apply/idP/eqP=> [/vlineP[k ->] | ->]; rewrite (scaler0, mem0v). Qed.

Lemma

memv0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "mem0v", "scaler0", "vlineP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subvf U : (U <= fullv)%VS.

Proof. by rewrite /subsetv vs2mxF submx1. Qed.

Lemma

subvf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "fullv", "submx1", "subsetv", "vs2mxF" ]

Full space

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memvf v : v \in fullv.

Proof. exact: subvf. Qed.

Lemma

memvf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "fullv", "subvf" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_pick U : vpick U \in U.

Proof. by rewrite mem_r2v nz_row_sub. Qed.

Lemma

memv_pick

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "mem_r2v", "nz_row_sub", "vpick" ]

Picking a non-zero vector in a subspace.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vpick0 U : (vpick U == 0) = (U == 0%VS).

Proof. by rewrite -memv0 mem_r2v -subv0 /subV vs2mx0 !submx0 nz_row_eq0. Qed.

Lemma

vpick0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "mem_r2v", "memv0", "nz_row_eq0", "subV", "submx0", "subv0", "vpick", "vs2mx0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subv_add U V W : (U + V <= W)%VS = (U <= W)%VS && (V <= W)%VS.

Proof. by rewrite /subV vs2mxD addsmx_sub. Qed.

Lemma

subv_add

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addsmx_sub", "subV", "vs2mxD" ]

Sum of subspaces.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addvS U1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 + V1 <= U2 + V2)%VS.

Proof. by rewrite /subV !vs2mxD; apply: addsmxS. Qed.

Lemma

addvS

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addsmxS", "apply", "subV", "vs2mxD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addvSl U V : (U <= U + V)%VS.

Proof. by rewrite /subV vs2mxD addsmxSl. Qed.

Lemma

addvSl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addsmxSl", "subV", "vs2mxD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addvSr U V : (V <= U + V)%VS.

Proof. by rewrite /subV vs2mxD addsmxSr. Qed.

Lemma

addvSr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addsmxSr", "subV", "vs2mxD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addvC : commutative addV.

Proof. by move=> U V; apply/vs2mxP; rewrite !vs2mxD addsmxC submx_refl. Qed.

Lemma

addvC

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addV", "addsmxC", "apply", "submx_refl", "vs2mxD", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addvA : associative addV.

Proof. by move=> U V W; apply/vs2mxP; rewrite !vs2mxD addsmxA submx_refl. Qed.

Lemma

addvA

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addV", "addsmxA", "apply", "submx_refl", "vs2mxD", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

addv_idPl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addsmx_idPl", "apply", "eqmxP", "subV", "vs2mxD", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_idPr {U V} : reflect (U + V = V)%VS (U <= V)%VS.

Proof. by rewrite addvC; apply: addv_idPl. Qed.

Lemma

addv_idPr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addvC", "addv_idPl", "apply" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addvv : idempotent_op addV.

Proof. by move=> U; apply/addv_idPl. Qed.

Lemma

addvv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addV", "addv_idPl", "apply", "idempotent_op" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add0v : left_id 0%VS addV.

Proof. by move=> U; apply/addv_idPr/sub0v. Qed.

Lemma

add0v

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addV", "addv_idPr", "apply", "sub0v" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv0 : right_id 0%VS addV.

Proof. by move=> U; apply/addv_idPl/sub0v. Qed.

Lemma

addv0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addV", "addv_idPl", "apply", "sub0v" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumfv : left_zero fullv addV.

Proof. by move=> U; apply/addv_idPl/subvf. Qed.

Lemma

sumfv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addV", "addv_idPl", "apply", "fullv", "subvf" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addvf : right_zero fullv addV.

Proof. by move=> U; apply/addv_idPr/subvf. Qed.

Lemma

addvf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addV", "addv_idPr", "apply", "fullv", "subvf" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_add u 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

memv_add

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addmx_sub_adds", "apply", "genmxE", "linearD", "memvK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

memv_addP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Uu", "apply", "genmxE", "last", "linearD", "mem_r2v", "memvK", "memv_add", "r2v", "sub_addsmxP", "submxMl", "v2rK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumv_sup i0 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.

