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	source_url stringclasses 1 value	commit stringclasses 1 value
aimgM (f : ahom aT rT) U V : (f @: (U * V) = f @: U * f @: V)%VS.	Proof. apply/eqP; rewrite eqEsubv; apply/andP; split; last first. apply/prodvP=> _ _ /memv_imgP[u Hu ->] /memv_imgP[v Hv ->]. by rewrite -rmorphM memv_img // memv_mul. apply/subvP=> _ /memv_imgP[w UVw ->]; rewrite memv_preim (subvP _ w UVw) //. by apply/prodvP=> u v Uu Vv; rewrite -memv_preim rmorphM memv_mul // me...	Lemma	aimgM	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", "aT", "ahom", "apply", "eqEsubv", "last", "memv_img", "memv_imgP", "memv_mul", "memv_preim", "prodvP", "rmorphM", "split", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aimg1 (f : ahom aT rT) : (f @: 1 = 1)%VS.	Proof. by rewrite limg_line rmorph1. Qed.	Lemma	aimg1	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", "ahom", "limg_line", "rmorph1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aimgX (f : ahom aT rT) U n : (f @: (U ^+ n) = f @: U ^+ n)%VS.	Proof. elim: n => [\|n IH]; first by rewrite !expv0 aimg1. by rewrite !expvSl aimgM IH. Qed.	Lemma	aimgX	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", "ahom", "aimg1", "aimgM", "expv0", "expvSl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aimg_agen (f : ahom aT rT) U : (f @: agenv U)%VS = agenv (f @: U).	Proof. apply/eqP; rewrite eqEsubv; apply/andP; split. by rewrite limg_sum; apply/subv_sumP => i _; rewrite aimgX subX_agenv. apply: agenv_sub_modl; first by rewrite -(aimg1 f) limgS // sub1_agenv. by rewrite -aimgM limgS // [rhs in (_ <= rhs)%VS]agenvEl addvSr. Qed.	Lemma	aimg_agen	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", "addvSr", "agenv", "agenvEl", "agenv_sub_modl", "ahom", "aimg1", "aimgM", "aimgX", "apply", "eqEsubv", "limgS", "limg_sum", "rhs", "split", "sub1_agenv", "subX_agenv", "subv_sumP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aimg_adjoin (f : ahom aT rT) U x : (f @: <<U; x>> = <<f @: U; f x>>)%VS.	Proof. by rewrite aimg_agen limgD limg_line. Qed.	Lemma	aimg_adjoin	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", "ahom", "aimg_agen", "limgD", "limg_line" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aimg_adjoin_seq (f : ahom aT rT) U xs : (f @: <<U & xs>> = <<f @: U & map f xs>>)%VS.	Proof. by rewrite aimg_agen limgD limg_span. Qed.	Lemma	aimg_adjoin_seq	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", "ahom", "aimg_agen", "limgD", "limg_span", "map" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_sub_ahom_is_aspace (f g : ahom aT rT) : is_aspace (lker (ahval f - ahval g)).	Proof. rewrite /is_aspace has_algid1; first by apply/eqlfunP; rewrite !rmorph1. apply/prodvP=> a b /eqlfunP Dfa /eqlfunP Dfb. by apply/eqlfunP; rewrite !rmorphM /= Dfa Dfb. Qed.	Fact	ker_sub_ahom_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" ]	[ "aT", "ahom", "apply", "eqlfunP", "has_algid1", "is_aspace", "lker", "prodvP", "rmorph1", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_sub_ahom_aspace f g	:= ASpace (ker_sub_ahom_is_aspace f g).	Canonical	ker_sub_ahom_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" ]	[ "ker_sub_ahom_is_aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fixedSpace_aspace aT (f : ahom aT aT)	:= [aspace of fixedSpace f].	Canonical	fixedSpace_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", "ahom", "aspace", "fixedSpace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''AHom' ( aT , rT )"	:= (ahom aT rT) : type_scope.	Notation	''AHom' ( aT , rT )	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", "ahom" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''AEnd' ( aT )"	:= (ahom aT aT) : type_scope.	Notation	''AEnd' ( 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", "ahom" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\1"	:= (@id_ahom _ _) : lrfun_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" ]	[ "id_ahom" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f \o g"	:= (comp_ahom f g) : lrfun_scope.	Notation	f \o g	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" ]	[ "comp_ahom" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'subfield' L }"	:= (aspace L) (* NB: was (@aspace_of _ (FalgType _) (Phant L)) *) (format "{ 'subfield' L }") : type_scope.	Notation	{ 'subfield' L }	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_cosetv U x : x != 0 -> \dim (U * <[x]>) = \dim U.	Proof. move=> nz_x; rewrite -limg_amulr limg_dim_eq //. apply/eqP; rewrite -subv0; apply/subvP=> y. by rewrite memv_cap memv0 memv_ker lfunE mulf_eq0 (negPf nz_x) orbF => /andP[]. Qed.	Lemma	dim_cosetv	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "dim", "lfunE", "limg_amulr", "limg_dim_eq", "memv0", "memv_cap", "memv_ker", "mulf_eq0", "subv0", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvC : commutative (@prodv F0 L).	Proof. move=> U V; without loss suffices subC: U V / (U * V <= V * U)%VS. by apply/eqP; rewrite eqEsubv !{1}subC. by apply/prodvP=> x y Ux Vy; rewrite mulrC memv_mul. Qed.	Lemma	prodvC	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "F0", "apply", "eqEsubv", "memv_mul", "mulrC", "prodv", "prodvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvCA : left_commutative (@prodv F0 L).	Proof. exact: Monoid.mulmCA. Qed.	Lemma	prodvCA	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "F0", "mulmCA", "prodv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodvAC : right_commutative (@prodv F0 L).	Proof. exact: Monoid.mulmAC. Qed.	Lemma	prodvAC	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "F0", "mulmAC", "prodv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
algid1 K : algid K = 1.	Proof. exact/skew_field_algid1/fieldP. Qed.	Lemma	algid1	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "algid", "fieldP", "skew_field_algid1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem1v K : 1 \in K.	Proof. by rewrite -algid_eq1 algid1. Qed.	Lemma	mem1v	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "algid1", "algid_eq1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub1v K : (1 <= K)%VS.	Proof. exact: mem1v. Qed.	Lemma	sub1v	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "mem1v" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subfield_closed K : agenv K = K.	Proof. by apply/eqP; rewrite eqEsubv sub_agenv agenv_sub_modr ?sub1v ?asubv. Qed.	Lemma	subfield_closed	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "agenv", "agenv_sub_modr", "apply", "asubv", "eqEsubv", "sub1v", "sub_agenv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
AHom_lker0 (rT : falgType F0) (f : 'AHom(L, rT)) : lker f == 0%VS.	Proof. by apply/lker0P; apply: fmorph_inj. Qed.	Lemma	AHom_lker0	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "F0", "apply", "fmorph_inj", "lker", "lker0P" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
AEnd_lker0 (f : 'AEnd(L)) : lker f == 0%VS.	Proof. exact: AHom_lker0. Qed.	Lemma	AEnd_lker0	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "AHom_lker0", "lker" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aimg_is_aspace (rT : falgType F0) (f : 'AHom(L, rT)) (E : {subfield L}) : is_aspace (f @: E).	Proof. rewrite /is_aspace -aimgM limgS ?prodv_id // has_algid1 //. by apply/memv_imgP; exists 1; rewrite ?mem1v ?rmorph1. Qed.	Fact	aimg_is_aspace	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "F0", "aimgM", "apply", "has_algid1", "is_aspace", "limgS", "mem1v", "memv_imgP", "prodv_id", "rmorph1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aimg_aspace rT f E	:= ASpace (@aimg_is_aspace rT f E).	Canonical	aimg_aspace	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "aimg_is_aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_idP {K x} : reflect (<<K; x>>%VS = K) (x \in K).	Proof. apply: (iffP idP) => [/addv_idPl-> \| <-]; first exact: subfield_closed. exact: memv_adjoin. Qed.	Lemma	Fadjoin_idP	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "addv_idPl", "apply", "memv_adjoin", "subfield_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin0 K : <<K; 0>>%VS = K.	Proof. by rewrite addv0 subfield_closed. Qed.	Lemma	Fadjoin0	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "addv0", "subfield_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_nil K : <<K & [::]>>%VS = K.	Proof. by rewrite adjoin_nil subfield_closed. Qed.	Lemma	Fadjoin_nil	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "adjoin_nil", "subfield_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FadjoinP {K x E} : reflect (K <= E /\ x \in E)%VS (<<K; x>>%AS <= E)%VS.	Proof. apply: (iffP idP) => [sKxE \| /andP]. by rewrite (subvP sKxE) ?memv_adjoin // (subv_trans _ sKxE) ?subv_adjoin. by rewrite -subv_add => /agenvS; rewrite subfield_closed. Qed.	Lemma	FadjoinP	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "agenvS", "apply", "memv_adjoin", "subfield_closed", "subvP", "subv_add", "subv_adjoin", "subv_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_seqP {K} {rs : seq L} {E} : reflect (K <= E /\ {subset rs <= E})%VS (<<K & rs>> <= E)%VS.	Proof. apply: (iffP idP) => [sKrsE \| [sKE /span_subvP/(conj sKE)/andP]]. split=> [\|x rs_x]; first exact: subv_trans (subv_adjoin_seq _ _) sKrsE. by rewrite (subvP sKrsE) ?seqv_sub_adjoin. by rewrite -subv_add => /agenvS; rewrite subfield_closed. Qed.	Lemma	Fadjoin_seqP	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "agenvS", "apply", "conj", "sKE", "seq", "seqv_sub_adjoin", "span_subvP", "split", "subfield_closed", "subvP", "subv_add", "subv_adjoin_seq", "subv_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
