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
directv_def S & phantom {vspace vT} (unwrap (addv_val S))	:= \dim (unwrap S) == unwrap (addv_dim S).	Definition	directv_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" ]	[ "dim", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv A	:= (directv_def (Phantom {vspace _} A%VS)).	Notation	directv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "directv_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directvE (S : addv_expr) : directv (unwrap S) = (\dim (unwrap S) == unwrap (addv_dim S)).	Proof. by []. Qed.	Lemma	directvE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_expr", "dim", "directv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directvP {S : proper_addv_expr} : reflect (\dim S = S :> nat) (directv S).	Proof. exact: eqnP. Qed.	Lemma	directvP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "directv", "eqnP", "nat", "proper_addv_expr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv_trivial U : directv (unwrap (@trivial_addv U)).	Proof. exact: eqxx. Qed.	Lemma	directv_trivial	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "directv", "eqxx", "trivial_addv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv_sum_leqif (S : addv_expr) : \dim (unwrap S) <= unwrap (addv_dim S) ?= iff directv (unwrap S).	Proof. rewrite directvE; case: S => [[U] [d] /= defUd]; split=> //=. rewrite /dimv; elim: {1}_ {U}_ d / defUd => // m1 m2 A1 A2 r1 r2 _ leA1 _ leA2. by apply: leq_trans (leq_add leA1 leA2); rewrite mxrank_adds_leqif. Qed.	Lemma	dimv_sum_leqif	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_expr", "apply", "dim", "dimv", "directv", "directvE", "leq_add", "leq_trans", "mxrank_adds_leqif", "r1", "r2", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directvEgeq (S : addv_expr) : directv (unwrap S) = (\dim (unwrap S) >= unwrap (addv_dim S)).	Proof. by rewrite leq_eqVlt ltnNge eq_sym !dimv_sum_leqif orbF. Qed.	Lemma	directvEgeq	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_expr", "dim", "dimv_sum_leqif", "directv", "eq_sym", "leq_eqVlt", "ltnNge" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv_addE (S1 S2 : addv_expr) : directv (unwrap S1 + unwrap S2) = [&& directv (unwrap S1), directv (unwrap S2) & unwrap S1 :&: unwrap S2 == 0]%VS.	Proof. by rewrite /directv_def /dimv vs2mxD -mxdirectE mxdirect_addsE -vs2mxI -vs2mx0. Qed.	Lemma	directv_addE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "S1", "S2", "addv_expr", "dimv", "directv", "directv_def", "mxdirectE", "mxdirect_addsE", "vs2mx0", "vs2mxD", "vs2mxI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv_addP {U V} : reflect (U :&: V = 0)%VS (directv (U + V)).	Proof. by rewrite directv_addE !directv_trivial; apply: eqP. Qed.	Lemma	directv_addP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "directv", "directv_addE", "directv_trivial" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv_add_unique {U V} : reflect (forall u1 u2 v1 v2, u1 \in U -> u2 \in U -> v1 \in V -> v2 \in V -> (u1 + v1 == u2 + v2) = ((u1, v1) == (u2, v2))) (directv (U + V)).	Proof. apply: (iffP directv_addP) => [dxUV u1 u2 v1 v2 Uu1 Uu2 Vv1 Vv2 \| dxUV]. apply/idP/idP=> [\| /eqP[-> ->] //]; rewrite -subr_eq0 opprD addrACA addr_eq0. move/eqP=> eq_uv; rewrite xpair_eqE -subr_eq0 eq_uv oppr_eq0 subr_eq0 andbb. by rewrite -subr_eq0 -memv0 -dxUV memv_cap -memvN -eq_uv !memvB. apply/eqP; rew...	Lemma	directv_add_unique	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addrACA", "addrC", "addr_eq0", "apply", "directv", "directv_addP", "eq_sym", "mem0v", "memv0", "memvB", "memvN", "memv_cap", "memv_capP", "opprD", "oppr_eq0", "subr_eq0", "subv0", "subvP", "xpair_eqE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv_sumP {Us : I -> {vspace vT}} : reflect (forall i, P i -> Us i :&: (\sum_(j \| P j && (j != i)) Us j) = 0)%VS (directv (\sum_(i \| P i) Us i)).	Proof. rewrite directvE /= /dimv vs2mx_sum -mxdirectE; apply: (equivP mxdirect_sumsP). by do [split=> dxU i /dxU; rewrite -vs2mx_sum -vs2mxI -vs2mx0] => [/val_inj\|->]. Qed.	Lemma	directv_sumP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "dimv", "directv", "directvE", "mxdirectE", "mxdirect_sumsP", "split", "vT", "val_inj", "vs2mx0", "vs2mxI", "vs2mx_sum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv_sumE {Ss : I -> addv_expr} (xunwrap := unwrap) : reflect [/\ forall i, P i -> directv (unwrap (Ss i)) & directv (\sum_(i \| P i) xunwrap (Ss i))] (directv (\sum_(i \| P i) unwrap (Ss i))).	Proof. by rewrite !directvE /= /dimv 2!{1}vs2mx_sum -!mxdirectE; apply: mxdirect_sumsE. Qed.	Lemma	directv_sumE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_expr", "apply", "dimv", "directv", "directvE", "mxdirectE", "mxdirect_sumsE", "vs2mx_sum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv_sum_independent {Us : I -> {vspace vT}} : reflect (forall us, (forall i, P i -> us i \in Us i) -> \sum_(i \| P i) us i = 0 -> (forall i, P i -> us i = 0)) (directv (\sum_(i \| P i) Us i)).	Proof. apply: (iffP directv_sumP) => [dxU us Uu u_0 i Pi \| dxU i Pi]. apply/eqP; rewrite -memv0 -(dxU i Pi) memv_cap Uu //= -memvN -sub0r -{1}u_0. by rewrite (bigD1 i) //= [_ - us i]addrC addKr memv_sumr // => j /andP[/Uu]. apply/eqP; rewrite -subv0; apply/subvP=> v. rewrite memv_cap memv0 => /andP[Uiv /memv_sumP[u...	Lemma	directv_sum_independent	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Uu", "addKr", "addrC", "apply", "big1", "bigD1", "directv", "directv_sumP", "eqVneq", "eqxx", "memv0", "memvN", "memv_cap", "memv_sumP", "memv_sumr", "oppr_eq0", "sub0r", "subrr", "subv0", "subvP", "sumrB", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv_sum_unique {Us : I -> {vspace vT}} : reflect (forall us vs, (forall i, P i -> us i \in Us i) -> (forall i, P i -> vs i \in Us i) -> (\sum_(i \| P i) us i == \sum_(i \| P i) vs i) = [forall (i \| P i), us i == vs i]) (directv (\sum_(i \| P i) Us i)).	Proof. apply: (iffP directv_sum_independent) => [dxU us vs Uu Uv \| dxU us Uu u_0 i Pi]. apply/idP/forall_inP=> [\|eq_uv]; last by apply/eqP/eq_bigr => i /eq_uv/eqP. rewrite -subr_eq0 -sumrB => /eqP/dxU eq_uv i Pi. by rewrite -subr_eq0 eq_uv // => j Pj; apply: memvB; move: j Pj. apply/eqP; have:= esym (dxU us \0 Uu...	Lemma	directv_sum_unique	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Uu", "apply", "big1_eq", "directv", "directv_sum_independent", "eq_bigr", "eqxx", "forall_inP", "last", "mem0v", "memvB", "subr_eq0", "sumrB", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_span X v : v \in X -> v \in <<X>>%VS.	Proof. by case/seq_tnthP=> i {v}->; rewrite unlock memvK genmxE (eq_row_sub i) // rowK. Qed.	Lemma	memv_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" ]	[ "eq_row_sub", "genmxE", "memvK", "rowK", "seq_tnthP" ]	Linear span generated by a list of vectors	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_span1 v : v \in <<[:: v]>>%VS.	Proof. by rewrite memv_span ?mem_head. Qed.	Lemma	memv_span1	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "mem_head", "memv_span" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_span X : \dim <<X>> <= size X.	Proof. by rewrite unlock /dimv genmxE rank_leq_row. Qed.	Lemma	dim_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" ]	[ "dim", "dimv", "genmxE", "rank_leq_row", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_subvP {X U} : reflect {subset X <= U} (<<X>> <= U)%VS.	Proof. rewrite /subV [@span _ _]unlock genmxE. apply: (iffP row_subP) => /= [sXU \| sXU i]. by move=> _ /seq_tnthP[i ->]; have:= sXU i; rewrite rowK memvK. by rewrite rowK -memvK sXU ?mem_tnth. Qed.	Lemma	span_subvP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "genmxE", "mem_tnth", "memvK", "rowK", "row_subP", "seq_tnthP", "span", "subV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_span X Y : {subset X <= Y} -> (<<X>> <= <<Y>>)%VS.	Proof. by move=> sXY; apply/span_subvP=> v /sXY/memv_span. Qed.	Lemma	sub_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" ]	[ "apply", "memv_span", "span_subvP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_span X Y : X =i Y -> (<<X>> = <<Y>>)%VS.	Proof. by move=> eqXY; apply: subv_anti; rewrite !sub_span // => u; rewrite eqXY. Qed.	Lemma	eq_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" ]	[ "apply", "sub_span", "subv_anti" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_def X : span X = (\sum_(u <- X) <[u]>)%VS.	Proof. apply/subv_anti/andP; split. by apply/span_subvP=> v Xv; rewrite (big_rem v) // memvE addvSl. by rewrite big_tnth; apply/subv_sumP=> i _; rewrite -memvE memv_span ?mem_tnth. Qed.	