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
mults_linear_r (t : 'T[R]_(u1_, d1_)) : GRing.linear_for :%R (fun u : 'T[R]_(u2_, d2_) => t t u).	Proof. move=> a x y; apply/val_inj/matrixP=> i j; rewrite /tensor_val/= !mxE/=. by rewrite mulrDr; congr (_ + _); rewrite mulrA (mulrCA a) mulrA. Qed.	Let	mults_linear_r	algebra	algebra/tensor.v	[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]	[ "apply", "linear_for", "matrixP", "mulrA", "mulrCA", "mulrDr", "mxE", "tensor_val", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mults_hmul (t1 u1 : 'T[R]_(u1_, d1_)) (t2 u2 : 'T[R]_(u2_, d2_)) : (t1 h u1) t (t2 h u2) = (t1 t t2) h (u1 t u2).	Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulrACA. Qed.	Lemma	mults_hmul	algebra	algebra/tensor.v	[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]	[ "apply", "matrixP", "mulrACA", "mxE", "tensor_val", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
multsNl {R : pzRingType} (t : 'T[R]_(u1_, d1_)) (u : 'T[R]_(u2_, d2_)) : (- t) t u = - (t t u).	Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulNr. Qed.	Lemma	multsNl	algebra	algebra/tensor.v	[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]	[ "apply", "matrixP", "mulNr", "mxE", "tensor_val", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
multsNr {R : pzRingType} (t : 'T[R]_(u1_, d1_)) (u : 'T[R]_(u2_, d2_)) : t t (- u) = - (t t u).	Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulrN. Qed.	Lemma	multsNr	algebra	algebra/tensor.v	[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]	[ "apply", "matrixP", "mulrN", "mxE", "tensor_val", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mults_scale {R : comPzSemiRingType} (a b : R) (t : 'T[R]_(u1_, d1_)) (u : 'T[R]_(u2_, d2_)) : (const_t a h t) t (const_t b h u) = const_t (a b)%R h (t t u).	Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulrACA. Qed.	Lemma	mults_scale	algebra	algebra/tensor.v	[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]	[ "apply", "const_t", "matrixP", "mulrACA", "mxE", "tensor_val", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mults_hmul_compat {R : comPzSemiRingType} (t1 t2 : 'T[R]_(u1_, d1_)) (u1 u2 : 'T[R]_(u2_, d2_)) : (t1 h t2) t (u1 h u2) = (t1 t u1) h (t2 t u2).	Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulrACA. Qed.	Lemma	mults_hmul_compat	algebra	algebra/tensor.v	[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]	[ "apply", "matrixP", "mulrACA", "mxE", "tensor_val", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semivector_axiom_def (R : nzSemiRingType) n (V : lSemiModType R)	:= { v2r : V -> 'rV[R]_n; v2r_semilinear : semilinear v2r; v2r_bijective : bijective v2r }.	Record	semivector_axiom_def	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" ]	[ "semilinear", "v2r" ]	Finite dimension vector space	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vector_axiom_def (R : nzRingType) n (V : lmodType R)	:= {v2r : V -> 'rV[R]_n \| linear v2r & bijective v2r}.	Definition	vector_axiom_def	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" ]	[ "linear", "v2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vector_axiom	:= Vector.axiom.	Notation	vector_axiom	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" ]	[ "axiom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
v2r	:= sval vector_subdef.	Definition	v2r	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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim : semiVectType >-> nat.		Coercion	dim	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
space (K : fieldType) (vT : vectType K)	:= Space (mx : 'M[K]_vT) & <<mx>>%MS == mx.	Inductive	space	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" ]	[ "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hom (R : nzSemiRingType) (vT wT : semiVectType R)	:= Hom of 'M[R]_(vT, wT).	Inductive	hom	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" ]	[ "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'vspace' vT }"	:= (space vT) : type_scope.	Notation	{ 'vspace' vT }	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" ]	[ "space", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Hom' ( aT , rT )"	:= (hom aT rT) : type_scope.	Notation	''Hom' ( aT , rT )	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" ]	[ "aT", "hom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''End' ( vT )"	:= (hom vT vT) : type_scope.	Notation	''End' ( vT )	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" ]	[ "hom", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
v2r_subproof : SemiVector.axiom vT vT.	Proof. exact: vector_subdef. Qed.	Fact	v2r_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" ]	[ "axiom", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
v2r	:= v2r v2r_subproof.	Definition	v2r	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" ]	[ "v2r_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