Lemma

sumv_sup

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Pi0", "addvSl", "bigD1", "i0", "subv_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

subv_sumP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "big_rec", "sub0v", "subv_add", "subv_trans", "sumv_sup" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_sumr P 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

memv_sumr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "rpred_sum", "sumv_sup", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

memv_sumP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "last", "linear_sum", "mem_r2v", "memvK", "memv_sumr", "r2v", "sub_sumsmxP", "submxMl", "v2rK", "vT", "vs2mx", "vs2mx_sum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subv_cap U V W : (U <= V :&: W)%VS = (U <= V)%VS && (U <= W)%VS.

Proof. by rewrite /subV vs2mxI sub_capmx. Qed.

Lemma

subv_cap

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "subV", "sub_capmx", "vs2mxI" ]

Intersection

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capvS U1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 :&: V1 <= U2 :&: V2)%VS.

Proof. by rewrite /subV !vs2mxI; apply: capmxS. Qed.

Lemma

capvS

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capmxS", "subV", "vs2mxI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capvSl U V : (U :&: V <= U)%VS.

Proof. by rewrite /subV vs2mxI capmxSl. Qed.

Lemma

capvSl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "capmxSl", "subV", "vs2mxI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capvSr U V : (U :&: V <= V)%VS.

Proof. by rewrite /subV vs2mxI capmxSr. Qed.

Lemma

capvSr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "capmxSr", "subV", "vs2mxI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capvC : commutative capV.

Proof. by move=> U V; apply/vs2mxP; rewrite !vs2mxI capmxC submx_refl. Qed.

Lemma

capvC

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capV", "capmxC", "submx_refl", "vs2mxI", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capvA : associative capV.

Proof. by move=> U V W; apply/vs2mxP; rewrite !vs2mxI capmxA submx_refl. Qed.

Lemma

capvA

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capV", "capmxA", "submx_refl", "vs2mxI", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

capv_idPl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capmx_idPl", "eqmxP", "subV", "vs2mxI", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capv_idPr {U V} : reflect (U :&: V = V)%VS (V <= U)%VS.

Proof. by rewrite capvC; apply: capv_idPl. Qed.

Lemma

capv_idPr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capvC", "capv_idPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capvv : idempotent_op capV.

Proof. by move=> U; apply/capv_idPl. Qed.

Lemma

capvv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capV", "capv_idPl", "idempotent_op" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cap0v : left_zero 0%VS capV.

Proof. by move=> U; apply/capv_idPl/sub0v. Qed.

Lemma

cap0v

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capV", "capv_idPl", "sub0v" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capv0 : right_zero 0%VS capV.

Proof. by move=> U; apply/capv_idPr/sub0v. Qed.

Lemma

capv0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capV", "capv_idPr", "sub0v" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capfv : left_id fullv capV.

Proof. by move=> U; apply/capv_idPr/subvf. Qed.

Lemma

capfv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capV", "capv_idPr", "fullv", "subvf" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capvf : right_id fullv capV.

Proof. by move=> U; apply/capv_idPl/subvf. Qed.

Lemma

capvf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capV", "capv_idPl", "fullv", "subvf" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_cap w U V : (w \in U :&: V)%VS = (w \in U) && (w \in V).

Proof. by rewrite !memvE subv_cap. Qed.

Lemma

memv_cap

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "memvE", "subv_cap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

memv_capP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "memv_cap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vspace_modl U 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

vspace_modl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "eqmxP", "matrix_modl", "vs2mxD", "vs2mxI", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vspace_modr U V W : (W <= U -> (U :&: V) + W = U :&: (V + W))%VS.

Proof. by rewrite -!(addvC W) !(capvC U); apply: vspace_modl. Qed.

Lemma

vspace_modr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addvC", "apply", "capvC", "vspace_modl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigcapv_inf i0 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

bigcapv_inf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Pi0", "apply", "bigD1", "capvSl", "i0", "subv_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

subv_bigcapP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "big_rec", "bigcapv_inf", "subv_cap", "subv_trans", "subvf" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_complf U : (U + U^C)%VS = fullv.

Proof. apply/vs2mxP; rewrite vs2mxD -gen_vs2mx -genmx_adds !genmxE submx1 sub1mx. exact: addsmx_compl_full. Qed.