Lemma	span_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" ]	[ "addvSl", "apply", "big_rem", "big_tnth", "mem_tnth", "memvE", "memv_span", "span", "span_subvP", "split", "subv_anti", "subv_sumP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_nil : (<<Nil vT>> = 0)%VS.	Proof. by rewrite span_def big_nil. Qed.	Lemma	span_nil	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Nil", "big_nil", "span_def", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_seq1 v : (<<[:: v]>> = <[v]>)%VS.	Proof. by rewrite span_def big_seq1. Qed.	Lemma	span_seq1	algebra	algebra/vector.v	[ "HB", "structures", "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_seq1", "span_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_cons v X : (<<v :: X>> = <[v]> + <<X>>)%VS.	Proof. by rewrite !span_def big_cons. Qed.	Lemma	span_cons	algebra	algebra/vector.v	[ "HB", "structures", "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_cons", "span_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_cat X Y : (<<X ++ Y>> = <<X>> + <<Y>>)%VS.	Proof. by rewrite !span_def big_cat. Qed.	Lemma	span_cat	algebra	algebra/vector.v	[ "HB", "structures", "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_cat", "span_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_expanded_def n (X : n.-tuple vT) i v	:= (v2r v *m pinvmx (b2mx X)) 0 i.	Definition	coord_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", "pinvmx", "tuple", "v2r", "vT" ]	Coordinates function; should perhaps be generalized to nat indices.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord	:= locked_with span_key coord_expanded_def.	Definition	coord	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "coord_expanded_def", "span_key" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_unlockable	:= [unlockable fun coord].	Canonical	coord_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" ]	[ "coord" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_is_scalar n (X : n.-tuple vT) i : scalar (coord X i).	Proof. by move=> k u v; rewrite unlock linearP mulmxDl -scalemxAl !mxE. Qed.	Fact	coord_is_scalar	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "coord", "linearP", "mulmxDl", "mxE", "scalar", "scalemxAl", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_span n (X : n.-tuple vT) v : v \in span X -> v = \sum_i coord X i v *: X`_i.	Proof. rewrite memvK span_b2mx genmxE => Xv. by rewrite unlock_with mul_b2mx mulmxKpV ?v2rK. Qed.	Lemma	coord_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" ]	[ "coord", "genmxE", "memvK", "mul_b2mx", "mulmxKpV", "span", "span_b2mx", "tuple", "v2rK", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord0 i v : coord [tuple 0] i v = 0.	Proof. rewrite unlock /pinvmx rank_rV; case: negP => [[] \| _]. by apply/eqP/rowP=> j; rewrite !mxE (tnth_nth 0) /= linear0 mxE. by rewrite pid_mx_0 !(mulmx0, mul0mx) mxE. Qed.	Lemma	coord0	algebra	algebra/vector.v	[ "HB", "structures", "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", "coord", "linear0", "mul0mx", "mulmx0", "mxE", "pid_mx_0", "pinvmx", "rank_rV", "rowP", "tnth_nth", "tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_free : free (Nil vT).	Proof. by rewrite /free span_nil dimv0. Qed.	Lemma	nil_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" ]	[ "Nil", "dimv0", "free", "span_nil", "vT" ]	Free generator sequences.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq1_free v : free [:: v] = (v != 0).	Proof. by rewrite /free span_seq1 dim_vline; case: (~~ _). Qed.	Lemma	seq1_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" ]	[ "dim_vline", "free", "span_seq1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_free X Y : perm_eq X Y -> free X = free Y.	Proof. by move=> eqXY; rewrite /free (perm_size eqXY) (eq_span (perm_mem eqXY)). Qed.	Lemma	perm_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" ]	[ "eq_span", "free", "perm_eq", "perm_mem", "perm_size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
free_directv X : free X = (0 \notin X) && directv (\sum_(v <- X) <[v]>).	Proof. have leXi i (v := tnth (in_tuple X) i): true -> \dim <[v]> <= 1 ?= iff (v != 0). by rewrite -seq1_free -span_seq1 => _; apply/leqif_eq/dim_span. have [_ /=] := leqif_trans (dimv_sum_leqif _) (leqif_sum leXi). rewrite sum1_card card_ord !directvE /= /free andbC span_def !(big_tnth _ _ X). by congr (_ = _ && _);...	Lemma	free_directv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "all_predC", "apply", "big_all", "big_andE", "big_tnth", "card_ord", "dim", "dim_span", "dimv_sum_leqif", "directv", "directvE", "free", "has_pred1", "in_tuple", "leqif_eq", "leqif_sum", "leqif_trans", "seq1_free", "span_def", "span_seq1", "sum1_card", "tnth" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
free_not0 v X : free X -> v \in X -> v != 0.	Proof. by rewrite free_directv andbC => /andP[_ /memPn]; apply. Qed.	Lemma	free_not0	algebra	algebra/vector.v	[ "HB", "structures", "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", "free", "free_directv", "memPn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
freeP n (X : n.-tuple vT) : reflect (forall k, \sum_(i < n) k i *: X`_i = 0 -> (forall i, k i = 0)) (free X).	Proof. rewrite free_b2mx; apply: (iffP idP) => [t_free k kt0 i \| t_free]. suffices /rowP/(_ i): \row_i k i = 0 by rewrite !mxE. by apply/(row_free_inj t_free)/r2v_inj; rewrite mul0mx -lin_b2mx kt0 linear0. rewrite -kermx_eq0; apply/rowV0P=> rk /sub_kermxP kt0. by apply/rowP=> i; rewrite mxE {}t_free // mul_b2mx kt0...	Lemma	freeP	algebra	algebra/vector.v	[ "HB", "structures", "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", "free", "free_b2mx", "kermx_eq0", "lin_b2mx", "linear0", "mul0mx", "mul_b2mx", "mxE", "r2v_inj", "rowP", "rowV0P", "row_free_inj", "sub_kermxP", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_free n (X : n.-tuple vT) (i j : 'I_n) : free X -> coord X j (X`_i) = (i == j)%:R.	Proof. rewrite unlock free_b2mx => /row_freeP[Ct CtK]; rewrite -row_b2mx. rewrite -row_mul -[pinvmx _]mulmx1 -CtK (mulmxA (b2mx X)) (mulmxA _ _ Ct). by rewrite mulmxKpV // CtK !mxE. Qed.	Lemma	coord_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" ]	[ "b2mx", "coord", "free", "free_b2mx", "mulmx1", "mulmxA", "mulmxKpV", "mxE", "pinvmx", "row_b2mx", "row_freeP", "row_mul", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_sum_free n (X : n.-tuple vT) k j : free X -> coord X j (\sum_(i < n) k i *: X`_i) = k j.	Proof. move=> Xfree; rewrite linear_sum (bigD1 j) 1?linearZ //= coord_free // eqxx. rewrite mulr1 big1 ?addr0 // => i /negPf j'i. by rewrite linearZ /= coord_free // j'i mulr0. Qed.	Lemma	coord_sum_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" ]	[ "addr0", "big1", "bigD1", "coord", "coord_free", "eqxx", "free", "linearZ", "linear_sum", "mulr0", "mulr1", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cat_free X Y : free (X ++ Y) = [&& free X, free Y & directv (<<X>> + <<Y>>)].	Proof. rewrite !free_directv mem_cat directvE /= !big_cat -directvE /= directv_addE /=. rewrite negb_or -!andbA; do !bool_congr; rewrite -!span_def. by rewrite (sameP eqP directv_addP). Qed.	Lemma	cat_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" ]	[ "big_cat", "directv", "directvE", "directv_addE", "directv_addP", "free", "free_directv", "mem_cat", "span_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
catl_free Y X : free (X ++ Y) -> free X.	Proof. by rewrite cat_free => /and3P[]. Qed.	Lemma	catl_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" ]	[ "cat_free", "free" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
catr_free X Y : free (X ++ Y) -> free Y.	Proof. by rewrite cat_free => /and3P[]. Qed.	Lemma	catr_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" ]	[ "cat_free", "free" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
filter_free p X : free X -> free (filter p X).	Proof. rewrite -(perm_free (etrans (perm_filterC p X _) (perm_refl X))). exact: catl_free. Qed.	Lemma	filter_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" ]	[ "catl_free", "filter", "free", "perm_filterC", "perm_free", "perm_refl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
free_cons v X : free (v :: X) = (v \notin <<X>>)%VS && free X.	Proof. rewrite (cat_free [:: v]) seq1_free directvEgeq /= span_seq1 dim_vline. case: eqP => [-> \| _] /=; first by rewrite mem0v. rewrite andbC ltnNge (geq_leqif (dimv_leqif_sup _)) ?addvSr //. by rewrite subv_add subvv andbT -memvE. Qed.	Lemma	free_cons	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addvSr", "cat_free", "dim_vline", "dimv_leqif_sup", "directvEgeq", "free", "geq_leqif", "ltnNge", "mem0v", "memvE", "seq1_free", "span_seq1", "subv_add", "subvv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