v2r_bij : bijective v2r	:= v2r_bijective v2r_subproof.	Let	v2r_bij	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" ]	[ "v2r", "v2r_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r2v_subproof : {r2v \| cancel r2v v2r}.	Proof. have r2vP r: {v \| v2r v = r}. by apply: sig_eqW; have [v _ vK] := v2r_bij; exists (v r). by exists (fun r => sval (r2vP r)) => r; case: (r2vP r). Qed.	Fact	r2v_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", "r2v", "sig_eqW", "v2r", "v2r_bij" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r2v	:= sval r2v_subproof.	Definition	r2v	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" ]	[ "r2v_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r2vK : cancel r2v v2r.	Proof. exact: svalP r2v_subproof. Qed.	Lemma	r2vK	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" ]	[ "r2v", "r2v_subproof", "v2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r2v_inj : injective r2v.	Proof. exact: can_inj r2vK. Qed.	Lemma	r2v_inj	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" ]	[ "r2v", "r2vK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
v2rK : cancel v2r r2v.	Proof. by have/bij_can_sym:= r2vK; apply. Qed.	Lemma	v2rK	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", "r2v", "r2vK", "v2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
v2r_inj : injective v2r.	Proof. exact: can_inj v2rK. Qed.	Lemma	v2r_inj	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" ]	[ "v2r", "v2rK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
b2mx n (X : n.-tuple vT)	:= \matrix_i v2r (tnth X i).	Definition	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" ]	[ "tnth", "tuple", "v2r", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
b2mxK n (X : n.-tuple vT) i : r2v (row i (b2mx X)) = X`_i.	Proof. by rewrite rowK v2rK -tnth_nth. Qed.	Lemma	b2mxK	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", "r2v", "row", "rowK", "tnth_nth", "tuple", "v2rK", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mx (U : @space K vT)	:= let: Space mx _ := U in mx.	Definition	vs2mx	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" ]	[ "space", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gen_vs2mx (U : {vspace vT}) : <<vs2mx U>>%MS = vs2mx U.	Proof. by apply/eqP; rewrite /vs2mx; case: U. Qed.	Lemma	gen_vs2mx	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", "vT", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mx2vs_subproof m (A : 'M[K]_(m, vT)) : <<(<<A>>)>>%MS == <<A>>%MS.	Proof. by rewrite genmx_id. Qed.	Fact	mx2vs_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" ]	[ "genmx_id", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mx2vs {m} A : {vspace vT}	:= Space (@mx2vs_subproof m A).	Definition	mx2vs	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" ]	[ "mx2vs_subproof", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mxK : cancel vs2mx mx2vs.	Proof. by move=> v; apply: val_inj; rewrite /= gen_vs2mx. Qed.	Lemma	vs2mxK	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", "gen_vs2mx", "mx2vs", "val_inj", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mx2vsK m (M : 'M_(m, vT)) : (vs2mx (mx2vs M) :=: M)%MS.	Proof. exact: genmxE. Qed.	Lemma	mx2vsK	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" ]	[ "genmxE", "mx2vs", "vT", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
f2mx (f : 'Hom(aT, rT))	:= let: Hom A := f in A.	Definition	f2mx	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" ]	[ "aT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv U	:= \rank (vs2mx U).	Definition	dimv	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" ]	[ "rank", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetv U V	:= (vs2mx U <= vs2mx V)%MS.	Definition	subsetv	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vline u	:= mx2vs (v2r u).	Definition	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" ]	[ "mx2vs", "v2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pred_of_vspace (U : space vT) : {pred vT}	:= fun v => (vs2mx (vline v) <= vs2mx U)%MS.	Definition	pred_of_vspace	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" ]	[ "space", "vT", "vline", "vs2mx" ]	Vspace membership is defined as line inclusion.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vspace_predType	:= @PredType _ (unkeyed {vspace vT}) pred_of_vspace.	Canonical	vspace_predType	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" ]	[ "pred_of_vspace", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fullv : {vspace vT}	:= mx2vs 1%:M.	Definition	fullv	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" ]	[ "mx2vs", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv U V	:= mx2vs (vs2mx U + vs2mx V).	Definition	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" ]	[ "mx2vs", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capv U V	:= mx2vs (vs2mx U :&: vs2mx V).	Definition	capv	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" ]	[ "mx2vs", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complv U	:= mx2vs (vs2mx U)^C.	Definition	complv	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" ]	[ "mx2vs", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diffv U V	:= mx2vs (vs2mx U :\: vs2mx V).	