fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
ahomP{f : 'Hom(aT, rT)} : reflect (multiplicative f) (ahom_in {:aT} f).
Proof. by apply: (iffP ahomP_tmp) => [][]. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | ahomP | |
ahom:= AHom {ahval :> 'Hom(aT, rT); _ : ahom_in {:aT} ahval}.
HB.instance Definition _ := [isSub for ahval].
HB.instance Definition _ := [Equality of ahom by <:].
HB.instance Definition _ := [Choice of ahom by <:].
Fact linfun_is_ahom (f : {lrmorphism aT -> rT}) : ahom_in {:aT} (linfun f).
Proof. by apply/ahom_inP; split=> [x y|]; rewrite !lfunE ?rmorphM ?rmorph1. Qed. | Structure | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | ahom | |
linfun_ahomf := AHom (linfun_is_ahom f). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | linfun_ahom | |
Definition_ (f : ahom aT rT) :=
GRing.isMonoidMorphism.Build aT rT f (ahom_is_monoid_morphism f).
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `ahom_is_monoid_morphism` instead")] | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | Definition | |
ahom_is_multiplicative(f : ahom aT rT) : multiplicative f :=
(fun p => (p.2, p.1)) (ahom_is_monoid_morphism f). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | ahom_is_multiplicative | |
ahomWin(f : ahom aT rT) U : ahom_in U f.
Proof.
by apply/ahom_inP; split; [apply: in2W (rmorphM _) | apply: rmorph1].
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | ahomWin | |
id_is_ahom(V : {vspace aT}) : ahom_in V \1.
Proof. by apply/ahom_inP; split=> [x y|] /=; rewrite !id_lfunE. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | id_is_ahom | |
id_ahom:= AHom (id_is_ahom (aspacef aT)). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | id_ahom | |
comp_is_ahom(V : {vspace aT}) (f : 'Hom(rT, sT)) (g : 'Hom(aT, rT)) :
ahom_in {:rT} f -> ahom_in V g -> ahom_in V (f \o g).
Proof.
move=> /ahom_inP fM /ahom_inP gM; apply/ahom_inP.
by split=> [x y Vx Vy|] /=; rewrite !comp_lfunE gM // fM ?memvf.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | comp_is_ahom | |
comp_ahom(f : ahom rT sT) (g : ahom aT rT) :=
AHom (comp_is_ahom (valP f) (valP g)). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | comp_ahom | |
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 // memv_img.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | aimgM | |
aimg1(f : ahom aT rT) : (f @: 1 = 1)%VS.
Proof. by rewrite limg_line rmorph1. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | aimg1 | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | aimgX | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | aimg_agen | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | aimg_adjoin | |
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.
Fact ker_sub_ahom_is_aspace (f g : ahom aT rT) :
is_aspace (lker (ahval f - ahval g)).
Proof.
rewrite /is_aspace has_algid1; last by apply/eqlfunP; rewrite !rmorph1.
apply/prodvP=> a b /eqlfunP Dfa /eqlfunP Dfb.
by apply/eqlfunP; rewrite !rmorphM /= Dfa Dfb.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | aimg_adjoin_seq | |
ker_sub_ahom_aspacef g := ASpace (ker_sub_ahom_is_aspace f g). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | ker_sub_ahom_aspace | |
fixedSpace_aspaceaT (f : ahom aT aT) := [aspace of fixedSpace f]. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop ssralg",
"From mathcomp Require Import finalg zmodp matrix vector poly"
] | field/falgebra.v | fixedSpace_aspace | |
DefinitionFieldExt (R : nzRingType) := {T of Falgebra R T &
GRing.PzRing_hasCommutativeMul T & GRing.Field T}. | HB.structure | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Definition | |
Definition_ (F : fieldType) := GRing.Field.on F^o. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Definition | |
dim_cosetvU 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | dim_cosetv | |
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.