freeE n (X : n.-tuple vT) : free X = [forall i : 'I_n, X`_i \notin <<drop i.+1 X>>%VS].	Proof. case: X => X /= /eqP <-{n}; rewrite -(big_andE xpredT) /=. elim: X => [\|v X IH_X] /=; first by rewrite nil_free big_ord0. by rewrite free_cons IH_X big_ord_recl drop0. Qed.	Lemma	freeE	algebra	algebra/vector.v	[ "HB", "structures", "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_andE", "big_ord0", "big_ord_recl", "drop", "drop0", "free", "free_cons", "nil_free", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
freeNE n (X : n.-tuple vT) : ~~ free X = [exists i : 'I_n, X`_i \in <<drop i.+1 X>>%VS].	Proof. by rewrite freeE -negb_exists negbK. Qed.	Lemma	freeNE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "drop", "free", "freeE", "negb_exists", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
free_uniq X : free X -> uniq X.	Proof. elim: X => //= v b IH_X; rewrite free_cons => /andP[X'v /IH_X->]. by rewrite (contra _ X'v) // => /memv_span. Qed.	Lemma	free_uniq	algebra	algebra/vector.v	[ "HB", "structures", "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", "free_cons", "memv_span", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
free_span X v (sumX := fun k => \sum_(x <- X) k x *: x) : free X -> v \in <<X>>%VS -> {k \| v = sumX k & forall k1, v = sumX k1 -> {in X, k1 =1 k}}.	Proof. rewrite -{2}[X]in_tupleE => freeX /coord_span def_v. pose k x := oapp (fun i => coord (in_tuple X) i v) 0 (insub (index x X)). exists k => [\|k1 {}def_v _ /(nthP 0)[i ltiX <-]]. rewrite /sumX (big_nth 0) big_mkord def_v; apply: eq_bigr => i _. by rewrite /k index_uniq ?free_uniq // valK. rewrite /k /= index_u...	Lemma	free_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" ]	[ "apply", "big_mkord", "big_nth", "coord", "coord_span", "coord_sum_free", "eq_bigr", "free", "free_uniq", "in_tuple", "in_tupleE", "index", "index_uniq", "insub", "insubT", "nthP", "valK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
linear_of_free (rT : lmodType K) X (fX : seq rT) : {f : {linear vT -> rT} \| free X -> size fX = size X -> map f X = fX}.	Proof. pose f u := \sum_i coord (in_tuple X) i u *: fX`_i. have lin_f: linear f. move=> k u v; rewrite scaler_sumr -big_split; apply: eq_bigr => i _. by rewrite /= scalerA -scalerDl linearP. pose flM := GRing.isLinear.Build _ _ _ _ f lin_f. pose fL : {linear _ -> _} := HB.pack f flM. exists fL => freeX eq_szX. appl...	Lemma	linear_of_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" ]	[ "Build", "addr0", "apply", "big1", "bigD1", "big_split", "coord", "coord_free", "eq_bigr", "eq_from_nth", "eq_sym", "eqxx", "free", "in_tuple", "linear", "linearP", "map", "nth_map", "scale0r", "scale1r", "scalerA", "scalerDl", "scaler_sumr", "seq", "size", "size_ma...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_basis U X : basis_of U X -> <<X>>%VS = U.	Proof. by case/andP=> /eqP. Qed.	Lemma	span_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of" ]	Subspace bases	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
basis_free U X : basis_of U X -> free X.	Proof. by case/andP. Qed.	Lemma	basis_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" ]	[ "basis_of", "free" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_basis U n (X : n.-tuple vT) v : basis_of U X -> v \in U -> v = \sum_i coord X i v *: X`_i.	Proof. by move/span_basis <-; apply: coord_span. Qed.	Lemma	coord_basis	algebra	algebra/vector.v	[ "HB", "structures", "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", "basis_of", "coord", "coord_span", "span_basis", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_basis : basis_of 0 (Nil vT).	Proof. by rewrite /basis_of span_nil eqxx nil_free. Qed.	Lemma	nil_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Nil", "basis_of", "eqxx", "nil_free", "span_nil", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq1_basis v : v != 0 -> basis_of <[v]> [:: v].	Proof. by move=> nz_v; rewrite /basis_of span_seq1 // eqxx seq1_free. Qed.	Lemma	seq1_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of", "eqxx", "seq1_free", "span_seq1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
basis_not0 x U X : basis_of U X -> x \in X -> x != 0.	Proof. by move/basis_free/free_not0; apply. Qed.	Lemma	basis_not0	algebra	algebra/vector.v	[ "HB", "structures", "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", "basis_free", "basis_of", "free_not0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
basis_mem x U X : basis_of U X -> x \in X -> x \in U.	Proof. by move/span_basis=> <- /memv_span. Qed.	Lemma	basis_mem	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of", "memv_span", "span_basis" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cat_basis U V X Y : directv (U + V) -> basis_of U X -> basis_of V Y -> basis_of (U + V) (X ++ Y).	Proof. move=> dxUV /andP[/eqP defU freeX] /andP[/eqP defV freeY]. by rewrite /basis_of span_cat cat_free defU defV // eqxx freeX freeY. Qed.	Lemma	cat_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of", "cat_free", "defU", "directv", "eqxx", "span_cat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_basis U n (X : n.-tuple vT) : basis_of U X -> \dim U = n.	Proof. by case/andP=> /eqP <- /eqnP->; apply: size_tuple. Qed.	Lemma	size_basis	algebra	algebra/vector.v	[ "HB", "structures", "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", "basis_of", "dim", "eqnP", "size_tuple", "tuple", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
basisEdim X U : basis_of U X = (U <= <<X>>)%VS && (size X <= \dim U).	Proof. apply/andP/idP=> [[defU /eqnP <-]\| ]; first by rewrite -eqEdim eq_sym. case/andP=> sUX leXU; have leXX := dim_span X. rewrite /free eq_sym eqEdim sUX eqn_leq !(leq_trans leXX) //. by rewrite (leq_trans leXU) ?dimvS. Qed.	Lemma	basisEdim	algebra	algebra/vector.v	[ "HB", "structures", "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", "basis_of", "defU", "dim", "dim_span", "dimvS", "eqEdim", "eq_sym", "eqnP", "eqn_leq", "free", "leq_trans", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
basisEfree X U : basis_of U X = [&& free X, (<<X>> <= U)%VS & \dim U <= size X].	Proof. by rewrite andbC; apply: andb_id2r => freeX; rewrite eqEdim (eqnP freeX). Qed.	Lemma	basisEfree	algebra	algebra/vector.v	[ "HB", "structures", "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", "basis_of", "dim", "eqEdim", "eqnP", "free", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_basis X Y U : perm_eq X Y -> basis_of U X = basis_of U Y.	Proof. move=> eqXY; congr ((_ == _) && _); last exact: perm_free. exact/eq_span/perm_mem. Qed.	Lemma	perm_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of", "eq_span", "last", "perm_eq", "perm_free", "perm_mem" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vbasisP U : basis_of U (vbasis U).	Proof. rewrite /basis_of free_b2mx span_b2mx (sameP eqP (vs2mxP _ _)) !genmxE. have ->: b2mx (vbasis U) = row_base (vs2mx U). by apply/row_matrixP=> i; rewrite unlock rowK tnth_mktuple r2vK. by rewrite row_base_free !eq_row_base submx_refl. Qed.	Lemma	vbasisP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "b2mx", "basis_of", "eq_row_base", "free_b2mx", "genmxE", "r2vK", "rowK", "row_base", "row_base_free", "row_matrixP", "span_b2mx", "submx_refl", "tnth_mktuple", "vbasis", "vs2mx", "vs2mxP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vbasis_mem v U : v \in (vbasis U) -> v \in U.	Proof. exact: basis_mem (vbasisP _). Qed.	Lemma	vbasis_mem	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_mem", "vbasis", "vbasisP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_vbasis v U : v \in U -> v = \sum_(i < \dim U) coord (vbasis U) i v *: (vbasis U)`_i.	Proof. exact: coord_basis (vbasisP U). Qed.	Lemma	coord_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" ]	[ "coord", "coord_basis", "dim", "vbasis", "vbasisP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_bigcat : (<<\big[cat/[::]]_(i \| P i) Xs i>> = \sum_(i \| P i) <<Xs i>>)%VS.	Proof. by rewrite (big_morph _ span_cat span_nil). Qed.	Lemma	span_bigcat	algebra	algebra/vector.v	[ "HB", "structures", "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", "cat", "span_cat", "span_nil" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigcat_free : directv (\sum_(i \| P i) <<Xs i>>) -> (forall i, P i -> free (Xs i)) -> free (\big[cat/[::]]_(i \| P i) Xs i).	Proof. rewrite /free directvE /= span_bigcat => /directvP-> /= freeXs. rewrite (big_morph _ (@size_cat _) (erefl _)) /=. by apply/eqP/eq_bigr=> i /freeXs/eqP. Qed.	Lemma	bigcat_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" ]	[ "apply", "big_morph", "cat", "directv", "directvE", "directvP", "eq_bigr", "free", "size_cat", "span_bigcat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigcat_basis Us (U := (\sum_(i \| P i) Us i)%VS) : directv U -> (forall i, P i -> basis_of (Us i) (Xs i)) -> basis_of U (\big[cat/[::]]_(i \| P i) Xs i).	Proof. move=> dxU XsUs; rewrite /basis_of span_bigcat. have defUs i: P i -> span (Xs i) = Us i by case/XsUs/andP=> /eqP. rewrite (eq_bigr _ defUs) eqxx bigcat_free // => [\|_ /XsUs/andP[]//]. apply/directvP; rewrite /= (eq_bigr _ defUs) (directvP dxU) /=. by apply/eq_bigr=> i /defUs->. Qed.	Lemma	bigcat_basis	algebra	algebra/vector.v	[ "HB", "structures", "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", "basis_of", "bigcat_free", "cat", "directv", "directvP", "eq_bigr", "eqxx", "span", "span_bigcat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