Definition	diffv	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" ]	[ "mx2vs", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vpick U	:= r2v (nz_row (vs2mx U)).	Definition	vpick	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" ]	[ "nz_row", "r2v", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_key : unit.	Proof. by []. Qed.	Fact	span_key	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" ]	[ "unit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_expanded_def X	:= mx2vs (b2mx (in_tuple X)).	Definition	span_expanded_def	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", "in_tuple", "mx2vs" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span	:= locked_with span_key span_expanded_def.	Definition	span	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" ]	[ "span_expanded_def", "span_key" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_unlockable	:= [unlockable fun span].	Canonical	span_unlockable	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" ]	[ "span" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vbasis_def U	:= [tuple r2v (row i (row_base (vs2mx U))) \| i < dimv U].	Definition	vbasis_def	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" ]	[ "dimv", "r2v", "row", "row_base", "tuple", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vbasis	:= locked_with span_key vbasis_def.	Definition	vbasis	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" ]	[ "span_key", "vbasis_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vbasis_unlockable	:= [unlockable fun vbasis].	Canonical	vbasis_unlockable	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" ]	[ "vbasis" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
free X	:= dimv (span X) == size X.	Definition	free	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" ]	[ "dimv", "size", "span" ]	coord and directv are defined in the VectorTheory section.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
basis_of U X	:= (span X == U) && free X.	Definition	basis_of	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" ]	[ "free", "span" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pred_of_vspace : space >-> pred_sort.		Coercion	pred_of_vspace	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" ]	[ "space" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\dim U"	:= (dimv U) : nat_scope.	Notation	\dim U	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" ]	[ "dimv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"U <= V"	:= (subsetv U V) : vspace_scope.	Notation	U <= V	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" ]	[ "subsetv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"U <= V <= W"	:= (subsetv U V && subsetv V W) : vspace_scope.	Notation	U <= V <= W	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" ]	[ "subsetv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<[ v ] >"	:= (vline v) : vspace_scope.	Notation	<[ v ] >	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" ]	[ "vline" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< X >>"	:= (span X) : vspace_scope.	Notation	<< X >>	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" ]	[ "span" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"0"	:= (vline 0) : vspace_scope.	Notation	0	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" ]	[ "vline" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"U + V"	:= (addv U V) : vspace_scope.	Notation	U + V	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"U :&: V"	:= (capv U V) : vspace_scope.	Notation	U :&: V	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" ]	[ "capv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"U ^C"	:= (complv U) : vspace_scope.	Notation	U ^C	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" ]	[ "complv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"U :\: V"	:= (diffv U V) : vspace_scope.	Notation	U :\: V	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" ]	[ "diffv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ : vT }"	:= (@fullv _ vT) (only parsing) : vspace_scope.	Notation	{ : vT }	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", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i <- r \| P ) U"	:= (\big[addv/0%VS]_(i <- r \| P%B) U%VS) : vspace_scope.	Notation	\sum_ ( i <- r \| P ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i <- r ) U"	:= (\big[addv/0%VS]_(i <- r) U%VS) : vspace_scope.	Notation	\sum_ ( i <- r ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( m <= i < n \| P ) U"	:= (\big[addv/0%VS]_(m <= i < n \| P%B) U%VS) : vspace_scope.	Notation	\sum_ ( m <= i < n \| P ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( m <= i < n ) U"	:= (\big[addv/0%VS]_(m <= i < n) U%VS) : vspace_scope.	Notation	\sum_ ( m <= i < n ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i \| P ) U"	:= (\big[addv/0%VS]_(i \| P%B) U%VS) : vspace_scope.	Notation	\sum_ ( i \| P ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ i U"	:= (\big[addv/0%VS]_i U%VS) : vspace_scope.	