HB.instance Definition _ :=
SemiGroup.isCommutativeLaw.Build {vspace L} (@prodv _ L) prodvC. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | prodvC | |
prodvCA: left_commutative (@prodv F0 L).
Proof. exact: Monoid.mulmCA. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | prodvCA | |
prodvAC: right_commutative (@prodv F0 L).
Proof. exact: Monoid.mulmAC. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | prodvAC | |
algid1K : algid K = 1. Proof. exact/skew_field_algid1/fieldP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | algid1 | |
mem1vK : 1 \in K. Proof. by rewrite -algid_eq1 algid1. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | mem1v | |
sub1vK : (1 <= K)%VS. Proof. exact: mem1v. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | sub1v | |
subfield_closedK : agenv K = K.
Proof.
by apply/eqP; rewrite eqEsubv sub_agenv agenv_sub_modr ?sub1v ?asubv.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfield_closed | |
AHom_lker0(rT : falgType F0) (f : 'AHom(L, rT)) : lker f == 0%VS.
Proof. by apply/lker0P; apply: fmorph_inj. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | AHom_lker0 | |
AEnd_lker0(f : 'AEnd(L)) : lker f == 0%VS. Proof. exact: AHom_lker0. Qed.
Fact 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. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | AEnd_lker0 | |
aimg_aspacerT f E := ASpace (@aimg_is_aspace rT f E). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | aimg_aspace | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_idP | |
Fadjoin0K : <<K; 0>>%VS = K.
Proof. by rewrite addv0 subfield_closed. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin0 | |
Fadjoin_nilK : <<K & [::]>>%VS = K.
Proof. by rewrite adjoin_nil subfield_closed. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_nil | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | FadjoinP | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_seqP | |
alg_polyOverE p : map_poly (in_alg L) p \is a polyOver E.
Proof. by apply/(polyOverS (subvP (sub1v _)))/polyOver1P; exists p. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | alg_polyOver | |
sub_adjoin1vx E : (<<1; x>> <= E)%VS = (x \in E)%VS.
Proof. by rewrite (sameP FadjoinP andP) sub1v. Qed.
Fact vsval_monoid_morphism K : monoid_morphism (vsval : subvs_of K -> L).
Proof. by split => //=; apply: algid1. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `subfx_eval_is_monoid_morphism` instead")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | sub_adjoin1v | |
vsval_is_multiplicativeK :=
(fun g => (g.2,g.1)) (vsval_monoid_morphism K).
HB.instance Definition _ (K : {subfield L}) :=
GRing.isMonoidMorphism.Build (subvs_of K) L vsval (vsval_monoid_morphism K). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | vsval_is_multiplicative | |
vsval_invfK (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.
Fact aspace_divr_closed K : divr_closed K.
Proof. by split=> [|u v Ku Kv]; rewrite ?mem1v ?memvM ?memvV. Qed.
HB.instance Definition _ K :=
GRing.isDivClosed.Build L (pred_of_vspace K) (aspace_divr_closed K).
HB.instance Definition _ (K : {subfield L}) :=
GRing.isSubPzSemiRing.Build L (pred_of_vspace K) (subvs_of K)
(rmorph1 _, rmorphM _). | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | vsval_invf | |
Definition_ (K : {subfield L}) :=
[SubNzRing_isSubComNzRing of subvs_of K by <:].
HB.instance Definition _ (K : {subfield L}) :=
[SubComUnitRing_isSubIntegralDomain of subvs_of K by <:]. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Definition | |
subvs_fieldMixinK : GRing.field_axiom (subvs_of K).
Proof.
by move=> w nz_w; rewrite unitrE -val_eqE /= vsval_invf algid1 divff.
Qed.
HB.instance Definition _ K := GRing.UnitRing_isField.Build (subvs_of K)
(@subvs_fieldMixin K). | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subvs_fieldMixin | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | polyOver_subvs | |
divp_polyOverK : {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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | divp_polyOver | |
modp_polyOverK : {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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | modp_polyOver | |
gcdp_polyOverK :
{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.