directv S	:= (directv_def (Phantom _ S%VS)).	Notation	directv	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "directv_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_key : unit.	Proof. by []. Qed.	Fact	lfun_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
fun_of_lfun_def aT rT (f : 'Hom(aT, rT))	:= r2v \o mulmxr (f2mx f) \o v2r.	Definition	fun_of_lfun_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" ]	[ "aT", "f2mx", "mulmxr", "r2v", "v2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fun_of_lfun	:= locked_with lfun_key fun_of_lfun_def.	Definition	fun_of_lfun	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "fun_of_lfun_def", "lfun_key" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fun_of_lfun_unlockable	:= [unlockable fun fun_of_lfun].	Canonical	fun_of_lfun_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" ]	[ "fun_of_lfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
linfun_def aT rT (f : aT -> rT)	:= Hom (lin1_mx (v2r \o f \o r2v)).	Definition	linfun_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" ]	[ "aT", "lin1_mx", "r2v", "v2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
linfun	:= locked_with lfun_key linfun_def.	Definition	linfun	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lfun_key", "linfun_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
linfun_unlockable	:= [unlockable fun linfun].	Canonical	linfun_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" ]	[ "linfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
id_lfun vT	:= @linfun vT vT idfun.	Definition	id_lfun	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "linfun", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_lfun aT vT rT (f : 'Hom(vT, rT)) (g : 'Hom(aT, vT))	:= linfun (fun_of_lfun f \o fun_of_lfun g).	Definition	comp_lfun	algebra	algebra/vector.v	[ "HB", "structures", "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", "fun_of_lfun", "linfun", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fun_of_lfun : hom >-> Funclass.		Coercion	fun_of_lfun	algebra	algebra/vector.v	[ "HB", "structures", "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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\1"	:= (@id_lfun _ _) : lfun_scope.	Notation	\1	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "id_lfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f \o g"	:= (comp_lfun f g) : lfun_scope.	Notation	f \o g	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "comp_lfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv_lfun aT rT (f : 'Hom(aT, rT))	:= Hom (pinvmx (f2mx f)).	Definition	inv_lfun	algebra	algebra/vector.v	[ "HB", "structures", "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", "f2mx", "pinvmx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker aT rT (f : 'Hom(aT, rT))	:= mx2vs (kermx (f2mx f)).	Definition	lker	algebra	algebra/vector.v	[ "HB", "structures", "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", "f2mx", "kermx", "mx2vs" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_img_key : unit.	Proof. by []. Qed.	Fact	lfun_img_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
lfun_img_def aT rT f (U : {vspace aT}) : {vspace rT}	:= mx2vs (vs2mx U *m f2mx f).	Definition	lfun_img_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" ]	[ "aT", "f2mx", "mx2vs", "vs2mx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_img	:= locked_with lfun_img_key lfun_img_def.	Definition	lfun_img	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lfun_img_def", "lfun_img_key" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_img_unlockable	:= [unlockable fun lfun_img].	Canonical	lfun_img_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" ]	[ "lfun_img" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_preim aT rT (f : 'Hom(aT, rT)) W	:= (lfun_img (inv_lfun f) (W :&: lfun_img f fullv) + lker f)%VS.	Definition	lfun_preim	algebra	algebra/vector.v	[ "HB", "structures", "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", "fullv", "inv_lfun", "lfun_img", "lker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f ^-1"	:= (inv_lfun f) : lfun_scope.	Notation	f ^-1	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "inv_lfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f @: U"	:= (lfun_img f%VF%R U) (at level 24) : vspace_scope.	Notation	f @: 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" ]	[ "lfun_img" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f @^-1: W"	:= (lfun_preim f%VF%R W) (at level 24) : vspace_scope.	Notation	f @^-1: 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" ]	[ "lfun_preim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg f	:= (lfun_img f fullv).	Notation	limg	algebra	algebra/vector.v	[ "HB", "structures", "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", "lfun_img" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_is_semilinear f : semilinear f.	Proof. by rewrite unlock; apply: semilinearP. Qed.	Fact	lfun_is_semilinear	algebra	algebra/vector.v	[ "HB", "structures", "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", "semilinear", "semilinearP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfunE (ff : {linear aT -> rT}) : linfun ff =1 ff.	Proof. by move=> v; rewrite 2!unlock /= mul_rV_lin1 /= !v2rK. Qed.	Lemma	lfunE	algebra	algebra/vector.v	[ "HB", "structures", "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", "ff", "linear", "linfun", "mul_rV_lin1", "v2rK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fun_of_lfunK : cancel (@fun_of_lfun R aT rT) linfun.	Proof. move=> f; apply/val_inj/row_matrixP=> i. by rewrite 2!unlock /= !rowE mul_rV_lin1 /= !r2vK. Qed.	Lemma	fun_of_lfunK	algebra	algebra/vector.v	[ "HB", "structures", "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", "apply", "fun_of_lfun", "linfun", "mul_rV_lin1", "r2vK", "rowE", "row_matrixP", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfunP f g : f =1 g <-> f = g.	Proof. split=> [eq_fg \| -> //]; rewrite -[f]fun_of_lfunK -[g]fun_of_lfunK unlock. by apply/val_inj/row_matrixP=> i; rewrite !rowE !mul_rV_lin1 /= eq_fg. Qed.	Lemma	lfunP	algebra	algebra/vector.v	[ "HB", "structures", "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", "fun_of_lfunK", "mul_rV_lin1", "rowE", "row_matrixP", "split", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
zero_lfun : 'Hom(aT, rT)	:= linfun \0.	Definition	zero_lfun	algebra	algebra/vector.v	[ "HB", "structures", "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", "linfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_lfun f g	:= linfun (f \+ g).	Definition	add_lfun	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "linfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_addA : associative add_lfun.	Proof. by move=> f g h; apply/lfunP=> v; rewrite !lfunE /= !lfunE addrA. Qed.	Fact	lfun_addA	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "add_lfun", "addrA", "apply", "lfunE", "lfunP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_addC : commutative add_lfun.	Proof. by move=> f g; apply/lfunP=> v; rewrite !lfunE /= addrC. Qed.	Fact	lfun_addC	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "add_lfun", "addrC", "apply", "lfunE", "lfunP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_add0 : left_id zero_lfun add_lfun.	Proof. by move=> f; apply/lfunP=> v; rewrite lfunE /= lfunE add0r. Qed.	Fact	lfun_add0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "add0r", "add_lfun", "apply", "lfunE", "lfunP", "zero_lfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_def S & phantom {vspace vT} (unwrap (addv_val S))