Notation	\sum_ i U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i : t \| P ) U"	:= (\big[addv/0%VS]_(i : t \| P%B) U%VS) (only parsing) : vspace_scope.	Notation	\sum_ ( i : t \| P ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i : t ) U"	:= (\big[addv/0%VS]_(i : t) U%VS) (only parsing) : vspace_scope.	Notation	\sum_ ( i : t ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i < n \| P ) U"	:= (\big[addv/0%VS]_(i < n \| P%B) U%VS) : vspace_scope.	Notation	\sum_ ( i < n \| P ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i < n ) U"	:= (\big[addv/0%VS]_(i < n) U%VS) : vspace_scope.	Notation	\sum_ ( i < n ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i 'in' A \| P ) U"	:= (\big[addv/0%VS]_(i in A \| P%B) U%VS) : vspace_scope.	Notation	\sum_ ( i 'in' A \| P ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i 'in' A ) U"	:= (\big[addv/0%VS]_(i in A) U%VS) : vspace_scope.	Notation	\sum_ ( i 'in' A ) U	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i <- r \| P ) U"	:= (\big[capv/fullv]_(i <- r \| P%B) U%VS) : vspace_scope.	Notation	\bigcap_ ( i <- r \| P ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i <- r ) U"	:= (\big[capv/fullv]_(i <- r) U%VS) : vspace_scope.	Notation	\bigcap_ ( i <- r ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( m <= i < n \| P ) U"	:= (\big[capv/fullv]_(m <= i < n \| P%B) U%VS) : vspace_scope.	Notation	\bigcap_ ( m <= i < n \| P ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( m <= i < n ) U"	:= (\big[capv/fullv]_(m <= i < n) U%VS) : vspace_scope.	Notation	\bigcap_ ( m <= i < n ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i \| P ) U"	:= (\big[capv/fullv]_(i \| P%B) U%VS) : vspace_scope.	Notation	\bigcap_ ( i \| P ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ i U"	:= (\big[capv/fullv]_i U%VS) : vspace_scope.	Notation	\bigcap_ i U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i : t \| P ) U"	:= (\big[capv/fullv]_(i : t \| P%B) U%VS) (only parsing) : vspace_scope.	Notation	\bigcap_ ( i : t \| P ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i : t ) U"	:= (\big[capv/fullv]_(i : t) U%VS) (only parsing) : vspace_scope.	Notation	\bigcap_ ( i : t ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i < n \| P ) U"	:= (\big[capv/fullv]_(i < n \| P%B) U%VS) : vspace_scope.	Notation	\bigcap_ ( i < n \| P ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i < n ) U"	:= (\big[capv/fullv]_(i < n) U%VS) : vspace_scope.	Notation	\bigcap_ ( i < n ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i 'in' A \| P ) U"	:= (\big[capv/fullv]_(i in A \| P%B) U%VS) : vspace_scope.	Notation	\bigcap_ ( i 'in' A \| P ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\bigcap_ ( i 'in' A ) U"	:= (\big[capv/fullv]_(i in A) U%VS) : vspace_scope.	Notation	\bigcap_ ( i 'in' A ) U	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" ]	[ "capv", "fullv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subV	:= (@subsetv K vT) (only parsing).	Notation	subV	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" ]	[ "subsetv", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addV	:= (@addv K vT) (only parsing).	Notation	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" ]	[ "addv", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capV	:= (@capv K vT) (only parsing).	Notation	capV	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" ]	[ "capv", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mxP U V : reflect (U = V) (vs2mx U == vs2mx V)%MS.	Proof. by rewrite (sameP genmxP eqP) !gen_vs2mx; apply: eqP. Qed.	Let	vs2mxP	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", "gen_vs2mx", "genmxP", "vs2mx" ]	Internal theory facts	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvK v U : (v \in U) = (v2r v <= vs2mx U)%MS.	Proof. by rewrite -genmxE. Qed.	Let	memvK	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" ]	[ "genmxE", "v2r", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_r2v rv U : (r2v rv \in U) = (rv <= vs2mx U)%MS.	Proof. by rewrite memvK r2vK. Qed.	Let	mem_r2v	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" ]	[ "memvK", "r2v", "r2vK", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mx0 : @vs2mx K vT 0 = 0.	Proof. by rewrite /= linear0 genmx0. Qed.	Let	vs2mx0	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" ]	[ "genmx0", "linear0", "vT", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mxD U V : vs2mx (U + V) = (vs2mx U + vs2mx V)%MS.	Proof. by rewrite /= genmx_adds !gen_vs2mx. Qed.	Let	vs2mxD	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" ]	[ "gen_vs2mx", "genmx_adds", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mx_sum	:= big_morph _ vs2mxD vs2mx0.	Let	vs2mx_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" ]	[ "big_morph", "vs2mx0", "vs2mxD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vs2mxI U V : vs2mx (U :&: V) = (vs2mx U :&: vs2mx V)%MS.	Proof. by rewrite /= genmx_cap !gen_vs2mx. Qed.	Let	vs2mxI	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" ]	[ "gen_vs2mx", "genmx_cap", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