Fact 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. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | gcdp_polyOver | |
prodv_aspaceE F : {subfield L} := ASpace (prodv_is_aspace E F).
Fact field_mem_algid E F : algid E \in F. Proof. by rewrite algid1 mem1v. Qed. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | prodv_aspace | |
capv_aspaceE F : {subfield L} := aspace_cap (field_mem_algid E F). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | capv_aspace | |
polyOverSvU V : (U <= V)%VS -> {subset polyOver U <= polyOver V}.
Proof. by move/subvP=> sUV; apply: polyOverS. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | polyOverSv | |
field_subvMlF U : (U <= F * U)%VS.
Proof. by rewrite -{1}[U]prod1v prodvSl ?sub1v. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | field_subvMl | |
field_subvMrU F : (U <= U * F)%VS.
Proof. by rewrite prodvC field_subvMl. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | field_subvMr | |
field_module_eqF M : (F * M <= M)%VS -> (F * M)%VS = M.
Proof. by move=> modM; apply/eqP; rewrite eqEsubv modM field_subvMl. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | field_module_eq | |
sup_field_moduleF 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | sup_field_module | |
field_module_dimSF M : (F * M <= M)%VS -> (\dim F %| \dim M)%N.
Proof. exact/skew_field_module_dimS/fieldP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | field_module_dimS | |
field_dimSF E : (F <= E)%VS -> (\dim F %| \dim E)%N.
Proof. exact/skew_field_dimS/fieldP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | field_dimS | |
dim_field_moduleF M : (F * M <= M)%VS -> \dim M = (\dim_F M * \dim F)%N.
Proof. by move/field_module_dimS/divnK. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | dim_field_module | |
dim_sup_fieldF E : (F <= E)%VS -> \dim E = (\dim_F E * \dim F)%N.
Proof. by move/field_dimS/divnK. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | dim_sup_field | |
field_module_semisimpleF 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 /=.
by rewrite mul1n dim_cosetv ?(memPn nzX).
rewrite directvE /= (big_nth 0) [eqbRHS](big_nth 0) in defM dxFX.
rewrite (eqP szX) !big_mkord -directvE in defM dxFX.
exists (Tuple szX) => //; split=> // _ /tnthP[i ->]; rewrite (tnth_nth 0) /=.
by rewrite -defM memvE (sumv_sup i) ?field_subvMl.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | field_module_semisimple | |
adjoin_degree:= (\dim_U <<U; x>>).-1.+1.
Local Notation n := adjoin_degree. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | adjoin_degree | |
Fadjoin_sum:= (\sum_(i < n) U * <[x ^+ i]>)%VS. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_sum | |
Fadjoin_polyv : {poly L} :=
\poly_(i < n) (sumv_pi Fadjoin_sum (inord i) v / x ^+ i). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_poly | |
minPoly: {poly L} := 'X^n - Fadjoin_poly (x ^+ n). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | minPoly | |
size_Fadjoin_polyv : size (Fadjoin_poly v) <= n.
Proof. exact: size_poly. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | size_Fadjoin_poly | |
Fadjoin_polyOverv : 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.
Fact 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.
HB.instance Definition _ :=
GRing.isSemilinear.Build F0 L {poly L} _ Fadjoin_poly
(GRing.semilinear_linear Fadjoin_poly_is_linear). | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_polyOver | |
size_minPoly: size minPoly = n.+1.
Proof. by rewrite size_polyDl ?size_polyXn // size_polyN ltnS size_poly. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | size_minPoly | |
monic_minPoly: minPoly \is monic.
Proof.
rewrite monicE /lead_coef size_minPoly coefB coefXn eqxx.
by rewrite nth_default ?subr0 ?size_poly.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | monic_minPoly | |
adjoin_degreeE: n = \dim_K <<K; x>>.
Proof. by rewrite [n]prednK // divn_gt0 ?adim_gt0 // dimvS ?subv_adjoin. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | adjoin_degreeE | |
dim_Fadjoin: \dim <<K; x>> = (n * \dim K)%N.