:= \dim (unwrap S) == unwrap (addv_dim S).

Definition

directv_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" ]

[ "dim", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv A

:= (directv_def (Phantom {vspace _} A%VS)).

Notation

directv

algebra

algebra/vector.v

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

[ "directv_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directvE (S : addv_expr) : directv (unwrap S) = (\dim (unwrap S) == unwrap (addv_dim S)).

Proof. by []. Qed.

Lemma

directvE

algebra

algebra/vector.v

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

[ "addv_expr", "dim", "directv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directvP {S : proper_addv_expr} : reflect (\dim S = S :> nat) (directv S).

Proof. exact: eqnP. Qed.

Lemma

directvP

algebra

algebra/vector.v

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

[ "dim", "directv", "eqnP", "nat", "proper_addv_expr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_trivial U : directv (unwrap (@trivial_addv U)).

Proof. exact: eqxx. Qed.

Lemma

directv_trivial

algebra

algebra/vector.v

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

[ "directv", "eqxx", "trivial_addv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv_sum_leqif (S : addv_expr) : \dim (unwrap S) <= unwrap (addv_dim S) ?= iff directv (unwrap S).

Proof. rewrite directvE; case: S => [[U] [d] /= defUd]; split=> //=. rewrite /dimv; elim: {1}_ {U}_ d / defUd => // m1 m2 A1 A2 r1 r2 _ leA1 _ leA2. by apply: leq_trans (leq_add leA1 leA2); rewrite mxrank_adds_leqif. Qed.

Lemma

dimv_sum_leqif

algebra

algebra/vector.v

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

[ "addv_expr", "apply", "dim", "dimv", "directv", "directvE", "leq_add", "leq_trans", "mxrank_adds_leqif", "r1", "r2", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directvEgeq (S : addv_expr) : directv (unwrap S) = (\dim (unwrap S) >= unwrap (addv_dim S)).

Proof. by rewrite leq_eqVlt ltnNge eq_sym !dimv_sum_leqif orbF. Qed.

Lemma

directvEgeq

algebra

algebra/vector.v

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

[ "addv_expr", "dim", "dimv_sum_leqif", "directv", "eq_sym", "leq_eqVlt", "ltnNge" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_addE (S1 S2 : addv_expr) : directv (unwrap S1 + unwrap S2) = [&& directv (unwrap S1), directv (unwrap S2) & unwrap S1 :&: unwrap S2 == 0]%VS.

Proof. by rewrite /directv_def /dimv vs2mxD -mxdirectE mxdirect_addsE -vs2mxI -vs2mx0. Qed.

Lemma

directv_addE

algebra

algebra/vector.v

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

[ "S1", "S2", "addv_expr", "dimv", "directv", "directv_def", "mxdirectE", "mxdirect_addsE", "vs2mx0", "vs2mxD", "vs2mxI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_addP {U V} : reflect (U :&: V = 0)%VS (directv (U + V)).

Proof. by rewrite directv_addE !directv_trivial; apply: eqP. Qed.

Lemma

directv_addP

algebra

algebra/vector.v

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

[ "apply", "directv", "directv_addE", "directv_trivial" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_add_unique {U V} : reflect (forall u1 u2 v1 v2, u1 \in U -> u2 \in U -> v1 \in V -> v2 \in V -> (u1 + v1 == u2 + v2) = ((u1, v1) == (u2, v2))) (directv (U + V)).

Proof. apply: (iffP directv_addP) => [dxUV u1 u2 v1 v2 Uu1 Uu2 Vv1 Vv2 | dxUV]. apply/idP/idP=> [| /eqP[-> ->] //]; rewrite -subr_eq0 opprD addrACA addr_eq0. move/eqP=> eq_uv; rewrite xpair_eqE -subr_eq0 eq_uv oppr_eq0 subr_eq0 andbb. by rewrite -subr_eq0 -memv0 -dxUV memv_cap -memvN -eq_uv !memvB. apply/eqP; rew...

Lemma

directv_add_unique

algebra

algebra/vector.v

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

[ "addrACA", "addrC", "addr_eq0", "apply", "directv", "directv_addP", "eq_sym", "mem0v", "memv0", "memvB", "memvN", "memv_cap", "memv_capP", "opprD", "oppr_eq0", "subr_eq0", "subv0", "subvP", "xpair_eqE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_sumP {Us : I -> {vspace vT}} : reflect (forall i, P i -> Us i :&: (\sum_(j | P j && (j != i)) Us j) = 0)%VS (directv (\sum_(i | P i) Us i)).

Proof. rewrite directvE /= /dimv vs2mx_sum -mxdirectE; apply: (equivP mxdirect_sumsP). by do [split=> dxU i /dxU; rewrite -vs2mx_sum -vs2mxI -vs2mx0] => [/val_inj|->]. Qed.

Lemma

directv_sumP

algebra

algebra/vector.v

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

[ "apply", "dimv", "directv", "directvE", "mxdirectE", "mxdirect_sumsP", "split", "vT", "val_inj", "vs2mx0", "vs2mxI", "vs2mx_sum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_sumE {Ss : I -> addv_expr} (xunwrap := unwrap) : reflect [/\ forall i, P i -> directv (unwrap (Ss i)) & directv (\sum_(i | P i) xunwrap (Ss i))] (directv (\sum_(i | P i) unwrap (Ss i))).

Proof. by rewrite !directvE /= /dimv 2!{1}vs2mx_sum -!mxdirectE; apply: mxdirect_sumsE. Qed.

Lemma

directv_sumE

algebra

algebra/vector.v

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

[ "addv_expr", "apply", "dimv", "directv", "directvE", "mxdirectE", "mxdirect_sumsE", "vs2mx_sum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_sum_independent {Us : I -> {vspace vT}} : reflect (forall us, (forall i, P i -> us i \in Us i) -> \sum_(i | P i) us i = 0 -> (forall i, P i -> us i = 0)) (directv (\sum_(i | P i) Us i)).

Proof. apply: (iffP directv_sumP) => [dxU us Uu u_0 i Pi | dxU i Pi]. apply/eqP; rewrite -memv0 -(dxU i Pi) memv_cap Uu //= -memvN -sub0r -{1}u_0. by rewrite (bigD1 i) //= [_ - us i]addrC addKr memv_sumr // => j /andP[/Uu]. apply/eqP; rewrite -subv0; apply/subvP=> v. rewrite memv_cap memv0 => /andP[Uiv /memv_sumP[u...

Lemma

directv_sum_independent

algebra

algebra/vector.v

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

[ "Uu", "addKr", "addrC", "apply", "big1", "bigD1", "directv", "directv_sumP", "eqVneq", "eqxx", "memv0", "memvN", "memv_cap", "memv_sumP", "memv_sumr", "oppr_eq0", "sub0r", "subrr", "subv0", "subvP", "sumrB", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv_sum_unique {Us : I -> {vspace vT}} : reflect (forall us vs, (forall i, P i -> us i \in Us i) -> (forall i, P i -> vs i \in Us i) -> (\sum_(i | P i) us i == \sum_(i | P i) vs i) = [forall (i | P i), us i == vs i]) (directv (\sum_(i | P i) Us i)).

Proof. apply: (iffP directv_sum_independent) => [dxU us vs Uu Uv | dxU us Uu u_0 i Pi]. apply/idP/forall_inP=> [|eq_uv]; last by apply/eqP/eq_bigr => i /eq_uv/eqP. rewrite -subr_eq0 -sumrB => /eqP/dxU eq_uv i Pi. by rewrite -subr_eq0 eq_uv // => j Pj; apply: memvB; move: j Pj. apply/eqP; have:= esym (dxU us \0 Uu...

Lemma

directv_sum_unique

algebra

algebra/vector.v

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

[ "Uu", "apply", "big1_eq", "directv", "directv_sum_independent", "eq_bigr", "eqxx", "forall_inP", "last", "mem0v", "memvB", "subr_eq0", "sumrB", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_span X v : v \in X -> v \in <<X>>%VS.

Proof. by case/seq_tnthP=> i {v}->; rewrite unlock memvK genmxE (eq_row_sub i) // rowK. Qed.

Lemma

memv_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" ]

[ "eq_row_sub", "genmxE", "memvK", "rowK", "seq_tnthP" ]

Linear span generated by a list of vectors

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_span1 v : v \in <<[:: v]>>%VS.

Proof. by rewrite memv_span ?mem_head. Qed.

Lemma

memv_span1

algebra

algebra/vector.v

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

[ "mem_head", "memv_span" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim_span X : \dim <<X>> <= size X.

Proof. by rewrite unlock /dimv genmxE rank_leq_row. Qed.

Lemma

dim_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" ]

[ "dim", "dimv", "genmxE", "rank_leq_row", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_subvP {X U} : reflect {subset X <= U} (<<X>> <= U)%VS.

Proof. rewrite /subV [@span _ _]unlock genmxE. apply: (iffP row_subP) => /= [sXU | sXU i]. by move=> _ /seq_tnthP[i ->]; have:= sXU i; rewrite rowK memvK. by rewrite rowK -memvK sXU ?mem_tnth. Qed.

Lemma

span_subvP

algebra

algebra/vector.v

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

[ "apply", "genmxE", "mem_tnth", "memvK", "rowK", "row_subP", "seq_tnthP", "span", "subV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_span X Y : {subset X <= Y} -> (<<X>> <= <<Y>>)%VS.

Proof. by move=> sXY; apply/span_subvP=> v /sXY/memv_span. Qed.

Lemma

sub_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" ]

[ "apply", "memv_span", "span_subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_span X Y : X =i Y -> (<<X>> = <<Y>>)%VS.

Proof. by move=> eqXY; apply: subv_anti; rewrite !sub_span // => u; rewrite eqXY. Qed.