alg_polyOver E p : map_poly (in_alg L) p \is a polyOver E.	Proof. by apply/(polyOverS (subvP (sub1v _)))/polyOver1P; exists p. Qed.	Lemma	alg_polyOver	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "in_alg", "map_poly", "polyOver", "polyOver1P", "polyOverS", "sub1v", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_adjoin1v x E : (<<1; x>> <= E)%VS = (x \in E)%VS.	Proof. by rewrite (sameP FadjoinP andP) sub1v. Qed.	Lemma	sub_adjoin1v	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "FadjoinP", "sub1v" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsval_monoid_morphism K : monoid_morphism (vsval : subvs_of K -> L).	Proof. by split => //=; apply: algid1. Qed.	Fact	vsval_monoid_morphism	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "algid1", "apply", "monoid_morphism", "split", "subvs_of", "vsval" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsval_is_multiplicative K	:= (fun g => (g.2,g.1)) (vsval_monoid_morphism K).	Definition	vsval_is_multiplicative	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "vsval_monoid_morphism" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsval_invf K (w : subvs_of K) : val w^-1 = (vsval w)^-1.	Proof. have [-> \| Uv] := eqVneq w 0; first by rewrite !invr0. by apply: vsval_invr; rewrite unitfE. Qed.	Lemma	vsval_invf	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "eqVneq", "invr0", "subvs_of", "unitfE", "val", "vsval", "vsval_invr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aspace_divr_closed K : divr_closed K.	Proof. by split=> [\|u v Ku Kv]; rewrite ?mem1v ?memvM ?memvV. Qed.	Fact	aspace_divr_closed	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "divr_closed", "mem1v", "memvM", "memvV", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_fieldMixin K : GRing.field_axiom (subvs_of K).	Proof. by move=> w nz_w; rewrite unitrE -val_eqE /= vsval_invf algid1 divff. Qed.	Lemma	subvs_fieldMixin	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "algid1", "divff", "field_axiom", "subvs_of", "unitrE", "val_eqE", "vsval_invf" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_subvs {K} {p : {poly L}} : reflect (exists q : {poly subvs_of K}, p = map_poly vsval q) (p \is a polyOver K).	Proof. apply: (iffP polyOverP) => [Hp \| [q ->] i]; last by rewrite coef_map // subvsP. exists (\poly_(i < size p) (Subvs (Hp i))); rewrite -{1}[p]coefK. by apply/polyP=> i; rewrite coef_map !coef_poly; case: ifP. Qed.	Lemma	polyOver_subvs	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "coefK", "coef_map", "coef_poly", "last", "map_poly", "poly", "polyOver", "polyOverP", "polyP", "size", "subvsP", "subvs_of", "vsval" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divp_polyOver K : {in polyOver K &, forall p q, p %/ q \is a polyOver K}.	Proof. move=> _ _ /polyOver_subvs[p ->] /polyOver_subvs[q ->]. by apply/polyOver_subvs; exists (p %/ q); rewrite map_divp. Qed.	Lemma	divp_polyOver	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "map_divp", "polyOver", "polyOver_subvs" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modp_polyOver K : {in polyOver K &, forall p q, p %% q \is a polyOver K}.	Proof. move=> _ _ /polyOver_subvs[p ->] /polyOver_subvs[q ->]. by apply/polyOver_subvs; exists (p %% q); rewrite map_modp. Qed.	Lemma	modp_polyOver	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "map_modp", "polyOver", "polyOver_subvs" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcdp_polyOver K : {in polyOver K &, forall p q, gcdp p q \is a polyOver K}.	Proof. move=> _ _ /polyOver_subvs[p ->] /polyOver_subvs[q ->]. by apply/polyOver_subvs; exists (gcdp p q); rewrite gcdp_map. Qed.	Lemma	gcdp_polyOver	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "gcdp", "gcdp_map", "polyOver", "polyOver_subvs" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv_is_aspace E F : is_aspace (E * F).	Proof. rewrite /is_aspace prodvCA -!prodvA prodvA !prodv_id has_algid1 //=. by rewrite -[1]mulr1 memv_mul ?mem1v. Qed.	Fact	prodv_is_aspace	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "has_algid1", "is_aspace", "mem1v", "memv_mul", "mulr1", "prodvA", "prodvCA", "prodv_id" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv_aspace E F : {subfield L}	:= ASpace (prodv_is_aspace E F).	Canonical	prodv_aspace	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "prodv_is_aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_mem_algid E F : algid E \in F.	Proof. by rewrite algid1 mem1v. Qed.	Fact	field_mem_algid	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "algid", "algid1", "mem1v" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
capv_aspace E F : {subfield L}	:= aspace_cap (field_mem_algid E F).	Canonical	capv_aspace	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "aspace_cap", "field_mem_algid" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverSv U V : (U <= V)%VS -> {subset polyOver U <= polyOver V}.	Proof. by move/subvP=> sUV; apply: polyOverS. Qed.	Lemma	polyOverSv	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "polyOver", "polyOverS", "subvP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_subvMl F U : (U <= F * U)%VS.	Proof. by rewrite -{1}[U]prod1v prodvSl ?sub1v. Qed.	Lemma	field_subvMl	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "prod1v", "prodvSl", "sub1v" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_subvMr U F : (U <= U * F)%VS.	Proof. by rewrite prodvC field_subvMl. Qed.	Lemma	field_subvMr	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "field_subvMl", "prodvC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_module_eq F M : (F * M <= M)%VS -> (F * M)%VS = M.	Proof. by move=> modM; apply/eqP; rewrite eqEsubv modM field_subvMl. Qed.	Lemma	field_module_eq	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "eqEsubv", "field_subvMl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sup_field_module F E : (F * E <= E)%VS = (F <= E)%VS.	Proof. apply/idP/idP; first exact: subv_trans (field_subvMr F E). by move/(prodvSl E)/subv_trans->; rewrite ?asubv. Qed.	Lemma	sup_field_module	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "asubv", "field_subvMr", "prodvSl", "subv_trans" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_module_dimS F M : (F * M <= M)%VS -> (\dim F %\| \dim M)%N.	Proof. exact/skew_field_module_dimS/fieldP. Qed.	Lemma	field_module_dimS	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "dim", "fieldP", "skew_field_module_dimS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_dimS F E : (F <= E)%VS -> (\dim F %\| \dim E)%N.	Proof. exact/skew_field_dimS/fieldP. Qed.	Lemma	field_dimS	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "dim", "fieldP", "skew_field_dimS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_field_module F M : (F * M <= M)%VS -> \dim M = (\dim_F M * \dim F)%N.	Proof. by move/field_module_dimS/divnK. Qed.	Lemma	dim_field_module	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "dim", "divnK", "field_module_dimS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_sup_field F E : (F <= E)%VS -> \dim E = (\dim_F E * \dim F)%N.	Proof. by move/field_dimS/divnK. Qed.	Lemma	dim_sup_field	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "dim", "divnK", "field_dimS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_module_semisimple F M (m := \dim_F M) : (F * M <= M)%VS -> {X : m.-tuple L \| {subset X <= M} /\ 0 \notin X & let FX := (\sum_(i < m) F * <[X`_i]>)%VS in FX = M /\ directv FX}.	Proof. move=> modM; have dimM: (m * \dim F)%N = \dim M by rewrite -dim_field_module. have [X [defM dxFX nzX]] := skew_field_module_semisimple (@fieldP L) modM. have szX: size X == m. rewrite -(eqn_pmul2r (adim_gt0 F)) dimM -defM (directvP dxFX) /=. rewrite -sum1_size big_distrl; apply/eqP/eq_big_seq => x Xx /=. b...	Lemma	field_module_semisimple	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "adim_gt0", "apply", "big_distrl", "big_mkord", "big_nth", "dim", "dim_cosetv", "dim_field_module", "directv", "directvE", "directvP", "eq_big_seq", "eqbRHS", "eqn_pmul2r", "fieldP", "field_subvMl", "memPn", "memvE", "mul1n", "size", "skew_field_module_semisimple", "split",...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_degree	:= (\dim_U <<U; x>>).-1.+1.	Definition	adjoin_degree	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
