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
regular_fullv (K : fieldType) : (fullv = 1 :> {vspace K^o})%VS.	Proof. by apply/esym/eqP; rewrite eqEdim subvf dim_vline oner_eq0 dimvf. Qed.	Lemma	regular_fullv	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "dim_vline", "dimvf", "eqEdim", "fullv", "oner_eq0", "subvf" ]	FIXME: remove once https://github.com/math-comp/hierarchy-builder/issues/197 is fixed	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FalgType_proper (R : nzRingType) (aT : falgType R) : dim aT > 0.	Proof. exact: dim_gt0. Qed.	Lemma	FalgType_proper	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "dim", "dim_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_mulE f g u : (f * g) u = g (f u).	Proof. exact: lfunE. Qed.	Lemma	lfun_mulE	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "lfunE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_compE f g : (g \o f)%VF = f * g.	Proof. by []. Qed.	Lemma	lfun_compE	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_invr f	:= if lker f == 0%VS then f^-1%VF else f.	Definition	lfun_invr	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "lker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_mulVr f : lker f == 0%VS -> f^-1%VF * f = 1.	Proof. exact: lker0_compfV. Qed.	Lemma	lfun_mulVr	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "lker", "lker0_compfV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_mulrV f : lker f == 0%VS -> f * f^-1%VF = 1.	Proof. exact: lker0_compVf. Qed.	Lemma	lfun_mulrV	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "lker", "lker0_compVf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_mulRVr f : lker f == 0%VS -> lfun_invr f * f = 1.	Proof. by move=> Uf; rewrite /lfun_invr Uf lfun_mulVr. Qed.	Fact	lfun_mulRVr	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "lfun_invr", "lfun_mulVr", "lker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_mulrRV f : lker f == 0%VS -> f * lfun_invr f = 1.	Proof. by move=> Uf; rewrite /lfun_invr Uf lfun_mulrV. Qed.	Fact	lfun_mulrRV	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "lfun_invr", "lfun_mulrV", "lker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_unitrP f g : g * f = 1 /\ f * g = 1 -> lker f == 0%VS.	Proof. case=> _ fK; apply/lker0P; apply: can_inj (g) _ => u. by rewrite -lfun_mulE fK lfunE. Qed.	Fact	lfun_unitrP	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "fK", "lfunE", "lfun_mulE", "lker", "lker0P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_invr_out f : lker f != 0%VS -> lfun_invr f = f.	Proof. by rewrite /lfun_invr => /negPf->. Qed.	Lemma	lfun_invr_out	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "lfun_invr", "lker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun_invE f : lker f == 0%VS -> f^-1%VF = f^-1.	Proof. by rewrite /f^-1 /= /lfun_invr => ->. Qed.	Lemma	lfun_invE	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "lfun_invr", "lker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amull u : 'End(aT)	:= linfun (u \*o @idfun aT).	Definition	amull	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "linfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amulr u : 'End(aT)	:= linfun (u \o* @idfun aT).	Definition	amulr	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "linfun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amull_inj : injective amull.	Proof. by move=> u v /lfunP/(_ 1); rewrite !lfunE /= !mulr1. Qed.	Lemma	amull_inj	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amull", "lfunE", "lfunP", "mulr1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amulr_inj : injective amulr.	Proof. by move=> u v /lfunP/(_ 1); rewrite !lfunE /= !mul1r. Qed.	Lemma	amulr_inj	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amulr", "lfunE", "lfunP", "mul1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amull_is_linear : linear amull.	Proof. move=> a u v; apply/lfunP => w. by rewrite !lfunE /= scale_lfunE !lfunE /= mulrDl scalerAl. Qed.	Fact	amull_is_linear	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amull", "apply", "lfunE", "lfunP", "linear", "mulrDl", "scale_lfunE", "scalerAl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amull1 : amull 1 = \1%VF.	Proof. by apply/lfunP => z; rewrite id_lfunE lfunE /= mul1r. Qed.	Lemma	amull1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amull", "apply", "id_lfunE", "lfunE", "lfunP", "mul1r" ]	amull is a converse ring morphism	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amullM u v : (amull (u * v) = amull v * amull u)%VF.	Proof. by apply/lfunP => w; rewrite comp_lfunE !lfunE /= mulrA. Qed.	Lemma	amullM	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amull", "apply", "comp_lfunE", "lfunE", "lfunP", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amulr_is_linear : linear amulr.	Proof. move=> a u v; apply/lfunP => w. by rewrite !lfunE /= !lfunE /= lfunE mulrDr /= scalerAr. Qed.	Lemma	amulr_is_linear	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amulr", "apply", "lfunE", "lfunP", "linear", "mulrDr", "scalerAr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amulr_is_monoid_morphism : monoid_morphism amulr.	Proof. split=> [\|x y]; first by apply/lfunP => w; rewrite id_lfunE !lfunE /= mulr1. by apply/lfunP=> w; rewrite comp_lfunE !lfunE /= mulrA. Qed.	Lemma	amulr_is_monoid_morphism	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amulr", "apply", "comp_lfunE", "id_lfunE", "lfunE", "lfunP", "monoid_morphism", "mulr1", "mulrA", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
amulr_is_multiplicative	:= (fun p => (p.2, p.1)) amulr_is_monoid_morphism.	Definition	amulr_is_multiplicative	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amulr_is_monoid_morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_amull u : u \is a GRing.unit -> lker (amull u) == 0%VS.	Proof. by move=> Uu; apply/lker0P=> v w; rewrite !lfunE; apply: mulrI. Qed.	Lemma	lker0_amull	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "Uu", "amull", "apply", "lfunE", "lker", "lker0P", "mulrI", "unit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker0_amulr u : u \is a GRing.unit -> lker (amulr u) == 0%VS.	Proof. by move=> Uu; apply/lker0P=> v w; rewrite !lfunE; apply: mulIr. Qed.	Lemma	lker0_amulr	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "Uu", "amulr", "apply", "lfunE", "lker", "lker0P", "mulIr", "unit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lfun1_poly (p : {poly aT}) : map_poly \1%VF p = p.	Proof. by apply: map_poly_id => u _; apply: id_lfunE. Qed.	Lemma	lfun1_poly	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "apply", "id_lfunE", "map_poly", "map_poly_id", "poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv_key : unit.	Proof. by []. Qed.	Fact	prodv_key	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "unit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv	:= locked_with prodv_key (fun U V => <<allpairs *%R (vbasis U) (vbasis V)>>%VS).	