mults_linear_r (t : 'T[R]_(u1_, d1_)) : GRing.linear_for *:%R (fun u : 'T[R]_(u2_, d2_) => t *t u).

Proof. move=> a x y; apply/val_inj/matrixP=> i j; rewrite /tensor_val/= !mxE/=. by rewrite mulrDr; congr (_ + _); rewrite mulrA (mulrCA a) mulrA. Qed.

Let

mults_linear_r

algebra

algebra/tensor.v

[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]

[ "apply", "linear_for", "matrixP", "mulrA", "mulrCA", "mulrDr", "mxE", "tensor_val", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mults_hmul (t1 u1 : 'T[R]_(u1_, d1_)) (t2 u2 : 'T[R]_(u2_, d2_)) : (t1 *h u1) *t (t2 *h u2) = (t1 *t t2) *h (u1 *t u2).

Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulrACA. Qed.

Lemma

mults_hmul

algebra

algebra/tensor.v

[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]

[ "apply", "matrixP", "mulrACA", "mxE", "tensor_val", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

multsNl {R : pzRingType} (t : 'T[R]_(u1_, d1_)) (u : 'T[R]_(u2_, d2_)) : (- t) *t u = - (t *t u).

Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulNr. Qed.

Lemma

multsNl

algebra

algebra/tensor.v

[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]

[ "apply", "matrixP", "mulNr", "mxE", "tensor_val", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

multsNr {R : pzRingType} (t : 'T[R]_(u1_, d1_)) (u : 'T[R]_(u2_, d2_)) : t *t (- u) = - (t *t u).

Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulrN. Qed.

Lemma

multsNr

algebra

algebra/tensor.v

[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]

[ "apply", "matrixP", "mulrN", "mxE", "tensor_val", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mults_scale {R : comPzSemiRingType} (a b : R) (t : 'T[R]_(u1_, d1_)) (u : 'T[R]_(u2_, d2_)) : (const_t a *h t) *t (const_t b *h u) = const_t (a * b)%R *h (t *t u).

Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulrACA. Qed.

Lemma

mults_scale

algebra

algebra/tensor.v

[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]

[ "apply", "const_t", "matrixP", "mulrACA", "mxE", "tensor_val", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mults_hmul_compat {R : comPzSemiRingType} (t1 t2 : 'T[R]_(u1_, d1_)) (u1 u2 : 'T[R]_(u2_, d2_)) : (t1 *h t2) *t (u1 *h u2) = (t1 *t u1) *h (t2 *t u2).

Proof. by apply/val_inj/matrixP => i j; rewrite /tensor_val /= !mxE /= mulrACA. Qed.

Lemma

mults_hmul_compat

algebra

algebra/tensor.v

[ "HB", "structures", "mathcomp", "ssreflect", "seq", "matrix", "bigop", "ssrbool", "eqtype", "choice", "fintype", "ssralg", "ssrnat", "ssrfun", "order", "finfun", "tuple", "finset", "sesquilinear", "interval", "interval_inference", "numdomain", "GRing.Theory" ]

[ "apply", "matrixP", "mulrACA", "mxE", "tensor_val", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semivector_axiom_def (R : nzSemiRingType) n (V : lSemiModType R)

:= { v2r : V -> 'rV[R]_n; v2r_semilinear : semilinear v2r; v2r_bijective : bijective v2r }.

Record

semivector_axiom_def

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" ]

[ "semilinear", "v2r" ]

Finite dimension vector space

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vector_axiom_def (R : nzRingType) n (V : lmodType R)

:= {v2r : V -> 'rV[R]_n | linear v2r & bijective v2r}.

Definition

vector_axiom_def

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" ]

[ "linear", "v2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vector_axiom

:= Vector.axiom.

Notation

vector_axiom

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" ]

[ "axiom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

v2r

:= sval vector_subdef.

Definition

v2r

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" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim : semiVectType >-> nat.

Coercion

dim

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

space (K : fieldType) (vT : vectType K)

:= Space (mx : 'M[K]_vT) & <<mx>>%MS == mx.

Inductive

space

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" ]

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hom (R : nzSemiRingType) (vT wT : semiVectType R)

:= Hom of 'M[R]_(vT, wT).

Inductive

hom

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" ]

[ "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'vspace' vT }"

:= (space vT) : type_scope.

Notation

{ 'vspace' vT }

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" ]

[ "space", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Hom' ( aT , rT )"

:= (hom aT rT) : type_scope.

Notation

''Hom' ( aT , rT )

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" ]

[ "aT", "hom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''End' ( vT )"

:= (hom vT vT) : type_scope.

Notation

''End' ( vT )

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" ]

[ "hom", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

v2r_subproof : SemiVector.axiom vT vT.