n	:= adjoin_degree.	Notation	n	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "adjoin_degree" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_sum	:= (\sum_(i < n) U * <[x ^+ i]>)%VS.	Definition	Fadjoin_sum	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_poly v : {poly L}	:= \poly_(i < n) (sumv_pi Fadjoin_sum (inord i) v / x ^+ i).	Definition	Fadjoin_poly	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_sum", "inord", "poly", "sumv_pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minPoly : {poly L}	:= 'X^n - Fadjoin_poly (x ^+ n).	Definition	minPoly	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "poly" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_Fadjoin_poly v : size (Fadjoin_poly v) <= n.	Proof. exact: size_poly. Qed.	Lemma	size_Fadjoin_poly	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "size", "size_poly" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_polyOver v : Fadjoin_poly v \is a polyOver U.	Proof. apply/(all_nthP 0) => i _; rewrite coef_poly /=. case: ifP => lti; last exact: mem0v. have /memv_cosetP[y Uy ->] := memv_sum_pi (erefl Fadjoin_sum) (inord i) v. rewrite inordK //; have [-> \| /mulfK-> //] := eqVneq (x ^+ i) 0. by rewrite mulr0 mul0r mem0v. Qed.	Lemma	Fadjoin_polyOver	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "Fadjoin_sum", "all_nthP", "apply", "coef_poly", "eqVneq", "inord", "inordK", "last", "mem0v", "memv_cosetP", "memv_sum_pi", "mul0r", "mulfK", "mulr0", "polyOver" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_poly_is_linear : linear_for (in_alg L \; *:%R) Fadjoin_poly.	Proof. move=> a u v; apply/polyP=> i; rewrite coefD coefZ !coef_poly. case: ifP => lti; last by rewrite mulr0 addr0. by rewrite linearP mulrA -mulrDl mulr_algl. Qed.	Fact	Fadjoin_poly_is_linear	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "addr0", "apply", "coefD", "coefZ", "coef_poly", "in_alg", "last", "linearP", "linear_for", "mulr0", "mulrA", "mulrDl", "mulr_algl", "polyP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_minPoly : size minPoly = n.+1.	Proof. by rewrite size_polyDl ?size_polyXn // size_polyN ltnS size_poly. Qed.	Lemma	size_minPoly	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "ltnS", "minPoly", "size", "size_poly", "size_polyDl", "size_polyN", "size_polyXn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_minPoly : minPoly \is monic.	Proof. rewrite monicE /lead_coef size_minPoly coefB coefXn eqxx. by rewrite nth_default ?subr0 ?size_poly. Qed.	Lemma	monic_minPoly	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "coefB", "coefXn", "eqxx", "lead_coef", "minPoly", "monic", "monicE", "nth_default", "size_minPoly", "size_poly", "subr0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
n	:= (adjoin_degree (asval K) x).	Notation	n	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "adjoin_degree" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumKx	:= (Fadjoin_sum (asval K) x).	Notation	sumKx	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_sum" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_degreeE : n = \dim_K <<K; x>>.	Proof. by rewrite [n]prednK // divn_gt0 ?adim_gt0 // dimvS ?subv_adjoin. Qed.	Lemma	adjoin_degreeE	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "adim_gt0", "dimvS", "divn_gt0", "prednK", "subv_adjoin" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_Fadjoin : \dim <<K; x>> = (n * \dim K)%N.	Proof. by rewrite adjoin_degreeE -dim_sup_field ?subv_adjoin. Qed.	Lemma	dim_Fadjoin	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "adjoin_degreeE", "dim", "dim_sup_field", "subv_adjoin" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin0_deg : adjoin_degree K 0 = 1.	Proof. by rewrite /adjoin_degree addv0 subfield_closed divnn adim_gt0. Qed.	Lemma	adjoin0_deg	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "addv0", "adim_gt0", "adjoin_degree", "divnn", "subfield_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_deg_eq1 : (n == 1) = (x \in K).	Proof. rewrite (sameP Fadjoin_idP eqP) adjoin_degreeE; have sK_Kx := subv_adjoin K x. apply/eqP/idP=> [dimKx1 \| /eqP->]; last by rewrite divnn adim_gt0. by rewrite eq_sym eqEdim sK_Kx /= (dim_sup_field sK_Kx) dimKx1 mul1n. Qed.	Lemma	adjoin_deg_eq1	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_idP", "adim_gt0", "adjoin_degreeE", "apply", "dim_sup_field", "divnn", "eqEdim", "eq_sym", "last", "mul1n", "subv_adjoin" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_sum_direct : directv sumKx.	Proof. rewrite directvE /=; case: (ubnPgeq n) (isT : n > 0) => -[//\|m] ltmn _. elim: m ltmn => [\|m IHm] ltm1n; rewrite ?big_ord1 // !(big_ord_recr m.+1) /=. do [move/(_ (ltnW ltm1n))/eqP; set S := (\sum_i _)%VS] in IHm *. rewrite -IHm dimv_add_leqif; apply/subvP=> z; rewrite memv_cap => /andP[Sz]. case/memv_cosetP=> y ...	Lemma	Fadjoin_sum_direct	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Sub", "adim_gt0", "agenv_sub_modl", "apply", "big_distrr", "big_ord1", "big_ord_recr", "card_ord", "dim", "dim_Fadjoin", "dim_cosetv", "dimvS", "dimv_add_leqif", "directv", "directvE", "eq_bigr", "eqn_leq", "expf_eq0", "expf_neq0", "expvSl", "expv_line", "leqNgt", "leq_o...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nz_x_i (i : 'I_n) : x ^+ i != 0.	Proof. by rewrite expf_eq0; case: eqP i => [->\|_] [[]] //; rewrite adjoin0_deg. Qed.	Let	nz_x_i	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "adjoin0_deg", "expf_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_eq_sum : <<K; x>>%VS = sumKx.	Proof. apply/esym/eqP; rewrite eqEdim eq_leq ?andbT. rewrite dim_Fadjoin -[n]card_ord -sum_nat_const (directvP Fadjoin_sum_direct). by apply: eq_bigr => i _; rewrite /= dim_cosetv. apply/subv_sumP=> i _; rewrite -agenvM prodvS ?subv_adjoin //. by rewrite -expv_line (subv_trans (subX_agenv _ _)) ?agenvS ?addvSr. Qed...	Lemma	Fadjoin_eq_sum	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_sum_direct", "addvSr", "agenvM", "agenvS", "apply", "card_ord", "dim_Fadjoin", "dim_cosetv", "directvP", "eqEdim", "eq_bigr", "eq_leq", "expv_line", "prodvS", "subX_agenv", "subv_adjoin", "subv_sumP", "subv_trans", "sumKx", "sum_nat_const" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_poly_eq v : v \in <<K; x>>%VS -> (Fadjoin_poly K x v).[x] = v.	Proof. move/(sumv_pi_sum Fadjoin_eq_sum)=> {2}<-; rewrite horner_poly. by apply: eq_bigr => i _; rewrite inord_val mulfVK. Qed.	Lemma	Fadjoin_poly_eq	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_eq_sum", "Fadjoin_poly", "apply", "eq_bigr", "horner_poly", "inord_val", "mulfVK", "sumv_pi_sum" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mempx_Fadjoin p : p \is a polyOver K -> p.[x] \in <<K; x>>%VS.	Proof. move=> Kp; rewrite rpred_horner ?memv_adjoin ?(polyOverS _ Kp) //. exact: subvP_adjoin. Qed.	Lemma	mempx_Fadjoin	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "memv_adjoin", "polyOver", "polyOverS", "rpred_horner", "subvP_adjoin" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_polyP {v} : reflect (exists2 p, p \in polyOver K & v = p.[x]) (v \in <<K; x>>%VS).	Proof. apply: (iffP idP) => [Kx_v \| [p Kp ->]]; last exact: mempx_Fadjoin. by exists (Fadjoin_poly K x v); rewrite ?Fadjoin_polyOver ?Fadjoin_poly_eq. Qed.	Lemma	Fadjoin_polyP	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "Fadjoin_polyOver", "Fadjoin_poly_eq", "apply", "last", "mempx_Fadjoin", "polyOver" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_poly_unique p v : p \is a polyOver K -> size p <= n -> p.[x] = v -> Fadjoin_poly K x v = p.	Proof. have polyKx q i: q \is a polyOver K -> q`_i * x ^+ i \in (K * <[x ^+ i]>)%VS. by move/polyOverP=> Kq; rewrite memv_mul ?Kq ?memv_line. move=> Kp szp Dv; have /Fadjoin_poly_eq/eqP := mempx_Fadjoin Kp. rewrite {1}Dv {Dv} !