Definition	prodv	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "allpairs", "prodv_key", "vbasis" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv_unlockable	:= [unlockable fun prodv].	Canonical	prodv_unlockable	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "prodv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A * B"	:= (prodv A B) : vspace_scope.	Notation	A * B	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "prodv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_mul U V : {in U & V, forall u v, u * v \in (U * V)%VS}.	Proof. move=> u v /coord_vbasis-> /coord_vbasis->. rewrite mulr_suml; apply: memv_suml => i _. rewrite mulr_sumr; apply: memv_suml => j _. rewrite -scalerAl -scalerAr !memvZ // [prodv]unlock memv_span //. by apply/allpairsP; exists ((vbasis U)`_i, (vbasis V)`_j); rewrite !memt_nth. Qed.	Lemma	memv_mul	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "allpairsP", "apply", "coord_vbasis", "memt_nth", "memvZ", "memv_span", "memv_suml", "mulr_suml", "mulr_sumr", "prodv", "scalerAl", "scalerAr", "vbasis" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvP {U V W} : reflect {in U & V, forall u v, u * v \in W} (U * V <= W)%VS.	Proof. apply: (iffP idP) => [sUVW u v Uu Vv \| sUVW]. by rewrite (subvP sUVW) ?memv_mul. rewrite [prodv]unlock; apply/span_subvP=> _ /allpairsP[[u v] /= [Uu Vv ->]]. by rewrite sUVW ?vbasis_mem. Qed.	Lemma	prodvP	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "Uu", "allpairsP", "apply", "memv_mul", "prodv", "span_subvP", "subvP", "vbasis_mem" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv_line u v : (<[u]> * <[v]> = <[u * v]>)%VS.	Proof. apply: subv_anti; rewrite -memvE memv_mul ?memv_line // andbT. apply/prodvP=> _ _ /vlineP[a ->] /vlineP[b ->]. by rewrite -scalerAr -scalerAl !memvZ ?memv_line. Qed.	Lemma	prodv_line	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "memvE", "memvZ", "memv_line", "memv_mul", "prodvP", "scalerAl", "scalerAr", "subv_anti", "vlineP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dimv1: \dim (1%VS : {vspace aT}) = 1.	Proof. by rewrite dim_vline oner_neq0. Qed.	Lemma	dimv1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "dim", "dim_vline", "oner_neq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_prodv U V : \dim (U * V) <= \dim U * \dim V.	Proof. by rewrite unlock (leq_trans (dim_span _)) ?size_tuple. Qed.	Lemma	dim_prodv	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "dim", "dim_span", "leq_trans", "size_tuple" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vspace1_neq0 : (1 != 0 :> {vspace aT})%VS.	Proof. by rewrite -dimv_eq0 dimv1. Qed.	Lemma	vspace1_neq0	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "dimv1", "dimv_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vbasis1 : exists2 k, k != 0 & vbasis 1 = [:: k%:A] :> seq aT.	Proof. move: (vbasis 1) (@vbasisP K aT 1); rewrite dim_vline oner_neq0. case/tupleP=> x X0; rewrite {X0}tuple0 => defX; have Xx := mem_head x nil. have /vlineP[k def_x] := basis_mem defX Xx; exists k; last by rewrite def_x. by have:= basis_not0 defX Xx; rewrite def_x scaler_eq0 oner_eq0 orbF. Qed.	Lemma	vbasis1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "basis_mem", "basis_not0", "dim_vline", "last", "mem_head", "oner_eq0", "oner_neq0", "scaler_eq0", "seq", "tuple0", "tupleP", "vbasis", "vbasisP", "vlineP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod0v : left_zero 0%VS prodv.	Proof. move=> U; apply/eqP; rewrite -dimv_eq0 -leqn0 (leq_trans (dim_prodv 0 U)) //. by rewrite dimv0. Qed.	Lemma	prod0v	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "dim_prodv", "dimv0", "dimv_eq0", "leq_trans", "leqn0", "prodv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv0 : right_zero 0%VS prodv.	Proof. move=> U; apply/eqP; rewrite -dimv_eq0 -leqn0 (leq_trans (dim_prodv U 0)) //. by rewrite dimv0 muln0. Qed.	Lemma	prodv0	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "dim_prodv", "dimv0", "dimv_eq0", "leq_trans", "leqn0", "muln0", "prodv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod1v : left_id 1%VS prodv.	Proof. move=> U; apply/subv_anti/andP; split. by apply/prodvP=> _ u /vlineP[a ->] Uu; rewrite mulr_algl memvZ. by apply/subvP=> u Uu; rewrite -[u]mul1r memv_mul ?memv_line. Qed.	Lemma	prod1v	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "Uu", "apply", "memvZ", "memv_line", "memv_mul", "mul1r", "mulr_algl", "prodv", "prodvP", "split", "subvP", "subv_anti", "vlineP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv1 : right_id 1%VS prodv.	Proof. move=> U; apply/subv_anti/andP; split. by apply/prodvP=> u _ Uu /vlineP[a ->]; rewrite mulr_algr memvZ. by apply/subvP=> u Uu; rewrite -[u]mulr1 memv_mul ?memv_line. Qed.	Lemma	prodv1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "Uu", "apply", "memvZ", "memv_line", "memv_mul", "mulr1", "mulr_algr", "prodv", "prodvP", "split", "subvP", "subv_anti", "vlineP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvS U1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 * V1 <= U2 * V2)%VS.	Proof. move/subvP=> sU12 /subvP sV12; apply/prodvP=> u v Uu Vv. by rewrite memv_mul ?sU12 ?sV12. Qed.	Lemma	prodvS	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "Uu", "apply", "memv_mul", "prodvP", "subvP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvSl U1 U2 V : (U1 <= U2 -> U1 * V <= U2 * V)%VS.	Proof. by move/prodvS->. Qed.	Lemma	prodvSl	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "prodvS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvSr U V1 V2 : (V1 <= V2 -> U * V1 <= U * V2)%VS.	Proof. exact: prodvS. Qed.	Lemma	prodvSr	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "prodvS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvDl : left_distributive prodv addv.	Proof. move=> U1 U2 V; apply/esym/subv_anti/andP; split. by rewrite subv_add 2?prodvS ?addvSl ?addvSr. apply/prodvP=> _ v /memv_addP[u1 Uu1 [u2 Uu2 ->]] Vv. by rewrite mulrDl memv_add ?memv_mul. Qed.	Lemma	prodvDl	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "addv", "addvSl", "addvSr", "apply", "memv_add", "memv_addP", "memv_mul", "mulrDl", "prodv", "prodvP", "prodvS", "split", "subv_add", "subv_anti" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvDr : right_distributive prodv addv.	Proof. move=> U V1 V2; apply/esym/subv_anti/andP; split. by rewrite subv_add 2?prodvS ?addvSl ?addvSr. apply/prodvP=> u _ Uu /memv_addP[v1 Vv1 [v2 Vv2 ->]]. by rewrite mulrDr memv_add ?memv_mul. Qed.	Lemma	prodvDr	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "Uu", "addv", "addvSl", "addvSr", "apply", "memv_add", "memv_addP", "memv_mul", "mulrDr", "prodv", "prodvP", "prodvS", "split", "subv_add", "subv_anti" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvA : associative prodv.	