Lemma

addv_complf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addsmx_compl_full", "apply", "fullv", "gen_vs2mx", "genmxE", "genmx_adds", "sub1mx", "submx1", "vs2mxD", "vs2mxP" ]

Complement

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capv_compl U : (U :&: U^C = 0)%VS.

Proof. apply/val_inj; rewrite [val]/= vs2mx0 vs2mxI -gen_vs2mx -genmx_cap. by rewrite capmx_compl genmx0. Qed.

Lemma

capv_compl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capmx_compl", "gen_vs2mx", "genmx0", "genmx_cap", "val", "val_inj", "vs2mx0", "vs2mxI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffvSl U V : (U :\: V <= U)%VS.

Proof. by rewrite /subV genmxE diffmxSl. Qed.

Lemma

diffvSl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "diffmxSl", "genmxE", "subV" ]

Difference

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capv_diff U 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

capv_diff

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "capmx_diff", "gen_vs2mx", "genmx0", "genmx_cap", "val", "val_inj", "vs2mx0", "vs2mxI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_diff_cap U V : (U :\: V + U :&: V)%VS = U.

Proof. apply/vs2mxP; rewrite vs2mxD -genmx_adds !genmxE. exact/eqmxP/addsmx_diff_cap_eq. Qed.

Lemma

addv_diff_cap

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addsmx_diff_cap_eq", "apply", "eqmxP", "genmxE", "genmx_adds", "vs2mxD", "vs2mxP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_diff U V : (U :\: V + V = U + V)%VS.

Proof. by rewrite -{2}(addv_diff_cap U V) -addvA (addv_idPr (capvSr U V)). Qed.

Lemma

addv_diff

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addvA", "addv_diff_cap", "addv_idPr", "capvSr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv0 : \dim (0%VS : {vspace vT}) = 0.

Proof. by rewrite /dimv vs2mx0 mxrank0. Qed.

Lemma

dimv0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "dimv", "mxrank0", "vT", "vs2mx0" ]

Subspace dimension.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_eq0 U : (\dim U == 0) = (U == 0%VS).

Proof. by rewrite /dimv /= mxrank_eq0 [in RHS]/eq_op /= linear0 genmx0. Qed.

Lemma

dimv_eq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "dimv", "genmx0", "linear0", "mxrank_eq0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimvf : \dim {:vT} = dim vT.

Proof. by rewrite /dimv vs2mxF mxrank1. Qed.

Lemma

dimvf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "dimv", "mxrank1", "vT", "vs2mxF" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim_vline v : \dim <[v]> = (v != 0).

Proof. by rewrite /dimv mxrank_gen rank_rV (can2_eq v2rK r2vK) linear0. Qed.

Lemma

dim_vline

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "can2_eq", "dim", "dimv", "linear0", "mxrank_gen", "r2vK", "rank_rV", "v2rK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimvS U V : (U <= V)%VS -> \dim U <= \dim V.

Proof. exact: mxrankS. Qed.

Lemma

dimvS

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "mxrankS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_leqif_sup U V : (U <= V)%VS -> \dim U <= \dim V ?= iff (V <= U)%VS.

Proof. exact: mxrank_leqif_sup. Qed.

Lemma

dimv_leqif_sup

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "mxrank_leqif_sup" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_leqif_eq U V : (U <= V)%VS -> \dim U <= \dim V ?= iff (U == V).

Proof. by rewrite eqEsubv; apply: mxrank_leqif_eq. Qed.

Lemma

dimv_leqif_eq

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "dim", "eqEsubv", "mxrank_leqif_eq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqEdim U V : (U == V) = (U <= V)%VS && (\dim V <= \dim U).

Proof. by apply/idP/andP=> [/eqP | [/dimv_leqif_eq/geq_leqif]] ->. Qed.

Lemma

eqEdim

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "dim", "dimv_leqif_eq", "geq_leqif" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_compl U : \dim U^C = (\dim {:vT} - \dim U)%N.

Proof. by rewrite dimvf /dimv mxrank_gen mxrank_compl. Qed.

Lemma

dimv_compl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "dimv", "dimvf", "mxrank_compl", "mxrank_gen", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_cap_compl U V : (\dim (U :&: V) + \dim (U :\: V))%N = \dim U.