Proof. exact: vector_subdef. Qed.

Fact

v2r_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" ]

[ "axiom", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

v2r

:= v2r v2r_subproof.

Definition

v2r

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" ]

[ "v2r_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

v2r_bij : bijective v2r

:= v2r_bijective v2r_subproof.

Let

v2r_bij

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" ]

[ "v2r", "v2r_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r2v_subproof : {r2v | cancel r2v v2r}.

Proof. have r2vP r: {v | v2r v = r}. by apply: sig_eqW; have [v _ vK] := v2r_bij; exists (v r). by exists (fun r => sval (r2vP r)) => r; case: (r2vP r). Qed.

Fact

r2v_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", "r2v", "sig_eqW", "v2r", "v2r_bij" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r2v

:= sval r2v_subproof.

Definition

r2v

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" ]

[ "r2v_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r2vK : cancel r2v v2r.

Proof. exact: svalP r2v_subproof. Qed.

Lemma

r2vK

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" ]

[ "r2v", "r2v_subproof", "v2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r2v_inj : injective r2v.

Proof. exact: can_inj r2vK. Qed.

Lemma

r2v_inj

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" ]

[ "r2v", "r2vK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

v2rK : cancel v2r r2v.

Proof. by have/bij_can_sym:= r2vK; apply. Qed.

Lemma

v2rK

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", "r2v", "r2vK", "v2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

v2r_inj : injective v2r.

Proof. exact: can_inj v2rK. Qed.

Lemma

v2r_inj

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" ]

[ "v2r", "v2rK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

b2mx n (X : n.-tuple vT)

:= \matrix_i v2r (tnth X i).

Definition

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" ]

[ "tnth", "tuple", "v2r", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

b2mxK n (X : n.-tuple vT) i : r2v (row i (b2mx X)) = X`_i.

Proof. by rewrite rowK v2rK -tnth_nth. Qed.

Lemma

b2mxK

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", "r2v", "row", "rowK", "tnth_nth", "tuple", "v2rK", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mx (U : @space K vT)

:= let: Space mx _ := U in mx.

Definition

vs2mx

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" ]

[ "space", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gen_vs2mx (U : {vspace vT}) : <<vs2mx U>>%MS = vs2mx U.

Proof. by apply/eqP; rewrite /vs2mx; case: U. Qed.

Lemma

gen_vs2mx

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", "vT", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mx2vs_subproof m (A : 'M[K]_(m, vT)) : <<(<<A>>)>>%MS == <<A>>%MS.

Proof. by rewrite genmx_id. Qed.

Fact

mx2vs_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" ]

[ "genmx_id", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mx2vs {m} A : {vspace vT}

:= Space (@mx2vs_subproof m A).

Definition

mx2vs

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" ]

[ "mx2vs_subproof", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mxK : cancel vs2mx mx2vs.

Proof. by move=> v; apply: val_inj; rewrite /= gen_vs2mx. Qed.

Lemma

vs2mxK

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", "gen_vs2mx", "mx2vs", "val_inj", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mx2vsK m (M : 'M_(m, vT)) : (vs2mx (mx2vs M) :=: M)%MS.

Proof. exact: genmxE. Qed.

Lemma

mx2vsK

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" ]

[ "genmxE", "mx2vs", "vT", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

f2mx (f : 'Hom(aT, rT))

:= let: Hom A := f in A.

Definition

f2mx

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" ]

[ "aT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv U

:= \rank (vs2mx U).

Definition

dimv

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" ]

[ "rank", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetv U V

:= (vs2mx U <= vs2mx V)%MS.

Definition

subsetv

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vline u

:= mx2vs (v2r u).

Definition

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" ]

[ "mx2vs", "v2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pred_of_vspace (U : space vT) : {pred vT}

:= fun v => (vs2mx (vline v) <= vs2mx U)%MS.

Definition

pred_of_vspace

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" ]

[ "space", "vT", "vline", "vs2mx" ]

Vspace membership is defined as line inclusion.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vspace_predType

:= @PredType _ (unkeyed {vspace vT}) pred_of_vspace.

Canonical

vspace_predType

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" ]

[ "pred_of_vspace", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fullv : {vspace vT}

:= mx2vs 1%:M.

Definition

fullv

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" ]

[ "mx2vs", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv U V

:= mx2vs (vs2mx U + vs2mx V).

Definition

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" ]

[ "mx2vs", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capv U V

:= mx2vs (vs2mx U :&: vs2mx V).