Proof. move=> U V W; rewrite -(span_basis (vbasisP U)) span_def !big_distrl /=. apply: eq_bigr => u _; rewrite -(span_basis (vbasisP W)) span_def !big_distrr. apply: eq_bigr => w _; rewrite -(span_basis (vbasisP V)) span_def /=. rewrite !(big_distrl, big_distrr) /=; apply: eq_bigr => v _. by rewrite !prodv_line mulrA. ...	Lemma	prodvA	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "big_distrl", "big_distrr", "eq_bigr", "mulrA", "prodv", "prodv_line", "span_basis", "span_def", "vbasisP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expv U n	:= iterop n.+1.-1 prodv U 1%VS.	Definition	expv	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "iterop", "prodv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A ^+ n"	:= (expv A n) : vspace_scope.	Notation	A ^+ n	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "expv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expv0 U : (U ^+ 0 = 1)%VS.	Proof. by []. Qed.	Lemma	expv0	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expv1 U : (U ^+ 1 = U)%VS.	Proof. by []. Qed.	Lemma	expv1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expv2 U : (U ^+ 2 = U * U)%VS.	Proof. by []. Qed.	Lemma	expv2	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expvSl U n : (U ^+ n.+1 = U * U ^+ n)%VS.	Proof. by case: n => //; rewrite prodv1. Qed.	Lemma	expvSl	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "prodv1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expv0n n : (0 ^+ n = if n is _.+1 then 0 else 1)%VS.	Proof. by case: n => // n; rewrite expvSl prod0v. Qed.	Lemma	expv0n	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "expvSl", "prod0v" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expv1n n : (1 ^+ n = 1)%VS.	Proof. by elim: n => // n IHn; rewrite expvSl IHn prodv1. Qed.	Lemma	expv1n	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "expvSl", "prodv1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expvD U m n : (U ^+ (m + n) = U ^+ m * U ^+ n)%VS.	Proof. by elim: m => [\|m IHm]; rewrite ?prod1v // !expvSl IHm prodvA. Qed.	Lemma	expvD	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "expvSl", "prod1v", "prodvA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expvSr U n : (U ^+ n.+1 = U ^+ n * U)%VS.	Proof. by rewrite -addn1 expvD. Qed.	Lemma	expvSr	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "addn1", "expvD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expvM U m n : (U ^+ (m * n) = U ^+ m ^+ n)%VS.	Proof. by elim: n => [\|n IHn]; rewrite ?muln0 // mulnS expvD IHn expvSl. Qed.	Lemma	expvM	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "expvD", "expvSl", "muln0", "mulnS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expvS U V n : (U <= V -> U ^+ n <= V ^+ n)%VS.	Proof. move=> sUV; elim: n => [\|n IHn]; first by rewrite !expv0 subvv. by rewrite !expvSl prodvS. Qed.	Lemma	expvS	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "expv0", "expvSl", "prodvS", "subvv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expv_line u n : (<[u]> ^+ n = <[u ^+ n]>)%VS.	Proof. elim: n => [\|n IH]; first by rewrite expr0 expv0. by rewrite exprS expvSl IH prodv_line. Qed.	Lemma	expv_line	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "expr0", "exprS", "expv0", "expvSl", "prodv_line" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centraliser1_vspace u	:= lker (amulr u - amull u).	Definition	centraliser1_vspace	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "amull", "amulr", "lker" ]	Centralisers and centers.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C [ u ]"	:= (centraliser1_vspace u) : vspace_scope.	Notation	'C [ u ]	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "centraliser1_vspace" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centraliser_vspace V	:= (\bigcap_i 'C[tnth (vbasis V) i])%VS.	Definition	centraliser_vspace	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "tnth", "vbasis" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C ( V )"	:= (centraliser_vspace V) : vspace_scope.	Notation	'C ( V )	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "centraliser_vspace" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_vspace V	:= (V :&: 'C(V))%VS.	Definition	center_vspace	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'Z ( V )"	:= (center_vspace V) : vspace_scope.	Notation	'Z ( V )	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "center_vspace" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1vP u v : reflect (u * v = v * u) (u \in 'C[v]%VS).	Proof. by rewrite (sameP eqlfunP eqP) !lfunE /=; apply: eqP. Qed.	Lemma	cent1vP	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "eqlfunP", "lfunE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1v1 u : 1 \in 'C[u]%VS.	Proof. by apply/cent1vP; rewrite commr1. Qed.	Lemma	cent1v1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "cent1vP", "commr1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1v_id u : u \in 'C[u]%VS.	Proof. exact/cent1vP. Qed.	Lemma	cent1v_id	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "cent1vP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1vX u n : u ^+ n \in 'C[u]%VS.	Proof. exact/cent1vP/esym/commrX. Qed.	Lemma	cent1vX	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "cent1vP", "commrX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1vC u v : (u \in 'C[v])%VS = (v \in 'C[u])%VS.	Proof. exact/cent1vP/cent1vP. Qed.	Lemma	cent1vC	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "cent1vP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centvP u V : reflect {in V, forall v, u * v = v * u} (u \in 'C(V))%VS.	Proof. apply: (iffP subv_bigcapP) => [cVu y /coord_vbasis-> \| cVu i _]. apply/esym/cent1vP/rpred_sum=> i _; apply: rpredZ. by rewrite -tnth_nth cent1vC memvE cVu. exact/cent1vP/cVu/vbasis_mem/mem_tnth. Qed.	Lemma	centvP	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "cent1vC", "cent1vP", "coord_vbasis", "mem_tnth", "memvE", "rpredZ", "rpred_sum", "subv_bigcapP", "tnth_nth", "vbasis_mem" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centvsP U V : reflect {in U & V, commutative *%R} (U <= 'C(V))%VS.	Proof. by apply: (iffP subvP) => [cUV u v \| cUV u] /cUV-/centvP; apply. Qed.	Lemma	centvsP	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "centvP", "subvP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subv_cent1 U v : (U <= 'C[v])%VS = (v \in 'C(U)%VS).	Proof. by apply/subvP/centvP=> cUv u Uu; apply/cent1vP; rewrite 1?cent1vC cUv. Qed.	Lemma	subv_cent1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "Uu", "apply", "cent1vC", "cent1vP", "centvP", "subvP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centv1 V : 1 \in 'C(V)%VS.	Proof. by apply/centvP=> v _; rewrite commr1. Qed.	Lemma	centv1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "centvP", "commr1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centvX V u n : u \in 'C(V)%VS -> u ^+ n \in 'C(V)%VS.	