Proof. by rewrite adjoin_degreeE -dim_sup_field ?subv_adjoin. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | dim_Fadjoin | |
adjoin0_deg: adjoin_degree K 0 = 1.
Proof. by rewrite /adjoin_degree addv0 subfield_closed divnn adim_gt0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | adjoin0_deg | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | adjoin_deg_eq1 | |
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 Ky Dz; rewrite memv0 Dz mulf_eq0 expf_eq0 /=.
apply: contraLR ltm1n => /norP[nz_y nz_x].
rewrite -leqNgt -(leq_pmul2r (adim_gt0 K)) -dim_Fadjoin.
have{IHm} ->: (m.+1 * \dim K)%N = \dim S.
rewrite -[m.+1]card_ord -sum_nat_const IHm.
by apply: eq_bigr => i; rewrite dim_cosetv ?expf_neq0.
apply/dimvS/agenv_sub_modl; first by rewrite (sumv_sup 0) //= prodv1 sub1v.
rewrite prodvDl subv_add -[S]big_distrr prodvA prodv_id subvv !big_distrr /=.
apply/subv_sumP=> i _; rewrite -expv_line prodvCA -expvSl expv_line.
have [ltim | lemi] := ltnP i m; first by rewrite (sumv_sup (Sub i.+1 _)).
have{lemi} /eqP->: i == m :> nat by rewrite eqn_leq leq_ord.
rewrite -big_distrr -2!{2}(prodv_id K) /= -!prodvA big_distrr -/S prodvSr //=.
by rewrite -(canLR (mulKf nz_y) Dz) -memvE memv_mul ?rpredV.
Qed.
Let nz_x_i (i : 'I_n) : x ^+ i != 0.
Proof.
by rewrite expf_eq0; case: eqP i => [->|_] [[]] //; rewrite adjoin0_deg.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_sum_direct | |
Fadjoin_eq_sum: <<K; x>>%VS = sumKx.
Proof.
apply/esym/eqP; rewrite eqEdim eq_leq ?andbT.
apply/subv_sumP=> i _; rewrite -agenvM prodvS ?subv_adjoin //.
by rewrite -expv_line (subv_trans (subX_agenv _ _)) ?agenvS ?addvSr.
rewrite dim_Fadjoin -[n]card_ord -sum_nat_const (directvP Fadjoin_sum_direct).
by apply: eq_bigr => i _; rewrite /= dim_cosetv.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_eq_sum | |
Fadjoin_poly_eqv : 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_poly_eq | |
mempx_Fadjoinp : 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | mempx_Fadjoin | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_polyP | |
Fadjoin_poly_uniquep 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_polyOver K x v.
rewrite (directv_sum_unique Fadjoin_sum_direct) // => /eqfunP eq_pq.
apply/polyP=> i; have [leni|?] := leqP n i; last exact: mulIf (eq_pq (Sub i _)).
by rewrite !nth_default ?(leq_trans _ leni) ?size_poly.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_poly_unique | |
Fadjoin_polyCv : 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_polyC | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_polyX | |
minPolyOver: minPoly K x \is a polyOver K.
Proof. by rewrite /minPoly rpredB ?rpredX ?polyOverX ?Fadjoin_polyOver. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | minPolyOver | |
minPolyxx: (minPoly K x).[x] = 0.
Proof.
by rewrite !hornerE Fadjoin_poly_eq ?subrr ?rpredX ?memv_adjoin.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | minPolyxx | |
root_minPoly: root (minPoly K x) x. Proof. exact/rootP/minPolyxx. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | root_minPoly | |
Fadjoin_poly_modp :
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin_poly_mod | |
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 rewrite big_seq1 (monicP (monic_minPoly K x)) scale1r.
by apply/eqP; rewrite size_minPoly eqSS adjoin_deg_eq1.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | minPoly_XsubC | |
root_small_adjoin_polyp :
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | root_small_adjoin_poly | |
minPoly_irrp :
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{pqx0} qx0: q p Kp Kq nzp nzq szpq / root q x.