Definition

capv

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" ]

[ "mx2vs", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complv U

:= mx2vs (vs2mx U)^C.

Definition

complv

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" ]

[ "mx2vs", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diffv U V

:= mx2vs (vs2mx U :\: vs2mx V).

Definition

diffv

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" ]

[ "mx2vs", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vpick U

:= r2v (nz_row (vs2mx U)).

Definition

vpick

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" ]

[ "nz_row", "r2v", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_key : unit.

Proof. by []. Qed.

Fact

span_key

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" ]

[ "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_expanded_def X

:= mx2vs (b2mx (in_tuple X)).

Definition

span_expanded_def

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", "in_tuple", "mx2vs" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span

:= locked_with span_key span_expanded_def.

Definition

span

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" ]

[ "span_expanded_def", "span_key" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_unlockable

:= [unlockable fun span].

Canonical

span_unlockable

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" ]

[ "span" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vbasis_def U

:= [tuple r2v (row i (row_base (vs2mx U))) | i < dimv U].

Definition

vbasis_def

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" ]

[ "dimv", "r2v", "row", "row_base", "tuple", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vbasis

:= locked_with span_key vbasis_def.

Definition

vbasis

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" ]

[ "span_key", "vbasis_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vbasis_unlockable

:= [unlockable fun vbasis].

Canonical

vbasis_unlockable

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" ]

[ "vbasis" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

free X

:= dimv (span X) == size X.

Definition

free

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" ]

[ "dimv", "size", "span" ]

coord and directv are defined in the VectorTheory section.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

basis_of U X

:= (span X == U) && free X.

Definition

basis_of

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" ]

[ "free", "span" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pred_of_vspace : space >-> pred_sort.

Coercion

pred_of_vspace

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" ]

[ "space" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\dim U"

:= (dimv U) : nat_scope.

Notation

\dim U

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" ]

[ "dimv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"U <= V"

:= (subsetv U V) : vspace_scope.

Notation

U <= V

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" ]

[ "subsetv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"U <= V <= W"

:= (subsetv U V && subsetv V W) : vspace_scope.

Notation

U <= V <= W

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" ]

[ "subsetv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<[ v ] >"

:= (vline v) : vspace_scope.

Notation

<[ v ] >

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" ]

[ "vline" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<< X >>"

:= (span X) : vspace_scope.

Notation

<< X >>

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" ]

[ "span" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"0"

:= (vline 0) : vspace_scope.

Notation

0

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" ]

[ "vline" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"U + V"

:= (addv U V) : vspace_scope.

Notation

U + V

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"U :&: V"

:= (capv U V) : vspace_scope.

Notation

U :&: V

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" ]

[ "capv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"U ^C"

:= (complv U) : vspace_scope.

Notation

U ^C

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" ]

[ "complv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"U :\: V"

:= (diffv U V) : vspace_scope.

Notation

U :\: V

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" ]

[ "diffv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ : vT }"

:= (@fullv _ vT) (only parsing) : vspace_scope.

Notation

{ : vT }

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", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i <- r | P ) U"

:= (\big[addv/0%VS]_(i <- r | P%B) U%VS) : vspace_scope.

Notation

\sum_ ( i <- r | P ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i <- r ) U"

:= (\big[addv/0%VS]_(i <- r) U%VS) : vspace_scope.

Notation

\sum_ ( i <- r ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( m <= i < n | P ) U"

:= (\big[addv/0%VS]_(m <= i < n | P%B) U%VS) : vspace_scope.

Notation

\sum_ ( m <= i < n | P ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( m <= i < n ) U"

:= (\big[addv/0%VS]_(m <= i < n) U%VS) : vspace_scope.

Notation

\sum_ ( m <= i < n ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i | P ) U"

:= (\big[addv/0%VS]_(i | P%B) U%VS) : vspace_scope.

Notation

\sum_ ( i | P ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ i U"

:= (\big[addv/0%VS]_i U%VS) : vspace_scope.

Notation

\sum_ i U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i : t | P ) U"

:= (\big[addv/0%VS]_(i : t | P%B) U%VS) (only parsing) : vspace_scope.

Notation

\sum_ ( i : t | P ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i : t ) U"

:= (\big[addv/0%VS]_(i : t) U%VS) (only parsing) : vspace_scope.

Notation

\sum_ ( i : t ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i < n | P ) U"

:= (\big[addv/0%VS]_(i < n | P%B) U%VS) : vspace_scope.