Proof. by move/centvP=> cVu; apply/centvP=> v /cVu/esym/commrX->. Qed.	Lemma	centvX	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "centvP", "commrX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centvC U V : (U <= 'C(V))%VS = (V <= 'C(U))%VS.	Proof. by apply/centvsP/centvsP=> cUV u v UVu /cUV->. Qed.	Lemma	centvC	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "centvsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centerv_sub V : ('Z(V) <= V)%VS.	Proof. exact: capvSl. Qed.	Lemma	centerv_sub	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "capvSl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_centerv V : (V <= 'C('Z(V)))%VS.	Proof. by rewrite centvC capvSr. Qed.	Lemma	cent_centerv	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "capvSr", "centvC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_algid e U	:= [/\ e \in U, e != 0 & {in U, forall u, e * u = u /\ u * e = u}].	Definition	is_algid	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[]	Building the predicate that checks is a vspace has a unit	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
algid_decidable U : decidable (exists e, is_algid e U).	Proof. have [-> \| nzU] := eqVneq U 0%VS. by right=> [[e []]]; rewrite memv0 => ->. pose X := vbasis U; pose feq f1 f2 := [tuple of map f1 X ++ map f2 X]. have feqL f i: tnth (feq _ f _) (lshift _ i) = f X`_i. set v := f _; rewrite (tnth_nth v) /= nth_cat size_map size_tuple. by rewrite ltn_ord (nth_map 0) ?size_t...	Fact	algid_decidable	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "addKn", "amull", "amulr", "apply", "contraNneq", "coord_vbasis", "decidable", "e0", "eqVneq", "eq_bigr", "f1", "f2", "id", "is_algid", "leq_addr", "lfunE", "lshift", "ltnNge", "ltn_ord", "map", "mem0v", "mem_tnth", "memv0", "mul0r", "mulr_suml", "mulr_sumr", "nth...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
has_algid : pred {vspace aT}	:= algid_decidable.	Definition	has_algid	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "algid_decidable" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
has_algidP {U} : reflect (exists e, is_algid e U) (has_algid U).	Proof. exact: sumboolP. Qed.	Lemma	has_algidP	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "has_algid", "is_algid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
has_algid1 U : 1 \in U -> has_algid U.	Proof. move=> U1; apply/has_algidP; exists 1; split; rewrite ?oner_eq0 // => u _. by rewrite mulr1 mul1r. Qed.	Lemma	has_algid1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "apply", "has_algid", "has_algidP", "mul1r", "mulr1", "oner_eq0", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_aspace U	:= has_algid U && (U * U <= U)%VS.	Definition	is_aspace	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "has_algid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aspace	:= ASpace {asval :> {vspace aT}; _ : is_aspace asval}.	Structure	aspace	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "is_aspace" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
clone_aspace U (A : aspace)	:= fun algU & phant_id algU (valP A) => @ASpace U algU : aspace.	Definition	clone_aspace	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aspace", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aspace1_subproof : is_aspace 1.	Proof. by rewrite /is_aspace prod1v -memvE has_algid1 memv_line. Qed.	Fact	aspace1_subproof	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "has_algid1", "is_aspace", "memvE", "memv_line", "prod1v" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aspace1 : aspace	:= ASpace aspace1_subproof.	Canonical	aspace1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aspace", "aspace1_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aspacef_subproof : is_aspace fullv.	Proof. by rewrite /is_aspace subvf has_algid1 ?memvf. Qed.	Lemma	aspacef_subproof	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "fullv", "has_algid1", "is_aspace", "memvf", "subvf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aspacef : aspace	:= ASpace aspacef_subproof.	Canonical	aspacef	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aspace", "aspacef_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver1P p : reflect (exists q, p = map_poly (in_alg aT) q) (p \is a polyOver 1%VS).	Proof. apply: (iffP idP) => [/allP/=Qp \| [q ->]]; last first. by apply/polyOverP=> j; rewrite coef_map rpredZ ?memv_line. exists (map_poly (coord [tuple 1] 0) p). rewrite -map_poly_comp map_poly_id // => _ /Qp/vlineP[a ->] /=. by rewrite linearZ /= (coord_free 0) ?mulr1 // seq1_free ?oner_eq0. Qed.	Lemma	polyOver1P	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "allP", "apply", "coef_map", "coord", "coord_free", "in_alg", "last", "linearZ", "map_poly", "map_poly_comp", "map_poly_id", "memv_line", "mulr1", "oner_eq0", "polyOver", "polyOverP", "rpredZ", "seq1_free", "tuple", "vlineP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'aspace' T }"	:= (aspace T) : type_scope.	Notation	{ 'aspace' T }	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aspace" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C_ U [ v ]"	:= (capv U 'C[v]) : vspace_scope.	Notation	'C_ U [ v ]	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "capv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C_ ( U ) [ v ]"	:= (capv U 'C[v]) (only parsing) : vspace_scope.	Notation	'C_ ( U ) [ v ]	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "capv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C_ U ( V )"	:= (capv U 'C(V)) : vspace_scope.	Notation	'C_ U ( V )	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "capv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C_ ( U ) ( V )"	:= (capv U 'C(V)) (only parsing) : vspace_scope.	Notation	'C_ ( U ) ( V )	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "capv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"1"	:= (aspace1 _) : aspace_scope.	Notation	1	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aspace1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ : aT }"	:= (aspacef aT) : aspace_scope.	Notation	{ : aT }	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "aT", "aspacef" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'aspace' 'of' U ]"	:= (@clone_aspace _ _ U _ _ id) (format "[ 'aspace' 'of' U ]") : form_scope.	Notation	[ 'aspace' 'of' U ]	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "clone_aspace", "id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'aspace' 'of' U 'for' A ]"	:= (@clone_aspace _ _ U A _ idfun) (format "[ 'aspace' 'of' U 'for' A ]") : form_scope.	Notation	[ 'aspace' 'of' U 'for' A ]	field	field/falgebra.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]	[ "clone_aspace" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