Lemma

eq_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" ]

[ "apply", "sub_span", "subv_anti" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_def X : span X = (\sum_(u <- X) <[u]>)%VS.

Proof. apply/subv_anti/andP; split. by apply/span_subvP=> v Xv; rewrite (big_rem v) // memvE addvSl. by rewrite big_tnth; apply/subv_sumP=> i _; rewrite -memvE memv_span ?mem_tnth. Qed.

Lemma

span_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" ]

[ "addvSl", "apply", "big_rem", "big_tnth", "mem_tnth", "memvE", "memv_span", "span", "span_subvP", "split", "subv_anti", "subv_sumP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_nil : (<<Nil vT>> = 0)%VS.

Proof. by rewrite span_def big_nil. Qed.

Lemma

span_nil

algebra

algebra/vector.v

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

[ "Nil", "big_nil", "span_def", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_seq1 v : (<<[:: v]>> = <[v]>)%VS.

Proof. by rewrite span_def big_seq1. Qed.

Lemma

span_seq1

algebra

algebra/vector.v

[ "HB", "structures", "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_seq1", "span_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_cons v X : (<<v :: X>> = <[v]> + <<X>>)%VS.

Proof. by rewrite !span_def big_cons. Qed.

Lemma

span_cons

algebra

algebra/vector.v

[ "HB", "structures", "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_cons", "span_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_cat X Y : (<<X ++ Y>> = <<X>> + <<Y>>)%VS.

Proof. by rewrite !span_def big_cat. Qed.

Lemma

span_cat

algebra

algebra/vector.v

[ "HB", "structures", "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_cat", "span_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_expanded_def n (X : n.-tuple vT) i v

:= (v2r v *m pinvmx (b2mx X)) 0 i.

Definition

coord_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", "pinvmx", "tuple", "v2r", "vT" ]

Coordinates function; should perhaps be generalized to nat indices.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord

:= locked_with span_key coord_expanded_def.

Definition

coord

algebra

algebra/vector.v

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

[ "coord_expanded_def", "span_key" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_unlockable

:= [unlockable fun coord].

Canonical

coord_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" ]

[ "coord" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_is_scalar n (X : n.-tuple vT) i : scalar (coord X i).

Proof. by move=> k u v; rewrite unlock linearP mulmxDl -scalemxAl !mxE. Qed.

Fact

coord_is_scalar

algebra

algebra/vector.v

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

[ "coord", "linearP", "mulmxDl", "mxE", "scalar", "scalemxAl", "tuple", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_span n (X : n.-tuple vT) v : v \in span X -> v = \sum_i coord X i v *: X`_i.

Proof. rewrite memvK span_b2mx genmxE => Xv. by rewrite unlock_with mul_b2mx mulmxKpV ?v2rK. Qed.

Lemma

coord_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" ]

[ "coord", "genmxE", "memvK", "mul_b2mx", "mulmxKpV", "span", "span_b2mx", "tuple", "v2rK", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord0 i v : coord [tuple 0] i v = 0.

Proof. rewrite unlock /pinvmx rank_rV; case: negP => [[] | _]. by apply/eqP/rowP=> j; rewrite !mxE (tnth_nth 0) /= linear0 mxE. by rewrite pid_mx_0 !(mulmx0, mul0mx) mxE. Qed.

Lemma

coord0

algebra

algebra/vector.v

[ "HB", "structures", "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", "coord", "linear0", "mul0mx", "mulmx0", "mxE", "pid_mx_0", "pinvmx", "rank_rV", "rowP", "tnth_nth", "tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nil_free : free (Nil vT).

Proof. by rewrite /free span_nil dimv0. Qed.

Lemma

nil_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" ]

[ "Nil", "dimv0", "free", "span_nil", "vT" ]

Free generator sequences.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq1_free v : free [:: v] = (v != 0).

Proof. by rewrite /free span_seq1 dim_vline; case: (~~ _). Qed.

Lemma

seq1_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" ]

[ "dim_vline", "free", "span_seq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_free X Y : perm_eq X Y -> free X = free Y.

Proof. by move=> eqXY; rewrite /free (perm_size eqXY) (eq_span (perm_mem eqXY)). Qed.

Lemma

perm_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" ]

[ "eq_span", "free", "perm_eq", "perm_mem", "perm_size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

free_directv X : free X = (0 \notin X) && directv (\sum_(v <- X) <[v]>).

Proof. have leXi i (v := tnth (in_tuple X) i): true -> \dim <[v]> <= 1 ?= iff (v != 0). by rewrite -seq1_free -span_seq1 => _; apply/leqif_eq/dim_span. have [_ /=] := leqif_trans (dimv_sum_leqif _) (leqif_sum leXi). rewrite sum1_card card_ord !directvE /= /free andbC span_def !(big_tnth _ _ X). by congr (_ = _ && _);...

Lemma

free_directv

algebra

algebra/vector.v

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

[ "all_predC", "apply", "big_all", "big_andE", "big_tnth", "card_ord", "dim", "dim_span", "dimv_sum_leqif", "directv", "directvE", "free", "has_pred1", "in_tuple", "leqif_eq", "leqif_sum", "leqif_trans", "seq1_free", "span_def", "span_seq1", "sum1_card", "tnth" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

free_not0 v X : free X -> v \in X -> v != 0.

Proof. by rewrite free_directv andbC => /andP[_ /memPn]; apply. Qed.

Lemma

free_not0

algebra

algebra/vector.v

[ "HB", "structures", "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", "free", "free_directv", "memPn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

freeP n (X : n.-tuple vT) : reflect (forall k, \sum_(i < n) k i *: X`_i = 0 -> (forall i, k i = 0)) (free X).

Proof. rewrite free_b2mx; apply: (iffP idP) => [t_free k kt0 i | t_free]. suffices /rowP/(_ i): \row_i k i = 0 by rewrite !mxE. by apply/(row_free_inj t_free)/r2v_inj; rewrite mul0mx -lin_b2mx kt0 linear0. rewrite -kermx_eq0; apply/rowV0P=> rk /sub_kermxP kt0. by apply/rowP=> i; rewrite mxE {}t_free // mul_b2mx kt0...

Lemma

freeP

algebra

algebra/vector.v

[ "HB", "structures", "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", "free", "free_b2mx", "kermx_eq0", "lin_b2mx", "linear0", "mul0mx", "mul_b2mx", "mxE", "r2v_inj", "rowP", "rowV0P", "row_free_inj", "sub_kermxP", "tuple", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_free n (X : n.-tuple vT) (i j : 'I_n) : free X -> coord X j (X`_i) = (i == j)%:R.

Proof. rewrite unlock free_b2mx => /row_freeP[Ct CtK]; rewrite -row_b2mx. rewrite -row_mul -[pinvmx _]mulmx1 -CtK (mulmxA (b2mx X)) (mulmxA _ _ Ct). by rewrite mulmxKpV // CtK !mxE. Qed.

Lemma

coord_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" ]

[ "b2mx", "coord", "free", "free_b2mx", "mulmx1", "mulmxA", "mulmxKpV", "mxE", "pinvmx", "row_b2mx", "row_freeP", "row_mul", "tuple", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_sum_free n (X : n.-tuple vT) k j : free X -> coord X j (\sum_(i < n) k i *: X`_i) = k j.

Proof. move=> Xfree; rewrite linear_sum (bigD1 j) 1?linearZ //= coord_free // eqxx. rewrite mulr1 big1 ?addr0 // => i /negPf j'i. by rewrite linearZ /= coord_free // j'i mulr0. Qed.

Lemma

coord_sum_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" ]

[ "addr0", "big1", "bigD1", "coord", "coord_free", "eqxx", "free", "linearZ", "linear_sum", "mulr0", "mulr1", "tuple", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cat_free X Y : free (X ++ Y) = [&& free X, free Y & directv (<<X>> + <<Y>>)].

Proof. rewrite !free_directv mem_cat directvE /= !big_cat -directvE /= directv_addE /=. rewrite negb_or -!andbA; do !bool_congr; rewrite -!span_def. by rewrite (sameP eqP directv_addP). Qed.

Lemma

cat_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" ]

[ "big_cat", "directv", "directvE", "directv_addE", "directv_addP", "free", "free_directv", "mem_cat", "span_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

catl_free Y X : free (X ++ Y) -> free X.

Proof. by rewrite cat_free => /and3P[]. Qed.

Lemma

catl_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" ]

[ "cat_free", "free" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

catr_free X Y : free (X ++ Y) -> free Y.

Proof. by rewrite cat_free => /and3P[]. Qed.

Lemma

catr_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" ]

[ "cat_free", "free" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

filter_free p X : free X -> free (filter p X).

Proof. rewrite -(perm_free (etrans (perm_filterC p X _) (perm_refl X))). exact: catl_free. Qed.

Lemma

filter_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" ]

[ "catl_free", "filter", "free", "perm_filterC", "perm_free", "perm_refl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

free_cons v X : free (v :: X) = (v \notin <<X>>)%VS && free X.