Notation

\sum_ ( i < n | P ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i < n ) U"

:= (\big[addv/0%VS]_(i < n) U%VS) : vspace_scope.

Notation

\sum_ ( i < n ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i 'in' A | P ) U"

:= (\big[addv/0%VS]_(i in A | P%B) U%VS) : vspace_scope.

Notation

\sum_ ( i 'in' A | P ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i 'in' A ) U"

:= (\big[addv/0%VS]_(i in A) U%VS) : vspace_scope.

Notation

\sum_ ( i 'in' A ) U

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i <- r | P ) U"

:= (\big[capv/fullv]_(i <- r | P%B) U%VS) : vspace_scope.

Notation

\bigcap_ ( i <- r | P ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i <- r ) U"

:= (\big[capv/fullv]_(i <- r) U%VS) : vspace_scope.

Notation

\bigcap_ ( i <- r ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( m <= i < n | P ) U"

:= (\big[capv/fullv]_(m <= i < n | P%B) U%VS) : vspace_scope.

Notation

\bigcap_ ( m <= i < n | P ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( m <= i < n ) U"

:= (\big[capv/fullv]_(m <= i < n) U%VS) : vspace_scope.

Notation

\bigcap_ ( m <= i < n ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i | P ) U"

:= (\big[capv/fullv]_(i | P%B) U%VS) : vspace_scope.

Notation

\bigcap_ ( i | P ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ i U"

:= (\big[capv/fullv]_i U%VS) : vspace_scope.

Notation

\bigcap_ i U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i : t | P ) U"

:= (\big[capv/fullv]_(i : t | P%B) U%VS) (only parsing) : vspace_scope.

Notation

\bigcap_ ( i : t | P ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i : t ) U"

:= (\big[capv/fullv]_(i : t) U%VS) (only parsing) : vspace_scope.

Notation

\bigcap_ ( i : t ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i < n | P ) U"

:= (\big[capv/fullv]_(i < n | P%B) U%VS) : vspace_scope.

Notation

\bigcap_ ( i < n | P ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i < n ) U"

:= (\big[capv/fullv]_(i < n) U%VS) : vspace_scope.

Notation

\bigcap_ ( i < n ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i 'in' A | P ) U"

:= (\big[capv/fullv]_(i in A | P%B) U%VS) : vspace_scope.

Notation

\bigcap_ ( i 'in' A | P ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\bigcap_ ( i 'in' A ) U"

:= (\big[capv/fullv]_(i in A) U%VS) : vspace_scope.

Notation

\bigcap_ ( i 'in' A ) U

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" ]

[ "capv", "fullv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subV

:= (@subsetv K vT) (only parsing).

Notation

subV

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" ]

[ "subsetv", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addV

:= (@addv K vT) (only parsing).

Notation

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" ]

[ "addv", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capV

:= (@capv K vT) (only parsing).

Notation

capV

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" ]

[ "capv", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mxP U V : reflect (U = V) (vs2mx U == vs2mx V)%MS.

Proof. by rewrite (sameP genmxP eqP) !gen_vs2mx; apply: eqP. Qed.

Let

vs2mxP

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", "gen_vs2mx", "genmxP", "vs2mx" ]

Internal theory facts

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memvK v U : (v \in U) = (v2r v <= vs2mx U)%MS.

Proof. by rewrite -genmxE. Qed.

Let

memvK

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" ]

[ "genmxE", "v2r", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_r2v rv U : (r2v rv \in U) = (rv <= vs2mx U)%MS.

Proof. by rewrite memvK r2vK. Qed.

Let

mem_r2v

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" ]

[ "memvK", "r2v", "r2vK", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mx0 : @vs2mx K vT 0 = 0.

Proof. by rewrite /= linear0 genmx0. Qed.

Let

vs2mx0

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" ]

[ "genmx0", "linear0", "vT", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mxD U V : vs2mx (U + V) = (vs2mx U + vs2mx V)%MS.

Proof. by rewrite /= genmx_adds !gen_vs2mx. Qed.

Let

vs2mxD

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" ]

[ "gen_vs2mx", "genmx_adds", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mx_sum

:= big_morph _ vs2mxD vs2mx0.

Let

vs2mx_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" ]

[ "big_morph", "vs2mx0", "vs2mxD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vs2mxI U V : vs2mx (U :&: V) = (vs2mx U :&: vs2mx V)%MS.

Proof. by rewrite /= genmx_cap !gen_vs2mx. Qed.

Let

vs2mxI

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" ]

[ "gen_vs2mx", "genmx_cap", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d