regular_fullv (K : fieldType) : (fullv = 1 :> {vspace K^o})%VS.

Proof. by apply/esym/eqP; rewrite eqEdim subvf dim_vline oner_eq0 dimvf. Qed.

Lemma

regular_fullv

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "dim_vline", "dimvf", "eqEdim", "fullv", "oner_eq0", "subvf" ]

FIXME: remove once https://github.com/math-comp/hierarchy-builder/issues/197 is fixed

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FalgType_proper (R : nzRingType) (aT : falgType R) : dim aT > 0.

Proof. exact: dim_gt0. Qed.

Lemma

FalgType_proper

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "dim", "dim_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_mulE f g u : (f * g) u = g (f u).

Proof. exact: lfunE. Qed.

Lemma

lfun_mulE

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_compE f g : (g \o f)%VF = f * g.

Proof. by []. Qed.

Lemma

lfun_compE

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_invr f

:= if lker f == 0%VS then f^-1%VF else f.

Definition

lfun_invr

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_mulVr f : lker f == 0%VS -> f^-1%VF * f = 1.

Proof. exact: lker0_compfV. Qed.

Lemma

lfun_mulVr

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "lker", "lker0_compfV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_mulrV f : lker f == 0%VS -> f * f^-1%VF = 1.

Proof. exact: lker0_compVf. Qed.

Lemma

lfun_mulrV

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "lker", "lker0_compVf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_mulRVr f : lker f == 0%VS -> lfun_invr f * f = 1.

Proof. by move=> Uf; rewrite /lfun_invr Uf lfun_mulVr. Qed.

Fact

lfun_mulRVr

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "lfun_invr", "lfun_mulVr", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_mulrRV f : lker f == 0%VS -> f * lfun_invr f = 1.

Proof. by move=> Uf; rewrite /lfun_invr Uf lfun_mulrV. Qed.

Fact

lfun_mulrRV

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "lfun_invr", "lfun_mulrV", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_unitrP f g : g * f = 1 /\ f * g = 1 -> lker f == 0%VS.

Proof. case=> _ fK; apply/lker0P; apply: can_inj (g) _ => u. by rewrite -lfun_mulE fK lfunE. Qed.

Fact

lfun_unitrP

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "fK", "lfunE", "lfun_mulE", "lker", "lker0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_invr_out f : lker f != 0%VS -> lfun_invr f = f.

Proof. by rewrite /lfun_invr => /negPf->. Qed.

Lemma

lfun_invr_out

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "lfun_invr", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun_invE f : lker f == 0%VS -> f^-1%VF = f^-1.

Proof. by rewrite /f^-1 /= /lfun_invr => ->. Qed.

Lemma

lfun_invE

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "lfun_invr", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amull u : 'End(aT)

:= linfun (u \*o @idfun aT).

Definition

amull

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "linfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amulr u : 'End(aT)

:= linfun (u \o* @idfun aT).

Definition

amulr

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "linfun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amull_inj : injective amull.

Proof. by move=> u v /lfunP/(_ 1); rewrite !lfunE /= !mulr1. Qed.

Lemma

amull_inj

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amull", "lfunE", "lfunP", "mulr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amulr_inj : injective amulr.

Proof. by move=> u v /lfunP/(_ 1); rewrite !lfunE /= !mul1r. Qed.

Lemma

amulr_inj

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amulr", "lfunE", "lfunP", "mul1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amull_is_linear : linear amull.

Proof. move=> a u v; apply/lfunP => w. by rewrite !lfunE /= scale_lfunE !lfunE /= mulrDl scalerAl. Qed.

Fact

amull_is_linear

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amull", "apply", "lfunE", "lfunP", "linear", "mulrDl", "scale_lfunE", "scalerAl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amull1 : amull 1 = \1%VF.

Proof. by apply/lfunP => z; rewrite id_lfunE lfunE /= mul1r. Qed.

Lemma

amull1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amull", "apply", "id_lfunE", "lfunE", "lfunP", "mul1r" ]

amull is a converse ring morphism

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amullM u v : (amull (u * v) = amull v * amull u)%VF.

Proof. by apply/lfunP => w; rewrite comp_lfunE !lfunE /= mulrA. Qed.

Lemma

amullM

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amull", "apply", "comp_lfunE", "lfunE", "lfunP", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amulr_is_linear : linear amulr.

Proof. move=> a u v; apply/lfunP => w. by rewrite !lfunE /= !lfunE /= lfunE mulrDr /= scalerAr. Qed.

Lemma

amulr_is_linear

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amulr", "apply", "lfunE", "lfunP", "linear", "mulrDr", "scalerAr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amulr_is_monoid_morphism : monoid_morphism amulr.

Proof. split=> [|x y]; first by apply/lfunP => w; rewrite id_lfunE !lfunE /= mulr1. by apply/lfunP=> w; rewrite comp_lfunE !lfunE /= mulrA. Qed.

Lemma

amulr_is_monoid_morphism

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amulr", "apply", "comp_lfunE", "id_lfunE", "lfunE", "lfunP", "monoid_morphism", "mulr1", "mulrA", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

amulr_is_multiplicative

:= (fun p => (p.2, p.1)) amulr_is_monoid_morphism.

Definition

amulr_is_multiplicative

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amulr_is_monoid_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_amull u : u \is a GRing.unit -> lker (amull u) == 0%VS.

Proof. by move=> Uu; apply/lker0P=> v w; rewrite !lfunE; apply: mulrI. Qed.

Lemma

lker0_amull

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "Uu", "amull", "apply", "lfunE", "lker", "lker0P", "mulrI", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker0_amulr u : u \is a GRing.unit -> lker (amulr u) == 0%VS.

Proof. by move=> Uu; apply/lker0P=> v w; rewrite !lfunE; apply: mulIr. Qed.

Lemma

lker0_amulr

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "Uu", "amulr", "apply", "lfunE", "lker", "lker0P", "mulIr", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lfun1_poly (p : {poly aT}) : map_poly \1%VF p = p.

Proof. by apply: map_poly_id => u _; apply: id_lfunE. Qed.

Lemma

lfun1_poly

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "apply", "id_lfunE", "map_poly", "map_poly_id", "poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodv_key : unit.

Proof. by []. Qed.

Fact

prodv_key

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodv

:= locked_with prodv_key (fun U V => <<allpairs *%R (vbasis U) (vbasis V)>>%VS).

Definition

prodv

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "allpairs", "prodv_key", "vbasis" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodv_unlockable

:= [unlockable fun prodv].