(@horner_coef_wide _ n) ?size_poly //. move/polyKx in Kp; have /polyKx K_pv := Fadjoin_poly...	Lemma	Fadjoin_poly_unique	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "Fadjoin_polyOver", "Fadjoin_poly_eq", "Fadjoin_sum_direct", "Sub", "apply", "directv_sum_unique", "eqfunP", "horner_coef_wide", "last", "leqP", "leq_trans", "mempx_Fadjoin", "memv_line", "memv_mul", "mulIf", "nth_default", "polyOver", "polyOverP", "polyP", "s...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_polyC v : v \in K -> Fadjoin_poly K x v = v%:P.	Proof. move=> Kv; apply: Fadjoin_poly_unique; rewrite ?polyOverC ?hornerC //. by rewrite size_polyC (leq_trans (leq_b1 _)). Qed.	Lemma	Fadjoin_polyC	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "Fadjoin_poly_unique", "apply", "hornerC", "leq_b1", "leq_trans", "polyOverC", "size_polyC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_polyX : x \notin K -> Fadjoin_poly K x x = 'X.	Proof. move=> K'x; apply: Fadjoin_poly_unique; rewrite ?polyOverX ?hornerX //. by rewrite size_polyX ltn_neqAle andbT eq_sym adjoin_deg_eq1. Qed.	Lemma	Fadjoin_polyX	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "Fadjoin_poly_unique", "adjoin_deg_eq1", "apply", "eq_sym", "hornerX", "ltn_neqAle", "polyOverX", "size_polyX" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minPolyOver : minPoly K x \is a polyOver K.	Proof. by rewrite /minPoly rpredB ?rpredX ?polyOverX ?Fadjoin_polyOver. Qed.	Lemma	minPolyOver	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_polyOver", "minPoly", "polyOver", "polyOverX", "rpredB", "rpredX" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minPolyxx : (minPoly K x).[x] = 0.	Proof. by rewrite !hornerE Fadjoin_poly_eq ?subrr ?rpredX ?memv_adjoin. Qed.	Lemma	minPolyxx	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly_eq", "hornerE", "memv_adjoin", "minPoly", "rpredX", "subrr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_minPoly : root (minPoly K x) x.	Proof. exact/rootP/minPolyxx. Qed.	Lemma	root_minPoly	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "minPoly", "minPolyxx", "root", "rootP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin_poly_mod p : p \is a polyOver K -> Fadjoin_poly K x p.[x] = p %% minPoly K x.	Proof. move=> Kp; rewrite {1}(divp_eq p (minPoly K x)) 2!hornerE minPolyxx mulr0 add0r. apply: Fadjoin_poly_unique => //; first by rewrite modp_polyOver // minPolyOver. by rewrite -ltnS -size_minPoly ltn_modp // monic_neq0 ?monic_minPoly. Qed.	Lemma	Fadjoin_poly_mod	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly", "Fadjoin_poly_unique", "add0r", "apply", "divp_eq", "hornerE", "ltnS", "ltn_modp", "minPoly", "minPolyOver", "minPolyxx", "modp_polyOver", "monic_minPoly", "monic_neq0", "mulr0", "polyOver", "size_minPoly" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minPoly_XsubC : reflect (minPoly K x = 'X - x%:P) (x \in K).	Proof. set p := minPoly K x; apply: (iffP idP) => [Kx \| Dp]; last first. suffices ->: x = - p`_0 by rewrite rpredN (polyOverP minPolyOver). by rewrite Dp coefB coefX coefC add0r opprK. rewrite (@all_roots_prod_XsubC _ p [:: x]) /= ?root_minPoly //. by apply/eqP; rewrite size_minPoly eqSS adjoin_deg_eq1. by rewrit...	Lemma	minPoly_XsubC	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "add0r", "adjoin_deg_eq1", "all_roots_prod_XsubC", "apply", "big_seq1", "coefB", "coefC", "coefX", "eqSS", "last", "minPoly", "minPolyOver", "monicP", "monic_minPoly", "opprK", "polyOverP", "root_minPoly", "rpredN", "scale1r", "size_minPoly" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_small_adjoin_poly p : p \is a polyOver K -> size p <= n -> root p x = (p == 0).	Proof. move=> Kp szp; apply/rootP/eqP=> [px0 \| ->]; last by rewrite horner0. rewrite -(Fadjoin_poly_unique Kp szp px0). by apply: Fadjoin_poly_unique; rewrite ?polyOver0 ?size_poly0 ?horner0. Qed.	Lemma	root_small_adjoin_poly	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_poly_unique", "apply", "horner0", "last", "polyOver", "polyOver0", "root", "rootP", "size", "size_poly0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minPoly_irr p : p \is a polyOver K -> p %\| minPoly K x -> (p %= minPoly K x) \|\| (p %= 1).	Proof. rewrite dvdp_eq; set q := _ %/ _ => Kp def_pq. have Kq: q \is a polyOver K by rewrite divp_polyOver // minPolyOver. move: q Kq def_pq root_minPoly (size_minPoly K x) => q Kq /eqP->. rewrite rootM => pqx0 szpq. have [nzq nzp]: q != 0 /\ p != 0. by apply/norP; rewrite -mulf_eq0 -size_poly_eq0 szpq. without loss{...	Lemma	minPoly_irr	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "divp_polyOver", "dvdp_eq", "eqn_leq", "eqp_mul2r", "eqp_sym", "leq_add2r", "leq_subLR", "lt0n", "ltnNge", "minPoly", "minPolyOver", "mul1r", "mulf_eq0", "mulrC", "polyOver", "root", "rootM", "root_minPoly", "root_small_adjoin_poly", "size", "size_minPoly", "size...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minPoly_dvdp p : p \is a polyOver K -> root p x -> (minPoly K x) %\| p.	Proof. move=> Kp rootp. have gcdK : gcdp (minPoly K x) p \is a polyOver K. by rewrite gcdp_polyOver ?minPolyOver. have /orP[gcd_eqK\|gcd_eq1] := minPoly_irr gcdK (dvdp_gcdl (minPoly K x) p). by rewrite -(eqp_dvdl _ gcd_eqK) dvdp_gcdr. case/negP: (root1 x). by rewrite -(eqp_root gcd_eq1) root_gcd rootp root_minPoly. ...	Lemma	minPoly_dvdp	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "dvdp_gcdl", "dvdp_gcdr", "eqp_dvdl", "eqp_root", "gcdp", "gcdp_polyOver", "minPoly", "minPolyOver", "minPoly_irr", "polyOver", "root", "root1", "root_gcd", "root_minPoly" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
minPolyS K E a : (K <= E)%VS -> minPoly E a %\| minPoly K a.	Proof. move=> sKE; apply: minPoly_dvdp; last exact: root_minPoly. by apply: (polyOverSv sKE); rewrite minPolyOver. Qed.	Lemma	minPolyS	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "apply", "last", "minPoly", "minPolyOver", "minPoly_dvdp", "polyOverSv", "root_minPoly", "sKE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fadjoin1_polyP x v : reflect (exists p, v = (map_poly (in_alg L) p).[x]) (v \in <<1; x>>%VS).	Proof. apply: (iffP Fadjoin_polyP) => [[_ /polyOver1P]\|] [p ->]; first by exists p. by exists (map_poly (in_alg L) p) => //; apply: alg_polyOver. Qed.	Lemma	Fadjoin1_polyP	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "Fadjoin_polyP", "alg_polyOver", "apply", "in_alg", "map_poly", "polyOver1P" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fieldExt_horner	:= horner_morph (fun x => mulrC z (in_alg L x)).	Definition	fieldExt_horner	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "horner_morph", "in_alg", "mulrC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fieldExt_hornerC b : fieldExt_horner b%:P = b%:A.	Proof. exact: horner_morphC. Qed.	Lemma	fieldExt_hornerC	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "fieldExt_horner", "horner_morphC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fieldExt_hornerX : fieldExt_horner 'X = z.	Proof. exact: horner_morphX. Qed.	Lemma	fieldExt_hornerX	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "fieldExt_horner", "horner_morphX" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fieldExt_hornerZ : scalable fieldExt_horner.	Proof. move=> a p; rewrite -mul_polyC rmorphM /= fieldExt_hornerC. by rewrite -scalerAl mul1r. Qed.	Fact	fieldExt_hornerZ	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "fieldExt_horner", "fieldExt_hornerC", "mul1r", "mul_polyC", "rmorphM", "scalable", "scalerAl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"E :&: F"	:= (capv_aspace E F) : aspace_scope.	Notation	E :&: F	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "capv_aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C_ E [ x ]"	:= (capv_aspace E 'C[x]) : aspace_scope.	Notation	'C_ E [ x ]	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "capv_aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C_ ( E ) [ x ]"	:= (capv_aspace E 'C[x]) (only parsing) : aspace_scope.	Notation	'C_ ( E ) [ x ]	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "capv_aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C_ E ( V )"	:= (capv_aspace E 'C(V)) : aspace_scope.	Notation	'C_ E ( V )	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "capv_aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C_ ( E ) ( V )"	:= (capv_aspace E 'C(V)) (only parsing) : aspace_scope.	Notation	'C_ ( E ) ( V )	field	field/fieldext.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...	[ "capv_aspace" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