Proof. rewrite (cat_free [:: v]) seq1_free directvEgeq /= span_seq1 dim_vline. case: eqP => [-> | _] /=; first by rewrite mem0v. rewrite andbC ltnNge (geq_leqif (dimv_leqif_sup _)) ?addvSr //. by rewrite subv_add subvv andbT -memvE. Qed.

Lemma

free_cons

algebra

algebra/vector.v

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

[ "addvSr", "cat_free", "dim_vline", "dimv_leqif_sup", "directvEgeq", "free", "geq_leqif", "ltnNge", "mem0v", "memvE", "seq1_free", "span_seq1", "subv_add", "subvv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

freeE n (X : n.-tuple vT) : free X = [forall i : 'I_n, X`_i \notin <<drop i.+1 X>>%VS].

Proof. case: X => X /= /eqP <-{n}; rewrite -(big_andE xpredT) /=. elim: X => [|v X IH_X] /=; first by rewrite nil_free big_ord0. by rewrite free_cons IH_X big_ord_recl drop0. Qed.

Lemma

freeE

algebra

algebra/vector.v

[ "HB", "structures", "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_andE", "big_ord0", "big_ord_recl", "drop", "drop0", "free", "free_cons", "nil_free", "tuple", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

freeNE n (X : n.-tuple vT) : ~~ free X = [exists i : 'I_n, X`_i \in <<drop i.+1 X>>%VS].

Proof. by rewrite freeE -negb_exists negbK. Qed.

Lemma

freeNE

algebra

algebra/vector.v

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

[ "drop", "free", "freeE", "negb_exists", "tuple", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

free_uniq X : free X -> uniq X.

Proof. elim: X => //= v b IH_X; rewrite free_cons => /andP[X'v /IH_X->]. by rewrite (contra _ X'v) // => /memv_span. Qed.

Lemma

free_uniq

algebra

algebra/vector.v

[ "HB", "structures", "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", "free_cons", "memv_span", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

free_span X v (sumX := fun k => \sum_(x <- X) k x *: x) : free X -> v \in <<X>>%VS -> {k | v = sumX k & forall k1, v = sumX k1 -> {in X, k1 =1 k}}.

Proof. rewrite -{2}[X]in_tupleE => freeX /coord_span def_v. pose k x := oapp (fun i => coord (in_tuple X) i v) 0 (insub (index x X)). exists k => [|k1 {}def_v _ /(nthP 0)[i ltiX <-]]. rewrite /sumX (big_nth 0) big_mkord def_v; apply: eq_bigr => i _. by rewrite /k index_uniq ?free_uniq // valK. rewrite /k /= index_u...

Lemma

free_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" ]

[ "apply", "big_mkord", "big_nth", "coord", "coord_span", "coord_sum_free", "eq_bigr", "free", "free_uniq", "in_tuple", "in_tupleE", "index", "index_uniq", "insub", "insubT", "nthP", "valK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

linear_of_free (rT : lmodType K) X (fX : seq rT) : {f : {linear vT -> rT} | free X -> size fX = size X -> map f X = fX}.

Proof. pose f u := \sum_i coord (in_tuple X) i u *: fX`_i. have lin_f: linear f. move=> k u v; rewrite scaler_sumr -big_split; apply: eq_bigr => i _. by rewrite /= scalerA -scalerDl linearP. pose flM := GRing.isLinear.Build _ _ _ _ f lin_f. pose fL : {linear _ -> _} := HB.pack f flM. exists fL => freeX eq_szX. appl...

Lemma

linear_of_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" ]

[ "Build", "addr0", "apply", "big1", "bigD1", "big_split", "coord", "coord_free", "eq_bigr", "eq_from_nth", "eq_sym", "eqxx", "free", "in_tuple", "linear", "linearP", "map", "nth_map", "scale0r", "scale1r", "scalerA", "scalerDl", "scaler_sumr", "seq", "size", "size_ma...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_basis U X : basis_of U X -> <<X>>%VS = U.

Proof. by case/andP=> /eqP. Qed.

Lemma

span_basis

algebra

algebra/vector.v

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

[ "basis_of" ]

Subspace bases

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

basis_free U X : basis_of U X -> free X.

Proof. by case/andP. Qed.

Lemma

basis_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" ]

[ "basis_of", "free" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_basis U n (X : n.-tuple vT) v : basis_of U X -> v \in U -> v = \sum_i coord X i v *: X`_i.

Proof. by move/span_basis <-; apply: coord_span. Qed.

Lemma

coord_basis

algebra

algebra/vector.v

[ "HB", "structures", "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", "basis_of", "coord", "coord_span", "span_basis", "tuple", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nil_basis : basis_of 0 (Nil vT).

Proof. by rewrite /basis_of span_nil eqxx nil_free. Qed.

Lemma

nil_basis

algebra

algebra/vector.v

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

[ "Nil", "basis_of", "eqxx", "nil_free", "span_nil", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq1_basis v : v != 0 -> basis_of <[v]> [:: v].

Proof. by move=> nz_v; rewrite /basis_of span_seq1 // eqxx seq1_free. Qed.

Lemma

seq1_basis

algebra

algebra/vector.v

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

[ "basis_of", "eqxx", "seq1_free", "span_seq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

basis_not0 x U X : basis_of U X -> x \in X -> x != 0.

Proof. by move/basis_free/free_not0; apply. Qed.

Lemma

basis_not0

algebra

algebra/vector.v

[ "HB", "structures", "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", "basis_free", "basis_of", "free_not0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

basis_mem x U X : basis_of U X -> x \in X -> x \in U.

Proof. by move/span_basis=> <- /memv_span. Qed.

Lemma

basis_mem

algebra

algebra/vector.v

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

[ "basis_of", "memv_span", "span_basis" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cat_basis U V X Y : directv (U + V) -> basis_of U X -> basis_of V Y -> basis_of (U + V) (X ++ Y).

Proof. move=> dxUV /andP[/eqP defU freeX] /andP[/eqP defV freeY]. by rewrite /basis_of span_cat cat_free defU defV // eqxx freeX freeY. Qed.

Lemma

cat_basis

algebra

algebra/vector.v

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

[ "basis_of", "cat_free", "defU", "directv", "eqxx", "span_cat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_basis U n (X : n.-tuple vT) : basis_of U X -> \dim U = n.

Proof. by case/andP=> /eqP <- /eqnP->; apply: size_tuple. Qed.

Lemma

size_basis

algebra

algebra/vector.v

[ "HB", "structures", "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", "basis_of", "dim", "eqnP", "size_tuple", "tuple", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

basisEdim X U : basis_of U X = (U <= <<X>>)%VS && (size X <= \dim U).

Proof. apply/andP/idP=> [[defU /eqnP <-]| ]; first by rewrite -eqEdim eq_sym. case/andP=> sUX leXU; have leXX := dim_span X. rewrite /free eq_sym eqEdim sUX eqn_leq !(leq_trans leXX) //. by rewrite (leq_trans leXU) ?dimvS. Qed.

Lemma

basisEdim

algebra

algebra/vector.v

[ "HB", "structures", "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", "basis_of", "defU", "dim", "dim_span", "dimvS", "eqEdim", "eq_sym", "eqnP", "eqn_leq", "free", "leq_trans", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

basisEfree X U : basis_of U X = [&& free X, (<<X>> <= U)%VS & \dim U <= size X].

Proof. by rewrite andbC; apply: andb_id2r => freeX; rewrite eqEdim (eqnP freeX). Qed.

Lemma

basisEfree

algebra

algebra/vector.v

[ "HB", "structures", "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", "basis_of", "dim", "eqEdim", "eqnP", "free", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_basis X Y U : perm_eq X Y -> basis_of U X = basis_of U Y.

Proof. move=> eqXY; congr ((_ == _) && _); last exact: perm_free. exact/eq_span/perm_mem. Qed.

Lemma

perm_basis

algebra

algebra/vector.v

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

[ "basis_of", "eq_span", "last", "perm_eq", "perm_free", "perm_mem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vbasisP U : basis_of U (vbasis U).

Proof. rewrite /basis_of free_b2mx span_b2mx (sameP eqP (vs2mxP _ _)) !genmxE. have ->: b2mx (vbasis U) = row_base (vs2mx U). by apply/row_matrixP=> i; rewrite unlock rowK tnth_mktuple r2vK. by rewrite row_base_free !eq_row_base submx_refl. Qed.

Lemma

vbasisP

algebra

algebra/vector.v

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

[ "apply", "b2mx", "basis_of", "eq_row_base", "free_b2mx", "genmxE", "r2vK", "rowK", "row_base", "row_base_free", "row_matrixP", "span_b2mx", "submx_refl", "tnth_mktuple", "vbasis", "vs2mx", "vs2mxP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vbasis_mem v U : v \in (vbasis U) -> v \in U.

Proof. exact: basis_mem (vbasisP _). Qed.