Canonical

prodv_unlockable

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "prodv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A * B"

:= (prodv A B) : vspace_scope.

Notation

A * B

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "prodv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_mul U V : {in U & V, forall u v, u * v \in (U * V)%VS}.

Proof. move=> u v /coord_vbasis-> /coord_vbasis->. rewrite mulr_suml; apply: memv_suml => i _. rewrite mulr_sumr; apply: memv_suml => j _. rewrite -scalerAl -scalerAr !memvZ // [prodv]unlock memv_span //. by apply/allpairsP; exists ((vbasis U)`_i, (vbasis V)`_j); rewrite !memt_nth. Qed.

Lemma

memv_mul

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "allpairsP", "apply", "coord_vbasis", "memt_nth", "memvZ", "memv_span", "memv_suml", "mulr_suml", "mulr_sumr", "prodv", "scalerAl", "scalerAr", "vbasis" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvP {U V W} : reflect {in U & V, forall u v, u * v \in W} (U * V <= W)%VS.

Proof. apply: (iffP idP) => [sUVW u v Uu Vv | sUVW]. by rewrite (subvP sUVW) ?memv_mul. rewrite [prodv]unlock; apply/span_subvP=> _ /allpairsP[[u v] /= [Uu Vv ->]]. by rewrite sUVW ?vbasis_mem. Qed.

Lemma

prodvP

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "Uu", "allpairsP", "apply", "memv_mul", "prodv", "span_subvP", "subvP", "vbasis_mem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodv_line u v : (<[u]> * <[v]> = <[u * v]>)%VS.

Proof. apply: subv_anti; rewrite -memvE memv_mul ?memv_line // andbT. apply/prodvP=> _ _ /vlineP[a ->] /vlineP[b ->]. by rewrite -scalerAr -scalerAl !memvZ ?memv_line. Qed.

Lemma

prodv_line

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "memvE", "memvZ", "memv_line", "memv_mul", "prodvP", "scalerAl", "scalerAr", "subv_anti", "vlineP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dimv1: \dim (1%VS : {vspace aT}) = 1.

Proof. by rewrite dim_vline oner_neq0. Qed.

Lemma

dimv1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "dim", "dim_vline", "oner_neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim_prodv U V : \dim (U * V) <= \dim U * \dim V.

Proof. by rewrite unlock (leq_trans (dim_span _)) ?size_tuple. Qed.

Lemma

dim_prodv

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "dim", "dim_span", "leq_trans", "size_tuple" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vspace1_neq0 : (1 != 0 :> {vspace aT})%VS.

Proof. by rewrite -dimv_eq0 dimv1. Qed.

Lemma

vspace1_neq0

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "dimv1", "dimv_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vbasis1 : exists2 k, k != 0 & vbasis 1 = [:: k%:A] :> seq aT.

Proof. move: (vbasis 1) (@vbasisP K aT 1); rewrite dim_vline oner_neq0. case/tupleP=> x X0; rewrite {X0}tuple0 => defX; have Xx := mem_head x nil. have /vlineP[k def_x] := basis_mem defX Xx; exists k; last by rewrite def_x. by have:= basis_not0 defX Xx; rewrite def_x scaler_eq0 oner_eq0 orbF. Qed.

Lemma

vbasis1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "basis_mem", "basis_not0", "dim_vline", "last", "mem_head", "oner_eq0", "oner_neq0", "scaler_eq0", "seq", "tuple0", "tupleP", "vbasis", "vbasisP", "vlineP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod0v : left_zero 0%VS prodv.

Proof. move=> U; apply/eqP; rewrite -dimv_eq0 -leqn0 (leq_trans (dim_prodv 0 U)) //. by rewrite dimv0. Qed.

Lemma

prod0v

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "dim_prodv", "dimv0", "dimv_eq0", "leq_trans", "leqn0", "prodv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodv0 : right_zero 0%VS prodv.

Proof. move=> U; apply/eqP; rewrite -dimv_eq0 -leqn0 (leq_trans (dim_prodv U 0)) //. by rewrite dimv0 muln0. Qed.

Lemma

prodv0

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "dim_prodv", "dimv0", "dimv_eq0", "leq_trans", "leqn0", "muln0", "prodv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod1v : left_id 1%VS prodv.

Proof. move=> U; apply/subv_anti/andP; split. by apply/prodvP=> _ u /vlineP[a ->] Uu; rewrite mulr_algl memvZ. by apply/subvP=> u Uu; rewrite -[u]mul1r memv_mul ?memv_line. Qed.

Lemma

prod1v

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "Uu", "apply", "memvZ", "memv_line", "memv_mul", "mul1r", "mulr_algl", "prodv", "prodvP", "split", "subvP", "subv_anti", "vlineP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodv1 : right_id 1%VS prodv.

Proof. move=> U; apply/subv_anti/andP; split. by apply/prodvP=> u _ Uu /vlineP[a ->]; rewrite mulr_algr memvZ. by apply/subvP=> u Uu; rewrite -[u]mulr1 memv_mul ?memv_line. Qed.

Lemma

prodv1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "Uu", "apply", "memvZ", "memv_line", "memv_mul", "mulr1", "mulr_algr", "prodv", "prodvP", "split", "subvP", "subv_anti", "vlineP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvS U1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 * V1 <= U2 * V2)%VS.

Proof. move/subvP=> sU12 /subvP sV12; apply/prodvP=> u v Uu Vv. by rewrite memv_mul ?sU12 ?sV12. Qed.

Lemma

prodvS

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "Uu", "apply", "memv_mul", "prodvP", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvSl U1 U2 V : (U1 <= U2 -> U1 * V <= U2 * V)%VS.

Proof. by move/prodvS->. Qed.

Lemma

prodvSl

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "prodvS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvSr U V1 V2 : (V1 <= V2 -> U * V1 <= U * V2)%VS.

Proof. exact: prodvS. Qed.

Lemma

prodvSr

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "prodvS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvDl : left_distributive prodv addv.

Proof. move=> U1 U2 V; apply/esym/subv_anti/andP; split. by rewrite subv_add 2?prodvS ?addvSl ?addvSr. apply/prodvP=> _ v /memv_addP[u1 Uu1 [u2 Uu2 ->]] Vv. by rewrite mulrDl memv_add ?memv_mul. Qed.

Lemma

prodvDl

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "addv", "addvSl", "addvSr", "apply", "memv_add", "memv_addP", "memv_mul", "mulrDl", "prodv", "prodvP", "prodvS", "split", "subv_add", "subv_anti" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvDr : right_distributive prodv addv.