aimgM (f : ahom aT rT) U V : (f @: (U * V) = f @: U * f @: V)%VS.

Proof. apply/eqP; rewrite eqEsubv; apply/andP; split; last first. apply/prodvP=> _ _ /memv_imgP[u Hu ->] /memv_imgP[v Hv ->]. by rewrite -rmorphM memv_img // memv_mul. apply/subvP=> _ /memv_imgP[w UVw ->]; rewrite memv_preim (subvP _ w UVw) //. by apply/prodvP=> u v Uu Vv; rewrite -memv_preim rmorphM memv_mul // me...

Lemma

aimgM

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", "aT", "ahom", "apply", "eqEsubv", "last", "memv_img", "memv_imgP", "memv_mul", "memv_preim", "prodvP", "rmorphM", "split", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aimg1 (f : ahom aT rT) : (f @: 1 = 1)%VS.

Proof. by rewrite limg_line rmorph1. Qed.

Lemma

aimg1

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", "ahom", "limg_line", "rmorph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aimgX (f : ahom aT rT) U n : (f @: (U ^+ n) = f @: U ^+ n)%VS.

Proof. elim: n => [|n IH]; first by rewrite !expv0 aimg1. by rewrite !expvSl aimgM IH. Qed.

Lemma

aimgX

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", "ahom", "aimg1", "aimgM", "expv0", "expvSl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aimg_agen (f : ahom aT rT) U : (f @: agenv U)%VS = agenv (f @: U).

Proof. apply/eqP; rewrite eqEsubv; apply/andP; split. by rewrite limg_sum; apply/subv_sumP => i _; rewrite aimgX subX_agenv. apply: agenv_sub_modl; first by rewrite -(aimg1 f) limgS // sub1_agenv. by rewrite -aimgM limgS // [rhs in (_ <= rhs)%VS]agenvEl addvSr. Qed.

Lemma

aimg_agen

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", "addvSr", "agenv", "agenvEl", "agenv_sub_modl", "ahom", "aimg1", "aimgM", "aimgX", "apply", "eqEsubv", "limgS", "limg_sum", "rhs", "split", "sub1_agenv", "subX_agenv", "subv_sumP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aimg_adjoin (f : ahom aT rT) U x : (f @: <<U; x>> = <<f @: U; f x>>)%VS.

Proof. by rewrite aimg_agen limgD limg_line. Qed.

Lemma

aimg_adjoin

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", "ahom", "aimg_agen", "limgD", "limg_line" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aimg_adjoin_seq (f : ahom aT rT) U xs : (f @: <<U & xs>> = <<f @: U & map f xs>>)%VS.

Proof. by rewrite aimg_agen limgD limg_span. Qed.

Lemma

aimg_adjoin_seq

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", "ahom", "aimg_agen", "limgD", "limg_span", "map" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_sub_ahom_is_aspace (f g : ahom aT rT) : is_aspace (lker (ahval f - ahval g)).

Proof. rewrite /is_aspace has_algid1; first by apply/eqlfunP; rewrite !rmorph1. apply/prodvP=> a b /eqlfunP Dfa /eqlfunP Dfb. by apply/eqlfunP; rewrite !rmorphM /= Dfa Dfb. Qed.

Fact

ker_sub_ahom_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" ]

[ "aT", "ahom", "apply", "eqlfunP", "has_algid1", "is_aspace", "lker", "prodvP", "rmorph1", "rmorphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_sub_ahom_aspace f g

:= ASpace (ker_sub_ahom_is_aspace f g).

Canonical

ker_sub_ahom_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" ]

[ "ker_sub_ahom_is_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fixedSpace_aspace aT (f : ahom aT aT)

:= [aspace of fixedSpace f].

Canonical

fixedSpace_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", "ahom", "aspace", "fixedSpace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''AHom' ( aT , rT )"

:= (ahom aT rT) : type_scope.

Notation

''AHom' ( aT , rT )

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", "ahom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''AEnd' ( aT )"

:= (ahom aT aT) : type_scope.

Notation

''AEnd' ( 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", "ahom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\1"

:= (@id_ahom _ _) : lrfun_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" ]

[ "id_ahom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f \o g"

:= (comp_ahom f g) : lrfun_scope.

Notation

f \o g

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" ]

[ "comp_ahom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'subfield' L }"

:= (aspace L) (* NB: was (@aspace_of _ (FalgType _) (Phant L)) *) (format "{ 'subfield' L }") : type_scope.

Notation

{ 'subfield' L }

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim_cosetv U x : x != 0 -> \dim (U * <[x]>) = \dim U.

Proof. move=> nz_x; rewrite -limg_amulr limg_dim_eq //. apply/eqP; rewrite -subv0; apply/subvP=> y. by rewrite memv_cap memv0 memv_ker lfunE mulf_eq0 (negPf nz_x) orbF => /andP[]. Qed.

Lemma

dim_cosetv

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "dim", "lfunE", "limg_amulr", "limg_dim_eq", "memv0", "memv_cap", "memv_ker", "mulf_eq0", "subv0", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvC : commutative (@prodv F0 L).

Proof. move=> U V; without loss suffices subC: U V / (U * V <= V * U)%VS. by apply/eqP; rewrite eqEsubv !{1}subC. by apply/prodvP=> x y Ux Vy; rewrite mulrC memv_mul. Qed.

Lemma

prodvC

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "F0", "apply", "eqEsubv", "memv_mul", "mulrC", "prodv", "prodvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvCA : left_commutative (@prodv F0 L).

Proof. exact: Monoid.mulmCA. Qed.

Lemma

prodvCA

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "F0", "mulmCA", "prodv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodvAC : right_commutative (@prodv F0 L).

Proof. exact: Monoid.mulmAC. Qed.

Lemma

prodvAC

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "F0", "mulmAC", "prodv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

algid1 K : algid K = 1.

Proof. exact/skew_field_algid1/fieldP. Qed.

Lemma

algid1

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "algid", "fieldP", "skew_field_algid1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem1v K : 1 \in K.

Proof. by rewrite -algid_eq1 algid1. Qed.

Lemma

mem1v

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "algid1", "algid_eq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub1v K : (1 <= K)%VS.

Proof. exact: mem1v. Qed.

Lemma

sub1v

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "mem1v" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subfield_closed K : agenv K = K.

Proof. by apply/eqP; rewrite eqEsubv sub_agenv agenv_sub_modr ?sub1v ?asubv. Qed.

Lemma

subfield_closed

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "agenv", "agenv_sub_modr", "apply", "asubv", "eqEsubv", "sub1v", "sub_agenv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

AHom_lker0 (rT : falgType F0) (f : 'AHom(L, rT)) : lker f == 0%VS.

Proof. by apply/lker0P; apply: fmorph_inj. Qed.

Lemma

AHom_lker0

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "F0", "apply", "fmorph_inj", "lker", "lker0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

AEnd_lker0 (f : 'AEnd(L)) : lker f == 0%VS.

Proof. exact: AHom_lker0. Qed.

Lemma

AEnd_lker0

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "AHom_lker0", "lker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aimg_is_aspace (rT : falgType F0) (f : 'AHom(L, rT)) (E : {subfield L}) : is_aspace (f @: E).

Proof. rewrite /is_aspace -aimgM limgS ?prodv_id // has_algid1 //. by apply/memv_imgP; exists 1; rewrite ?mem1v ?rmorph1. Qed.

Fact

aimg_is_aspace

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "F0", "aimgM", "apply", "has_algid1", "is_aspace", "limgS", "mem1v", "memv_imgP", "prodv_id", "rmorph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aimg_aspace rT f E

:= ASpace (@aimg_is_aspace rT f E).

Canonical

aimg_aspace

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "aimg_is_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_idP {K x} : reflect (<<K; x>>%VS = K) (x \in K).

Proof. apply: (iffP idP) => [/addv_idPl-> | <-]; first exact: subfield_closed. exact: memv_adjoin. Qed.

Lemma

Fadjoin_idP

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "addv_idPl", "apply", "memv_adjoin", "subfield_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin0 K : <<K; 0>>%VS = K.

Proof. by rewrite addv0 subfield_closed. Qed.

Lemma

Fadjoin0

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "addv0", "subfield_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_nil K : <<K & [::]>>%VS = K.

Proof. by rewrite adjoin_nil subfield_closed. Qed.

Lemma

Fadjoin_nil

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "adjoin_nil", "subfield_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FadjoinP {K x E} : reflect (K <= E /\ x \in E)%VS (<<K; x>>%AS <= E)%VS.

Proof. apply: (iffP idP) => [sKxE | /andP]. by rewrite (subvP sKxE) ?memv_adjoin // (subv_trans _ sKxE) ?subv_adjoin. by rewrite -subv_add => /agenvS; rewrite subfield_closed. Qed.

Lemma

FadjoinP

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "agenvS", "apply", "memv_adjoin", "subfield_closed", "subvP", "subv_add", "subv_adjoin", "subv_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_seqP {K} {rs : seq L} {E} : reflect (K <= E /\ {subset rs <= E})%VS (<<K & rs>> <= E)%VS.