Proof. by rewrite /dimv !mxrank_gen mxrank_cap_compl. Qed.

Lemma

dimv_cap_compl

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "dimv", "mxrank_cap_compl", "mxrank_gen" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_sum_cap U V : (\dim (U + V) + \dim (U :&: V) = \dim U + \dim V)%N.

Proof. by rewrite /dimv !mxrank_gen mxrank_sum_cap. Qed.

Lemma

dimv_sum_cap

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "dimv", "mxrank_gen", "mxrank_sum_cap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_disjoint_sum U 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

dimv_disjoint_sum

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addn0", "dim", "dimv0", "dimv_sum_cap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_add_leqif U 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

dimv_add_leqif

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "dim", "dimv", "genmxE", "mxrank_adds_leqif", "mxrank_gen", "subV", "vs2mx0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffv_eq0 U 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

diffv_eq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addn0", "capvSl", "capv_idPl", "dim", "dimv_cap_compl", "dimv_eq0", "dimv_leqif_eq", "eq_sym", "eqn_add2l" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_leq_sum I 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

dimv_leq_sum

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "big_rec2", "dim", "dim_vline", "dimv_add_leqif", "eqxx", "leq_add2l", "leq_trans", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_expr

:= Sumv { addv_val :> wrapped {vspace vT}; addv_dim : wrapped nat; _ : mxsum_spec (vs2mx (unwrap addv_val)) (unwrap addv_dim) }.

Structure

addv_expr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "mxsum_spec", "nat", "vT", "vs2mx", "wrapped" ]

nevetheless reuses much of the matrix theory.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mx_sum_expr_subproof (S : addv_expr) : mxsum_spec (vs2mx (unwrap S)) (unwrap (addv_dim S)).

Proof. by case: S. Qed.

Definition

vs2mx_sum_expr_subproof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_expr", "mxsum_spec", "vs2mx" ]

Piggyback on mxalgebra theory.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mx_sum_expr S

:= ProperMxsumExpr (vs2mx_sum_expr_subproof S).

Canonical

vs2mx_sum_expr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vs2mx_sum_expr_subproof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivial_addv U

:= @Sumv (Wrap U) (Wrap (\dim U)) (TrivialMxsum _).

Canonical

trivial_addv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_addv_expr

:= ProperSumvExpr { proper_addv_val :> {vspace vT}; proper_addv_dim :> nat; _ : mxsum_spec (vs2mx proper_addv_val) proper_addv_dim }.

Structure

proper_addv_expr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "mxsum_spec", "nat", "vT", "vs2mx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_addvP (S : proper_addv_expr)

:= let: ProperSumvExpr _ _ termS := S return mxsum_spec (vs2mx S) S in termS.

Definition

proper_addvP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "mxsum_spec", "proper_addv_expr", "vs2mx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_addv (S : proper_addv_expr)

:= @Sumv (wrap (S : {vspace vT})) (wrap (S : nat)) (proper_addvP S).

Canonical

proper_addv

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "nat", "proper_addvP", "proper_addv_expr", "vT", "wrap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

binary_addv_subproof : mxsum_spec (vs2mx (unwrap S1 + unwrap S2)) (unwrap (addv_dim S1) + unwrap (addv_dim S2)).

Proof. by rewrite vs2mxD; apply: proper_mxsumP. Qed.

Fact

binary_addv_subproof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "S1", "S2", "apply", "mxsum_spec", "proper_mxsumP", "vs2mx", "vs2mxD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

binary_addv_expr

:= ProperSumvExpr binary_addv_subproof.

Canonical

binary_addv_expr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "binary_addv_subproof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nary_addv_subproof : mxsum_spec (vs2mx (\sum_(i <- r | P i) unwrap (S_ i))) (\sum_(i <- r | P i) unwrap (addv_dim (S_ i))).

Proof. by rewrite vs2mx_sum; apply: proper_mxsumP. Qed.

Fact

nary_addv_subproof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "mxsum_spec", "proper_mxsumP", "vs2mx", "vs2mx_sum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nary_addv_expr

:= ProperSumvExpr nary_addv_subproof.

Canonical

nary_addv_expr

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "nary_addv_subproof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d