Lemma

vbasis_mem

algebra

algebra/vector.v

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

[ "basis_mem", "vbasis", "vbasisP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_vbasis v U : v \in U -> v = \sum_(i < \dim U) coord (vbasis U) i v *: (vbasis U)`_i.

Proof. exact: coord_basis (vbasisP U). Qed.

Lemma

coord_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" ]

[ "coord", "coord_basis", "dim", "vbasis", "vbasisP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_bigcat : (<<\big[cat/[::]]_(i | P i) Xs i>> = \sum_(i | P i) <<Xs i>>)%VS.

Proof. by rewrite (big_morph _ span_cat span_nil). Qed.

Lemma

span_bigcat

algebra

algebra/vector.v

[ "HB", "structures", "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", "cat", "span_cat", "span_nil" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigcat_free : directv (\sum_(i | P i) <<Xs i>>) -> (forall i, P i -> free (Xs i)) -> free (\big[cat/[::]]_(i | P i) Xs i).

Proof. rewrite /free directvE /= span_bigcat => /directvP-> /= freeXs. rewrite (big_morph _ (@size_cat _) (erefl _)) /=. by apply/eqP/eq_bigr=> i /freeXs/eqP. Qed.

Lemma

bigcat_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" ]

[ "apply", "big_morph", "cat", "directv", "directvE", "directvP", "eq_bigr", "free", "size_cat", "span_bigcat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigcat_basis Us (U := (\sum_(i | P i) Us i)%VS) : directv U -> (forall i, P i -> basis_of (Us i) (Xs i)) -> basis_of U (\big[cat/[::]]_(i | P i) Xs i).

Proof. move=> dxU XsUs; rewrite /basis_of span_bigcat. have defUs i: P i -> span (Xs i) = Us i by case/XsUs/andP=> /eqP. rewrite (eq_bigr _ defUs) eqxx bigcat_free // => [|_ /XsUs/andP[]//]. apply/directvP; rewrite /= (eq_bigr _ defUs) (directvP dxU) /=. by apply/eq_bigr=> i /defUs->. Qed.

Lemma

bigcat_basis

algebra

algebra/vector.v

[ "HB", "structures", "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", "basis_of", "bigcat_free", "cat", "directv", "directvP", "eq_bigr", "eqxx", "span", "span_bigcat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

directv S

:= (directv_def (Phantom _ S%VS)).

Notation

directv

algebra

algebra/vector.v

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

[ "directv_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_key : unit.

Proof. by []. Qed.

Fact

lfun_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

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

:= r2v \o mulmxr (f2mx f) \o v2r.

Definition

fun_of_lfun_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" ]

[ "aT", "f2mx", "mulmxr", "r2v", "v2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fun_of_lfun

:= locked_with lfun_key fun_of_lfun_def.

Definition

fun_of_lfun

algebra

algebra/vector.v

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

[ "fun_of_lfun_def", "lfun_key" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fun_of_lfun_unlockable

:= [unlockable fun fun_of_lfun].

Canonical

fun_of_lfun_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" ]

[ "fun_of_lfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

linfun_def aT rT (f : aT -> rT)

:= Hom (lin1_mx (v2r \o f \o r2v)).

Definition

linfun_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" ]

[ "aT", "lin1_mx", "r2v", "v2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

linfun

:= locked_with lfun_key linfun_def.

Definition

linfun

algebra

algebra/vector.v

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

[ "lfun_key", "linfun_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

linfun_unlockable

:= [unlockable fun linfun].

Canonical

linfun_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" ]

[ "linfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

id_lfun vT

:= @linfun vT vT idfun.

Definition

id_lfun

algebra

algebra/vector.v

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

[ "linfun", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_lfun aT vT rT (f : 'Hom(vT, rT)) (g : 'Hom(aT, vT))

:= linfun (fun_of_lfun f \o fun_of_lfun g).

Definition

comp_lfun

algebra

algebra/vector.v

[ "HB", "structures", "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", "fun_of_lfun", "linfun", "vT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fun_of_lfun : hom >-> Funclass.

Coercion

fun_of_lfun

algebra

algebra/vector.v

[ "HB", "structures", "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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\1"

:= (@id_lfun _ _) : lfun_scope.

Notation

\1

algebra

algebra/vector.v

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

[ "id_lfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f \o g"

:= (comp_lfun f g) : lfun_scope.

Notation

f \o g

algebra

algebra/vector.v

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

[ "comp_lfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

:= Hom (pinvmx (f2mx f)).

Definition

inv_lfun

algebra

algebra/vector.v

[ "HB", "structures", "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", "f2mx", "pinvmx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

:= mx2vs (kermx (f2mx f)).

Definition

lker

algebra

algebra/vector.v

[ "HB", "structures", "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", "f2mx", "kermx", "mx2vs" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_img_key : unit.

Proof. by []. Qed.

Fact

lfun_img_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

lfun_img_def aT rT f (U : {vspace aT}) : {vspace rT}

:= mx2vs (vs2mx U *m f2mx f).

Definition

lfun_img_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" ]

[ "aT", "f2mx", "mx2vs", "vs2mx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_img

:= locked_with lfun_img_key lfun_img_def.

Definition

lfun_img

algebra

algebra/vector.v

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

[ "lfun_img_def", "lfun_img_key" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_img_unlockable

:= [unlockable fun lfun_img].

Canonical

lfun_img_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" ]

[ "lfun_img" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_preim aT rT (f : 'Hom(aT, rT)) W

:= (lfun_img (inv_lfun f) (W :&: lfun_img f fullv) + lker f)%VS.

Definition

lfun_preim

algebra

algebra/vector.v

[ "HB", "structures", "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", "fullv", "inv_lfun", "lfun_img", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f ^-1"

:= (inv_lfun f) : lfun_scope.

Notation

f ^-1

algebra

algebra/vector.v

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

[ "inv_lfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f @: U"

:= (lfun_img f%VF%R U) (at level 24) : vspace_scope.

Notation

f @: 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" ]

[ "lfun_img" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f @^-1: W"

:= (lfun_preim f%VF%R W) (at level 24) : vspace_scope.

Notation

f @^-1: 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" ]

[ "lfun_preim" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg f

:= (lfun_img f fullv).

Notation

limg

algebra

algebra/vector.v

[ "HB", "structures", "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", "lfun_img" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_is_semilinear f : semilinear f.

Proof. by rewrite unlock; apply: semilinearP. Qed.

Fact

lfun_is_semilinear

algebra

algebra/vector.v

[ "HB", "structures", "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", "semilinear", "semilinearP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfunE (ff : {linear aT -> rT}) : linfun ff =1 ff.

Proof. by move=> v; rewrite 2!unlock /= mul_rV_lin1 /= !v2rK. Qed.

Lemma

lfunE

algebra

algebra/vector.v

[ "HB", "structures", "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", "ff", "linear", "linfun", "mul_rV_lin1", "v2rK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fun_of_lfunK : cancel (@fun_of_lfun R aT rT) linfun.

Proof. move=> f; apply/val_inj/row_matrixP=> i. by rewrite 2!unlock /= !rowE mul_rV_lin1 /= !r2vK. Qed.

Lemma

fun_of_lfunK

algebra

algebra/vector.v

[ "HB", "structures", "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", "apply", "fun_of_lfun", "linfun", "mul_rV_lin1", "r2vK", "rowE", "row_matrixP", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfunP f g : f =1 g <-> f = g.

Proof. split=> [eq_fg | -> //]; rewrite -[f]fun_of_lfunK -[g]fun_of_lfunK unlock. by apply/val_inj/row_matrixP=> i; rewrite !rowE !mul_rV_lin1 /= eq_fg. Qed.

Lemma

lfunP

algebra

algebra/vector.v

[ "HB", "structures", "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", "fun_of_lfunK", "mul_rV_lin1", "rowE", "row_matrixP", "split", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

zero_lfun : 'Hom(aT, rT)

:= linfun \0.

Definition

zero_lfun

algebra

algebra/vector.v

[ "HB", "structures", "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", "linfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add_lfun f g

:= linfun (f \+ g).

Definition

add_lfun

algebra

algebra/vector.v

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

[ "linfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_addA : associative add_lfun.

Proof. by move=> f g h; apply/lfunP=> v; rewrite !lfunE /= !lfunE addrA. Qed.

Fact

lfun_addA

algebra

algebra/vector.v

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

[ "add_lfun", "addrA", "apply", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_addC : commutative add_lfun.

Proof. by move=> f g; apply/lfunP=> v; rewrite !lfunE /= addrC. Qed.

Fact

lfun_addC

algebra

algebra/vector.v

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

[ "add_lfun", "addrC", "apply", "lfunE", "lfunP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_add0 : left_id zero_lfun add_lfun.

Proof. by move=> f; apply/lfunP=> v; rewrite lfunE /= lfunE add0r. Qed.

Fact

lfun_add0

algebra

algebra/vector.v

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

[ "add0r", "add_lfun", "apply", "lfunE", "lfunP", "zero_lfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d