Proof. apply: (iffP idP) => [sKrsE | [sKE /span_subvP/(conj sKE)/andP]]. split=> [|x rs_x]; first exact: subv_trans (subv_adjoin_seq _ _) sKrsE. by rewrite (subvP sKrsE) ?seqv_sub_adjoin. by rewrite -subv_add => /agenvS; rewrite subfield_closed. Qed.

Lemma

Fadjoin_seqP

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "agenvS", "apply", "conj", "sKE", "seq", "seqv_sub_adjoin", "span_subvP", "split", "subfield_closed", "subvP", "subv_add", "subv_adjoin_seq", "subv_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

alg_polyOver E p : map_poly (in_alg L) p \is a polyOver E.

Proof. by apply/(polyOverS (subvP (sub1v _)))/polyOver1P; exists p. Qed.

Lemma

alg_polyOver

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "in_alg", "map_poly", "polyOver", "polyOver1P", "polyOverS", "sub1v", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_adjoin1v x E : (<<1; x>> <= E)%VS = (x \in E)%VS.

Proof. by rewrite (sameP FadjoinP andP) sub1v. Qed.

Lemma

sub_adjoin1v

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "FadjoinP", "sub1v" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsval_monoid_morphism K : monoid_morphism (vsval : subvs_of K -> L).

Proof. by split => //=; apply: algid1. Qed.

Fact

vsval_monoid_morphism

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "algid1", "apply", "monoid_morphism", "split", "subvs_of", "vsval" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsval_is_multiplicative K

:= (fun g => (g.2,g.1)) (vsval_monoid_morphism K).

Definition

vsval_is_multiplicative

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "vsval_monoid_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsval_invf K (w : subvs_of K) : val w^-1 = (vsval w)^-1.

Proof. have [-> | Uv] := eqVneq w 0; first by rewrite !invr0. by apply: vsval_invr; rewrite unitfE. Qed.

Lemma

vsval_invf

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "eqVneq", "invr0", "subvs_of", "unitfE", "val", "vsval", "vsval_invr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aspace_divr_closed K : divr_closed K.

Proof. by split=> [|u v Ku Kv]; rewrite ?mem1v ?memvM ?memvV. Qed.

Fact

aspace_divr_closed

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "divr_closed", "mem1v", "memvM", "memvV", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subvs_fieldMixin K : GRing.field_axiom (subvs_of K).

Proof. by move=> w nz_w; rewrite unitrE -val_eqE /= vsval_invf algid1 divff. Qed.

Lemma

subvs_fieldMixin

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "algid1", "divff", "field_axiom", "subvs_of", "unitrE", "val_eqE", "vsval_invf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_subvs {K} {p : {poly L}} : reflect (exists q : {poly subvs_of K}, p = map_poly vsval q) (p \is a polyOver K).

Proof. apply: (iffP polyOverP) => [Hp | [q ->] i]; last by rewrite coef_map // subvsP. exists (\poly_(i < size p) (Subvs (Hp i))); rewrite -{1}[p]coefK. by apply/polyP=> i; rewrite coef_map !coef_poly; case: ifP. Qed.

Lemma

polyOver_subvs

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "coefK", "coef_map", "coef_poly", "last", "map_poly", "poly", "polyOver", "polyOverP", "polyP", "size", "subvsP", "subvs_of", "vsval" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divp_polyOver K : {in polyOver K &, forall p q, p %/ q \is a polyOver K}.

Proof. move=> _ _ /polyOver_subvs[p ->] /polyOver_subvs[q ->]. by apply/polyOver_subvs; exists (p %/ q); rewrite map_divp. Qed.

Lemma

divp_polyOver

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "map_divp", "polyOver", "polyOver_subvs" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modp_polyOver K : {in polyOver K &, forall p q, p %% q \is a polyOver K}.

Proof. move=> _ _ /polyOver_subvs[p ->] /polyOver_subvs[q ->]. by apply/polyOver_subvs; exists (p %% q); rewrite map_modp. Qed.

Lemma

modp_polyOver

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "map_modp", "polyOver", "polyOver_subvs" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcdp_polyOver K : {in polyOver K &, forall p q, gcdp p q \is a polyOver K}.

Proof. move=> _ _ /polyOver_subvs[p ->] /polyOver_subvs[q ->]. by apply/polyOver_subvs; exists (gcdp p q); rewrite gcdp_map. Qed.

Lemma

gcdp_polyOver

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "gcdp", "gcdp_map", "polyOver", "polyOver_subvs" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodv_is_aspace E F : is_aspace (E * F).

Proof. rewrite /is_aspace prodvCA -!prodvA prodvA !prodv_id has_algid1 //=. by rewrite -[1]mulr1 memv_mul ?mem1v. Qed.

Fact

prodv_is_aspace

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "has_algid1", "is_aspace", "mem1v", "memv_mul", "mulr1", "prodvA", "prodvCA", "prodv_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodv_aspace E F : {subfield L}

:= ASpace (prodv_is_aspace E F).

Canonical

prodv_aspace

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "prodv_is_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_mem_algid E F : algid E \in F.

Proof. by rewrite algid1 mem1v. Qed.

Fact

field_mem_algid

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "algid", "algid1", "mem1v" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

capv_aspace E F : {subfield L}

:= aspace_cap (field_mem_algid E F).

Canonical

capv_aspace

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "aspace_cap", "field_mem_algid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverSv U V : (U <= V)%VS -> {subset polyOver U <= polyOver V}.

Proof. by move/subvP=> sUV; apply: polyOverS. Qed.

Lemma

polyOverSv

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "polyOver", "polyOverS", "subvP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_subvMl F U : (U <= F * U)%VS.

Proof. by rewrite -{1}[U]prod1v prodvSl ?sub1v. Qed.

Lemma

field_subvMl

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "prod1v", "prodvSl", "sub1v" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_subvMr U F : (U <= U * F)%VS.

Proof. by rewrite prodvC field_subvMl. Qed.

Lemma

field_subvMr

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "field_subvMl", "prodvC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_module_eq F M : (F * M <= M)%VS -> (F * M)%VS = M.

Proof. by move=> modM; apply/eqP; rewrite eqEsubv modM field_subvMl. Qed.

Lemma

field_module_eq

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "eqEsubv", "field_subvMl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sup_field_module F E : (F * E <= E)%VS = (F <= E)%VS.

Proof. apply/idP/idP; first exact: subv_trans (field_subvMr F E). by move/(prodvSl E)/subv_trans->; rewrite ?asubv. Qed.

Lemma

sup_field_module

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "asubv", "field_subvMr", "prodvSl", "subv_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_module_dimS F M : (F * M <= M)%VS -> (\dim F %| \dim M)%N.

Proof. exact/skew_field_module_dimS/fieldP. Qed.

Lemma

field_module_dimS

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "dim", "fieldP", "skew_field_module_dimS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_dimS F E : (F <= E)%VS -> (\dim F %| \dim E)%N.

Proof. exact/skew_field_dimS/fieldP. Qed.

Lemma

field_dimS

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "dim", "fieldP", "skew_field_dimS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim_field_module F M : (F * M <= M)%VS -> \dim M = (\dim_F M * \dim F)%N.

Proof. by move/field_module_dimS/divnK. Qed.

Lemma

dim_field_module

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "dim", "divnK", "field_module_dimS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim_sup_field F E : (F <= E)%VS -> \dim E = (\dim_F E * \dim F)%N.

Proof. by move/field_dimS/divnK. Qed.

Lemma

dim_sup_field

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "dim", "divnK", "field_dimS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_module_semisimple F M (m := \dim_F M) : (F * M <= M)%VS -> {X : m.-tuple L | {subset X <= M} /\ 0 \notin X & let FX := (\sum_(i < m) F * <[X`_i]>)%VS in FX = M /\ directv FX}.

Proof. move=> modM; have dimM: (m * \dim F)%N = \dim M by rewrite -dim_field_module. have [X [defM dxFX nzX]] := skew_field_module_semisimple (@fieldP L) modM. have szX: size X == m. rewrite -(eqn_pmul2r (adim_gt0 F)) dimM -defM (directvP dxFX) /=. rewrite -sum1_size big_distrl; apply/eqP/eq_big_seq => x Xx /=. b...

Lemma

field_module_semisimple

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "adim_gt0", "apply", "big_distrl", "big_mkord", "big_nth", "dim", "dim_cosetv", "dim_field_module", "directv", "directvE", "directvP", "eq_big_seq", "eqbRHS", "eqn_pmul2r", "fieldP", "field_subvMl", "memPn", "memvE", "mul1n", "size", "skew_field_module_semisimple", "split",...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

adjoin_degree

:= (\dim_U <<U; x>>).-1.+1.

Definition

adjoin_degree

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

n

:= adjoin_degree.