Proof. move=> U V1 V2; apply/esym/subv_anti/andP; split. by rewrite subv_add 2?prodvS ?addvSl ?addvSr. apply/prodvP=> u _ Uu /memv_addP[v1 Vv1 [v2 Vv2 ->]]. by rewrite mulrDr memv_add ?memv_mul. Qed.

Lemma

prodvDr

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "Uu", "addv", "addvSl", "addvSr", "apply", "memv_add", "memv_addP", "memv_mul", "mulrDr", "prodv", "prodvP", "prodvS", "split", "subv_add", "subv_anti" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvA : associative prodv.

Proof. move=> U V W; rewrite -(span_basis (vbasisP U)) span_def !big_distrl /=. apply: eq_bigr => u _; rewrite -(span_basis (vbasisP W)) span_def !big_distrr. apply: eq_bigr => w _; rewrite -(span_basis (vbasisP V)) span_def /=. rewrite !(big_distrl, big_distrr) /=; apply: eq_bigr => v _. by rewrite !prodv_line mulrA. ...

Lemma

prodvA

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "big_distrl", "big_distrr", "eq_bigr", "mulrA", "prodv", "prodv_line", "span_basis", "span_def", "vbasisP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expv U n

:= iterop n.+1.-1 prodv U 1%VS.

Definition

expv

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "iterop", "prodv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A ^+ n"

:= (expv A n) : vspace_scope.

Notation

A ^+ n

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "expv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expv0 U : (U ^+ 0 = 1)%VS.

Proof. by []. Qed.

Lemma

expv0

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expv1 U : (U ^+ 1 = U)%VS.

Proof. by []. Qed.

Lemma

expv1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expv2 U : (U ^+ 2 = U * U)%VS.

Proof. by []. Qed.

Lemma

expv2

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expvSl U n : (U ^+ n.+1 = U * U ^+ n)%VS.

Proof. by case: n => //; rewrite prodv1. Qed.

Lemma

expvSl

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "prodv1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expv0n n : (0 ^+ n = if n is _.+1 then 0 else 1)%VS.

Proof. by case: n => // n; rewrite expvSl prod0v. Qed.

Lemma

expv0n

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "expvSl", "prod0v" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expv1n n : (1 ^+ n = 1)%VS.

Proof. by elim: n => // n IHn; rewrite expvSl IHn prodv1. Qed.

Lemma

expv1n

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "expvSl", "prodv1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expvD U m n : (U ^+ (m + n) = U ^+ m * U ^+ n)%VS.

Proof. by elim: m => [|m IHm]; rewrite ?prod1v // !expvSl IHm prodvA. Qed.

Lemma

expvD

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "expvSl", "prod1v", "prodvA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expvSr U n : (U ^+ n.+1 = U ^+ n * U)%VS.

Proof. by rewrite -addn1 expvD. Qed.

Lemma

expvSr

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "addn1", "expvD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expvM U m n : (U ^+ (m * n) = U ^+ m ^+ n)%VS.

Proof. by elim: n => [|n IHn]; rewrite ?muln0 // mulnS expvD IHn expvSl. Qed.

Lemma

expvM

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "expvD", "expvSl", "muln0", "mulnS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expvS U V n : (U <= V -> U ^+ n <= V ^+ n)%VS.

Proof. move=> sUV; elim: n => [|n IHn]; first by rewrite !expv0 subvv. by rewrite !expvSl prodvS. Qed.

Lemma

expvS

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "expv0", "expvSl", "prodvS", "subvv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expv_line u n : (<[u]> ^+ n = <[u ^+ n]>)%VS.

Proof. elim: n => [|n IH]; first by rewrite expr0 expv0. by rewrite exprS expvSl IH prodv_line. Qed.

Lemma

expv_line

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "expr0", "exprS", "expv0", "expvSl", "prodv_line" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centraliser1_vspace u

:= lker (amulr u - amull u).

Definition

centraliser1_vspace

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "amull", "amulr", "lker" ]

Centralisers and centers.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C [ u ]"

:= (centraliser1_vspace u) : vspace_scope.

Notation

'C [ u ]

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "centraliser1_vspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centraliser_vspace V

:= (\bigcap_i 'C[tnth (vbasis V) i])%VS.

Definition

centraliser_vspace

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "tnth", "vbasis" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C ( V )"

:= (centraliser_vspace V) : vspace_scope.

Notation

'C ( V )

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "centraliser_vspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_vspace V

:= (V :&: 'C(V))%VS.

Definition

center_vspace

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'Z ( V )"

:= (center_vspace V) : vspace_scope.

Notation

'Z ( V )

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "center_vspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1vP u v : reflect (u * v = v * u) (u \in 'C[v]%VS).

Proof. by rewrite (sameP eqlfunP eqP) !lfunE /=; apply: eqP. Qed.

Lemma

cent1vP

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "eqlfunP", "lfunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1v1 u : 1 \in 'C[u]%VS.

Proof. by apply/cent1vP; rewrite commr1. Qed.

Lemma

cent1v1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "cent1vP", "commr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1v_id u : u \in 'C[u]%VS.

Proof. exact/cent1vP. Qed.

Lemma

cent1v_id

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "cent1vP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1vX u n : u ^+ n \in 'C[u]%VS.

Proof. exact/cent1vP/esym/commrX. Qed.

Lemma

cent1vX

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "cent1vP", "commrX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1vC u v : (u \in 'C[v])%VS = (v \in 'C[u])%VS.

Proof. exact/cent1vP/cent1vP. Qed.

Lemma

cent1vC

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "cent1vP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centvP u V : reflect {in V, forall v, u * v = v * u} (u \in 'C(V))%VS.

Proof. apply: (iffP subv_bigcapP) => [cVu y /coord_vbasis-> | cVu i _]. apply/esym/cent1vP/rpred_sum=> i _; apply: rpredZ. by rewrite -tnth_nth cent1vC memvE cVu. exact/cent1vP/cVu/vbasis_mem/mem_tnth. Qed.

Lemma

centvP

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "cent1vC", "cent1vP", "coord_vbasis", "mem_tnth", "memvE", "rpredZ", "rpred_sum", "subv_bigcapP", "tnth_nth", "vbasis_mem" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centvsP U V : reflect {in U & V, commutative *%R} (U <= 'C(V))%VS.

Proof. by apply: (iffP subvP) => [cUV u v | cUV u] /cUV-/centvP; apply. Qed.

Lemma

centvsP

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "centvP", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subv_cent1 U v : (U <= 'C[v])%VS = (v \in 'C(U)%VS).

Proof. by apply/subvP/centvP=> cUv u Uu; apply/cent1vP; rewrite 1?cent1vC cUv. Qed.

Lemma

subv_cent1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "Uu", "apply", "cent1vC", "cent1vP", "centvP", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centv1 V : 1 \in 'C(V)%VS.

Proof. by apply/centvP=> v _; rewrite commr1. Qed.

Lemma

centv1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "centvP", "commr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centvX V u n : u \in 'C(V)%VS -> u ^+ n \in 'C(V)%VS.

Proof. by move/centvP=> cVu; apply/centvP=> v /cVu/esym/commrX->. Qed.

Lemma

centvX

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "centvP", "commrX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centvC U V : (U <= 'C(V))%VS = (V <= 'C(U))%VS.

Proof. by apply/centvsP/centvsP=> cUV u v UVu /cUV->. Qed.

Lemma

centvC

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "centvsP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centerv_sub V : ('Z(V) <= V)%VS.

Proof. exact: capvSl. Qed.

Lemma

centerv_sub

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "capvSl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_centerv V : (V <= 'C('Z(V)))%VS.

Proof. by rewrite centvC capvSr. Qed.

Lemma

cent_centerv

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "capvSr", "centvC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

is_algid e U

:= [/\ e \in U, e != 0 & {in U, forall u, e * u = u /\ u * e = u}].

Definition

is_algid

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[]

Building the predicate that checks is a vspace has a unit

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

algid_decidable U : decidable (exists e, is_algid e U).

Proof. have [-> | nzU] := eqVneq U 0%VS. by right=> [[e []]]; rewrite memv0 => ->. pose X := vbasis U; pose feq f1 f2 := [tuple of map f1 X ++ map f2 X]. have feqL f i: tnth (feq _ f _) (lshift _ i) = f X`_i. set v := f _; rewrite (tnth_nth v) /= nth_cat size_map size_tuple. by rewrite ltn_ord (nth_map 0) ?size_t...

Fact

algid_decidable

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "addKn", "amull", "amulr", "apply", "contraNneq", "coord_vbasis", "decidable", "e0", "eqVneq", "eq_bigr", "f1", "f2", "id", "is_algid", "leq_addr", "lfunE", "lshift", "ltnNge", "ltn_ord", "map", "mem0v", "mem_tnth", "memv0", "mul0r", "mulr_suml", "mulr_sumr", "nth...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

has_algid : pred {vspace aT}

:= algid_decidable.

Definition

has_algid

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "algid_decidable" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

has_algidP {U} : reflect (exists e, is_algid e U) (has_algid U).

Proof. exact: sumboolP. Qed.

Lemma

has_algidP

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "has_algid", "is_algid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

has_algid1 U : 1 \in U -> has_algid U.

Proof. move=> U1; apply/has_algidP; exists 1; split; rewrite ?oner_eq0 // => u _. by rewrite mulr1 mul1r. Qed.

Lemma

has_algid1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "apply", "has_algid", "has_algidP", "mul1r", "mulr1", "oner_eq0", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

is_aspace U

:= has_algid U && (U * U <= U)%VS.

Definition

is_aspace

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "has_algid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aspace

:= ASpace {asval :> {vspace aT}; _ : is_aspace asval}.

Structure

aspace

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "is_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

clone_aspace U (A : aspace)

:= fun algU & phant_id algU (valP A) => @ASpace U algU : aspace.

Definition

clone_aspace

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aspace", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aspace1_subproof : is_aspace 1.

Proof. by rewrite /is_aspace prod1v -memvE has_algid1 memv_line. Qed.

Fact

aspace1_subproof

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "has_algid1", "is_aspace", "memvE", "memv_line", "prod1v" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aspace1 : aspace

:= ASpace aspace1_subproof.

Canonical

aspace1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aspace", "aspace1_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aspacef_subproof : is_aspace fullv.

Proof. by rewrite /is_aspace subvf has_algid1 ?memvf. Qed.

Lemma

aspacef_subproof

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "fullv", "has_algid1", "is_aspace", "memvf", "subvf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aspacef : aspace

:= ASpace aspacef_subproof.

Canonical

aspacef

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aspace", "aspacef_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver1P p : reflect (exists q, p = map_poly (in_alg aT) q) (p \is a polyOver 1%VS).

Proof. apply: (iffP idP) => [/allP/=Qp | [q ->]]; last first. by apply/polyOverP=> j; rewrite coef_map rpredZ ?memv_line. exists (map_poly (coord [tuple 1] 0) p). rewrite -map_poly_comp map_poly_id // => _ /Qp/vlineP[a ->] /=. by rewrite linearZ /= (coord_free 0) ?mulr1 // seq1_free ?oner_eq0. Qed.

Lemma

polyOver1P

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "allP", "apply", "coef_map", "coord", "coord_free", "in_alg", "last", "linearZ", "map_poly", "map_poly_comp", "map_poly_id", "memv_line", "mulr1", "oner_eq0", "polyOver", "polyOverP", "rpredZ", "seq1_free", "tuple", "vlineP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'aspace' T }"

:= (aspace T) : type_scope.

Notation

{ 'aspace' T }

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C_ U [ v ]"

:= (capv U 'C[v]) : vspace_scope.

Notation

'C_ U [ v ]

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "capv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C_ ( U ) [ v ]"

:= (capv U 'C[v]) (only parsing) : vspace_scope.

Notation

'C_ ( U ) [ v ]

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "capv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C_ U ( V )"

:= (capv U 'C(V)) : vspace_scope.

Notation

'C_ U ( V )

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "capv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C_ ( U ) ( V )"

:= (capv U 'C(V)) (only parsing) : vspace_scope.

Notation

'C_ ( U ) ( V )

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "capv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"1"

:= (aspace1 _) : aspace_scope.

Notation

1

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aspace1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ : aT }"

:= (aspacef aT) : aspace_scope.

Notation

{ : aT }

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "aT", "aspacef" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'aspace' 'of' U ]"

:= (@clone_aspace _ _ U _ _ id) (format "[ 'aspace' 'of' U ]") : form_scope.

Notation

[ 'aspace' 'of' U ]

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "clone_aspace", "id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'aspace' 'of' U 'for' A ]"

:= (@clone_aspace _ _ U A _ idfun) (format "[ 'aspace' 'of' U 'for' A ]") : form_scope.

Notation

[ 'aspace' 'of' U 'for' A ]

field

field/falgebra.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]

[ "clone_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d