Notation

n

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "adjoin_degree" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_sum

:= (\sum_(i < n) U * <[x ^+ i]>)%VS.

Definition

Fadjoin_sum

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_poly v : {poly L}

:= \poly_(i < n) (sumv_pi Fadjoin_sum (inord i) v / x ^+ i).

Definition

Fadjoin_poly

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_sum", "inord", "poly", "sumv_pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minPoly : {poly L}

:= 'X^n - Fadjoin_poly (x ^+ n).

Definition

minPoly

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_Fadjoin_poly v : size (Fadjoin_poly v) <= n.

Proof. exact: size_poly. Qed.

Lemma

size_Fadjoin_poly

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "size", "size_poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_polyOver v : Fadjoin_poly v \is a polyOver U.

Proof. apply/(all_nthP 0) => i _; rewrite coef_poly /=. case: ifP => lti; last exact: mem0v. have /memv_cosetP[y Uy ->] := memv_sum_pi (erefl Fadjoin_sum) (inord i) v. rewrite inordK //; have [-> | /mulfK-> //] := eqVneq (x ^+ i) 0. by rewrite mulr0 mul0r mem0v. Qed.

Lemma

Fadjoin_polyOver

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "Fadjoin_sum", "all_nthP", "apply", "coef_poly", "eqVneq", "inord", "inordK", "last", "mem0v", "memv_cosetP", "memv_sum_pi", "mul0r", "mulfK", "mulr0", "polyOver" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_poly_is_linear : linear_for (in_alg L \; *:%R) Fadjoin_poly.

Proof. move=> a u v; apply/polyP=> i; rewrite coefD coefZ !coef_poly. case: ifP => lti; last by rewrite mulr0 addr0. by rewrite linearP mulrA -mulrDl mulr_algl. Qed.

Fact

Fadjoin_poly_is_linear

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "addr0", "apply", "coefD", "coefZ", "coef_poly", "in_alg", "last", "linearP", "linear_for", "mulr0", "mulrA", "mulrDl", "mulr_algl", "polyP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_minPoly : size minPoly = n.+1.

Proof. by rewrite size_polyDl ?size_polyXn // size_polyN ltnS size_poly. Qed.

Lemma

size_minPoly

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "ltnS", "minPoly", "size", "size_poly", "size_polyDl", "size_polyN", "size_polyXn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_minPoly : minPoly \is monic.

Proof. rewrite monicE /lead_coef size_minPoly coefB coefXn eqxx. by rewrite nth_default ?subr0 ?size_poly. Qed.

Lemma

monic_minPoly

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "coefB", "coefXn", "eqxx", "lead_coef", "minPoly", "monic", "monicE", "nth_default", "size_minPoly", "size_poly", "subr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

n

:= (adjoin_degree (asval K) x).

Notation

n

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "adjoin_degree" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumKx

:= (Fadjoin_sum (asval K) x).

Notation

sumKx

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_sum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

adjoin_degreeE : n = \dim_K <<K; x>>.

Proof. by rewrite [n]prednK // divn_gt0 ?adim_gt0 // dimvS ?subv_adjoin. Qed.

Lemma

adjoin_degreeE

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "adim_gt0", "dimvS", "divn_gt0", "prednK", "subv_adjoin" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim_Fadjoin : \dim <<K; x>> = (n * \dim K)%N.

Proof. by rewrite adjoin_degreeE -dim_sup_field ?subv_adjoin. Qed.

Lemma

dim_Fadjoin

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "adjoin_degreeE", "dim", "dim_sup_field", "subv_adjoin" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

adjoin0_deg : adjoin_degree K 0 = 1.

Proof. by rewrite /adjoin_degree addv0 subfield_closed divnn adim_gt0. Qed.

Lemma

adjoin0_deg

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "addv0", "adim_gt0", "adjoin_degree", "divnn", "subfield_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

adjoin_deg_eq1 : (n == 1) = (x \in K).

Proof. rewrite (sameP Fadjoin_idP eqP) adjoin_degreeE; have sK_Kx := subv_adjoin K x. apply/eqP/idP=> [dimKx1 | /eqP->]; last by rewrite divnn adim_gt0. by rewrite eq_sym eqEdim sK_Kx /= (dim_sup_field sK_Kx) dimKx1 mul1n. Qed.

Lemma

adjoin_deg_eq1

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_idP", "adim_gt0", "adjoin_degreeE", "apply", "dim_sup_field", "divnn", "eqEdim", "eq_sym", "last", "mul1n", "subv_adjoin" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_sum_direct : directv sumKx.

Proof. rewrite directvE /=; case: (ubnPgeq n) (isT : n > 0) => -[//|m] ltmn _. elim: m ltmn => [|m IHm] ltm1n; rewrite ?big_ord1 // !(big_ord_recr m.+1) /=. do [move/(_ (ltnW ltm1n))/eqP; set S := (\sum_i _)%VS] in IHm *. rewrite -IHm dimv_add_leqif; apply/subvP=> z; rewrite memv_cap => /andP[Sz]. case/memv_cosetP=> y ...

Lemma

Fadjoin_sum_direct

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Sub", "adim_gt0", "agenv_sub_modl", "apply", "big_distrr", "big_ord1", "big_ord_recr", "card_ord", "dim", "dim_Fadjoin", "dim_cosetv", "dimvS", "dimv_add_leqif", "directv", "directvE", "eq_bigr", "eqn_leq", "expf_eq0", "expf_neq0", "expvSl", "expv_line", "leqNgt", "leq_o...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nz_x_i (i : 'I_n) : x ^+ i != 0.

Proof. by rewrite expf_eq0; case: eqP i => [->|_] [[]] //; rewrite adjoin0_deg. Qed.

Let

nz_x_i

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "adjoin0_deg", "expf_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_eq_sum : <<K; x>>%VS = sumKx.

Proof. apply/esym/eqP; rewrite eqEdim eq_leq ?andbT. rewrite dim_Fadjoin -[n]card_ord -sum_nat_const (directvP Fadjoin_sum_direct). by apply: eq_bigr => i _; rewrite /= dim_cosetv. apply/subv_sumP=> i _; rewrite -agenvM prodvS ?subv_adjoin //. by rewrite -expv_line (subv_trans (subX_agenv _ _)) ?agenvS ?addvSr. Qed...

Lemma

Fadjoin_eq_sum

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_sum_direct", "addvSr", "agenvM", "agenvS", "apply", "card_ord", "dim_Fadjoin", "dim_cosetv", "directvP", "eqEdim", "eq_bigr", "eq_leq", "expv_line", "prodvS", "subX_agenv", "subv_adjoin", "subv_sumP", "subv_trans", "sumKx", "sum_nat_const" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_poly_eq v : v \in <<K; x>>%VS -> (Fadjoin_poly K x v).[x] = v.

Proof. move/(sumv_pi_sum Fadjoin_eq_sum)=> {2}<-; rewrite horner_poly. by apply: eq_bigr => i _; rewrite inord_val mulfVK. Qed.

Lemma

Fadjoin_poly_eq

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_eq_sum", "Fadjoin_poly", "apply", "eq_bigr", "horner_poly", "inord_val", "mulfVK", "sumv_pi_sum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mempx_Fadjoin p : p \is a polyOver K -> p.[x] \in <<K; x>>%VS.

Proof. move=> Kp; rewrite rpred_horner ?memv_adjoin ?(polyOverS _ Kp) //. exact: subvP_adjoin. Qed.

Lemma

mempx_Fadjoin

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "memv_adjoin", "polyOver", "polyOverS", "rpred_horner", "subvP_adjoin" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_polyP {v} : reflect (exists2 p, p \in polyOver K & v = p.[x]) (v \in <<K; x>>%VS).

Proof. apply: (iffP idP) => [Kx_v | [p Kp ->]]; last exact: mempx_Fadjoin. by exists (Fadjoin_poly K x v); rewrite ?Fadjoin_polyOver ?Fadjoin_poly_eq. Qed.

Lemma

Fadjoin_polyP

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "Fadjoin_polyOver", "Fadjoin_poly_eq", "apply", "last", "mempx_Fadjoin", "polyOver" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_poly_unique p v : p \is a polyOver K -> size p <= n -> p.[x] = v -> Fadjoin_poly K x v = p.

Proof. have polyKx q i: q \is a polyOver K -> q`_i * x ^+ i \in (K * <[x ^+ i]>)%VS. by move/polyOverP=> Kq; rewrite memv_mul ?Kq ?memv_line. move=> Kp szp Dv; have /Fadjoin_poly_eq/eqP := mempx_Fadjoin Kp. rewrite {1}Dv {Dv} !(@horner_coef_wide _ n) ?size_poly //. move/polyKx in Kp; have /polyKx K_pv := Fadjoin_poly...

Lemma

Fadjoin_poly_unique

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "Fadjoin_polyOver", "Fadjoin_poly_eq", "Fadjoin_sum_direct", "Sub", "apply", "directv_sum_unique", "eqfunP", "horner_coef_wide", "last", "leqP", "leq_trans", "mempx_Fadjoin", "memv_line", "memv_mul", "mulIf", "nth_default", "polyOver", "polyOverP", "polyP", "s...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_polyC v : v \in K -> Fadjoin_poly K x v = v%:P.

Proof. move=> Kv; apply: Fadjoin_poly_unique; rewrite ?polyOverC ?hornerC //. by rewrite size_polyC (leq_trans (leq_b1 _)). Qed.

Lemma

Fadjoin_polyC

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "Fadjoin_poly_unique", "apply", "hornerC", "leq_b1", "leq_trans", "polyOverC", "size_polyC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_polyX : x \notin K -> Fadjoin_poly K x x = 'X.

Proof. move=> K'x; apply: Fadjoin_poly_unique; rewrite ?polyOverX ?hornerX //. by rewrite size_polyX ltn_neqAle andbT eq_sym adjoin_deg_eq1. Qed.

Lemma

Fadjoin_polyX

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "Fadjoin_poly_unique", "adjoin_deg_eq1", "apply", "eq_sym", "hornerX", "ltn_neqAle", "polyOverX", "size_polyX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minPolyOver : minPoly K x \is a polyOver K.

Proof. by rewrite /minPoly rpredB ?rpredX ?polyOverX ?Fadjoin_polyOver. Qed.

Lemma

minPolyOver

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_polyOver", "minPoly", "polyOver", "polyOverX", "rpredB", "rpredX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minPolyxx : (minPoly K x).[x] = 0.

Proof. by rewrite !hornerE Fadjoin_poly_eq ?subrr ?rpredX ?memv_adjoin. Qed.

Lemma

minPolyxx

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly_eq", "hornerE", "memv_adjoin", "minPoly", "rpredX", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root_minPoly : root (minPoly K x) x.

Proof. exact/rootP/minPolyxx. Qed.

Lemma

root_minPoly

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "minPoly", "minPolyxx", "root", "rootP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin_poly_mod p : p \is a polyOver K -> Fadjoin_poly K x p.[x] = p %% minPoly K x.

Proof. move=> Kp; rewrite {1}(divp_eq p (minPoly K x)) 2!hornerE minPolyxx mulr0 add0r. apply: Fadjoin_poly_unique => //; first by rewrite modp_polyOver // minPolyOver. by rewrite -ltnS -size_minPoly ltn_modp // monic_neq0 ?monic_minPoly. Qed.

Lemma

Fadjoin_poly_mod

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly", "Fadjoin_poly_unique", "add0r", "apply", "divp_eq", "hornerE", "ltnS", "ltn_modp", "minPoly", "minPolyOver", "minPolyxx", "modp_polyOver", "monic_minPoly", "monic_neq0", "mulr0", "polyOver", "size_minPoly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minPoly_XsubC : reflect (minPoly K x = 'X - x%:P) (x \in K).

Proof. set p := minPoly K x; apply: (iffP idP) => [Kx | Dp]; last first. suffices ->: x = - p`_0 by rewrite rpredN (polyOverP minPolyOver). by rewrite Dp coefB coefX coefC add0r opprK. rewrite (@all_roots_prod_XsubC _ p [:: x]) /= ?root_minPoly //. by apply/eqP; rewrite size_minPoly eqSS adjoin_deg_eq1. by rewrit...

Lemma

minPoly_XsubC

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "add0r", "adjoin_deg_eq1", "all_roots_prod_XsubC", "apply", "big_seq1", "coefB", "coefC", "coefX", "eqSS", "last", "minPoly", "minPolyOver", "monicP", "monic_minPoly", "opprK", "polyOverP", "root_minPoly", "rpredN", "scale1r", "size_minPoly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root_small_adjoin_poly p : p \is a polyOver K -> size p <= n -> root p x = (p == 0).

Proof. move=> Kp szp; apply/rootP/eqP=> [px0 | ->]; last by rewrite horner0. rewrite -(Fadjoin_poly_unique Kp szp px0). by apply: Fadjoin_poly_unique; rewrite ?polyOver0 ?size_poly0 ?horner0. Qed.

Lemma

root_small_adjoin_poly

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_poly_unique", "apply", "horner0", "last", "polyOver", "polyOver0", "root", "rootP", "size", "size_poly0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minPoly_irr p : p \is a polyOver K -> p %| minPoly K x -> (p %= minPoly K x) || (p %= 1).

Proof. rewrite dvdp_eq; set q := _ %/ _ => Kp def_pq. have Kq: q \is a polyOver K by rewrite divp_polyOver // minPolyOver. move: q Kq def_pq root_minPoly (size_minPoly K x) => q Kq /eqP->. rewrite rootM => pqx0 szpq. have [nzq nzp]: q != 0 /\ p != 0. by apply/norP; rewrite -mulf_eq0 -size_poly_eq0 szpq. without loss{...

Lemma

minPoly_irr

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "divp_polyOver", "dvdp_eq", "eqn_leq", "eqp_mul2r", "eqp_sym", "leq_add2r", "leq_subLR", "lt0n", "ltnNge", "minPoly", "minPolyOver", "mul1r", "mulf_eq0", "mulrC", "polyOver", "root", "rootM", "root_minPoly", "root_small_adjoin_poly", "size", "size_minPoly", "size...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minPoly_dvdp p : p \is a polyOver K -> root p x -> (minPoly K x) %| p.

Proof. move=> Kp rootp. have gcdK : gcdp (minPoly K x) p \is a polyOver K. by rewrite gcdp_polyOver ?minPolyOver. have /orP[gcd_eqK|gcd_eq1] := minPoly_irr gcdK (dvdp_gcdl (minPoly K x) p). by rewrite -(eqp_dvdl _ gcd_eqK) dvdp_gcdr. case/negP: (root1 x). by rewrite -(eqp_root gcd_eq1) root_gcd rootp root_minPoly. ...

Lemma

minPoly_dvdp

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "dvdp_gcdl", "dvdp_gcdr", "eqp_dvdl", "eqp_root", "gcdp", "gcdp_polyOver", "minPoly", "minPolyOver", "minPoly_irr", "polyOver", "root", "root1", "root_gcd", "root_minPoly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

minPolyS K E a : (K <= E)%VS -> minPoly E a %| minPoly K a.

Proof. move=> sKE; apply: minPoly_dvdp; last exact: root_minPoly. by apply: (polyOverSv sKE); rewrite minPolyOver. Qed.

Lemma

minPolyS

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "apply", "last", "minPoly", "minPolyOver", "minPoly_dvdp", "polyOverSv", "root_minPoly", "sKE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fadjoin1_polyP x v : reflect (exists p, v = (map_poly (in_alg L) p).[x]) (v \in <<1; x>>%VS).

Proof. apply: (iffP Fadjoin_polyP) => [[_ /polyOver1P]|] [p ->]; first by exists p. by exists (map_poly (in_alg L) p) => //; apply: alg_polyOver. Qed.

Lemma

Fadjoin1_polyP

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "Fadjoin_polyP", "alg_polyOver", "apply", "in_alg", "map_poly", "polyOver1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fieldExt_horner

:= horner_morph (fun x => mulrC z (in_alg L x)).

Definition

fieldExt_horner

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "horner_morph", "in_alg", "mulrC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fieldExt_hornerC b : fieldExt_horner b%:P = b%:A.

Proof. exact: horner_morphC. Qed.

Lemma

fieldExt_hornerC

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "fieldExt_horner", "horner_morphC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fieldExt_hornerX : fieldExt_horner 'X = z.

Proof. exact: horner_morphX. Qed.

Lemma

fieldExt_hornerX

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "fieldExt_horner", "horner_morphX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fieldExt_hornerZ : scalable fieldExt_horner.

Proof. move=> a p; rewrite -mul_polyC rmorphM /= fieldExt_hornerC. by rewrite -scalerAl mul1r. Qed.

Fact

fieldExt_hornerZ

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "fieldExt_horner", "fieldExt_hornerC", "mul1r", "mul_polyC", "rmorphM", "scalable", "scalerAl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"E :&: F"

:= (capv_aspace E F) : aspace_scope.

Notation

E :&: F

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "capv_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C_ E [ x ]"

:= (capv_aspace E 'C[x]) : aspace_scope.

Notation

'C_ E [ x ]

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "capv_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C_ ( E ) [ x ]"

:= (capv_aspace E 'C[x]) (only parsing) : aspace_scope.

Notation

'C_ ( E ) [ x ]

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "capv_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C_ E ( V )"

:= (capv_aspace E 'C(V)) : aspace_scope.

Notation

'C_ E ( V )

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "capv_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'C_ ( E ) ( V )"

:= (capv_aspace E 'C(V)) (only parsing) : aspace_scope.

Notation

'C_ ( E ) ( V )

field

field/fieldext.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "ssralg", "countalg", "finalg", "zmodp", "matrix", "vector", "falgebra", "poly", "polydiv", "mxpoly", "generic_quotie...

[ "capv_aspace" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d