move=> IH; case/orP: pqx0 => /IH{}IH; first exact: IH.
have{IH} /orP[]: (q %= p * q) || (q %= 1) by apply: IH => //; rewrite mulrC.
by rewrite orbC -{1}[q]mul1r eqp_mul2r // eqp_sym => ->.
by rewrite -{1}[p]mul1r eqp_sym eqp_mul2r // => ->.
apply/orP; right; rewrite -size_poly_eq1 eqn_leq lt0n size_poly_eq0 nzp andbT.
rewrite -(leq_add2r (size q)) -leq_subLR subn1 -size_mul // mulrC szpq.
by rewrite ltnNge; apply: contra nzq => /(root_small_adjoin_poly Kq) <-.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | minPoly_irr | |
minPoly_dvdpp : 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.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | minPoly_dvdp | |
minPolySK 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.
Arguments Fadjoin_polyP {K x v}. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | minPolyS | |
Fadjoin1_polyPx 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Fadjoin1_polyP | |
fieldExt_horner:= horner_morph (fun x => mulrC z (in_alg L x)).
HB.instance Definition _ := GRing.RMorphism.on fieldExt_horner. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | fieldExt_horner | |
fieldExt_hornerCb : fieldExt_horner b%:P = b%:A.
Proof. exact: horner_morphC. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | fieldExt_hornerC | |
fieldExt_hornerX: fieldExt_horner 'X = z.
Proof. exact: horner_morphX. Qed.
Fact fieldExt_hornerZ : scalable fieldExt_horner.
Proof.
move=> a p; rewrite -mul_polyC rmorphM /= fieldExt_hornerC.
by rewrite -scalerAl mul1r.
Qed.
HB.instance Definition _ :=
GRing.isScalable.Build F0 {poly F0} L *:%R fieldExt_horner fieldExt_hornerZ. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | fieldExt_hornerX | |
adjoin_degree_aimg: adjoin_degree (f @: K) (f x) = adjoin_degree K x.
Proof.
rewrite !adjoin_degreeE -aimg_adjoin.
by rewrite !limg_dim_eq ?(eqP (AHom_lker0 f)) ?capv0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | adjoin_degree_aimg | |
map_minPoly: map_poly f (minPoly K x) = minPoly (f @: K) (f x).
Proof.
set fp := minPoly (f @: K) (f x).
have [p Kp Dp]: exists2 p, p \is a polyOver K & map_poly f p = fp.
have Kfp: fp \is a polyOver (f @: K)%VS by apply: minPolyOver.
exists (map_poly f^-1%VF fp).
apply/polyOver_poly=> j _; have /memv_imgP[y Ky ->] := polyOverP Kfp j.
by rewrite lker0_lfunK ?AHom_lker0.
rewrite -map_poly_comp map_poly_id // => _ /(allP Kfp)/memv_imgP[y _ ->].
by rewrite /= limg_lfunVK ?memv_img ?memvf.
apply/eqP; rewrite -eqp_monic ?monic_map ?monic_minPoly // -Dp eqp_map.
have: ~~ (p %= 1) by rewrite -size_poly_eq1 -(size_map_poly f) Dp size_minPoly.
apply: implyP; rewrite implyNb orbC eqp_sym minPoly_irr //.
rewrite -(dvdp_map f) Dp minPoly_dvdp ?fmorph_root ?root_minPoly //.
by apply/polyOver_poly=> j _; apply/memv_img/polyOverP/minPolyOver.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | map_minPoly | |
fieldOverof {vspace L} : Type := L.
Local Notation K_F := (subvs_of F).
Local Notation L_F := (fieldOver F).
HB.instance Definition _ := GRing.Field.on L_F. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | fieldOver | |
fieldOver_scale(a : K_F) (u : L_F) : L_F := vsval a * u.
Local Infix "*F:" := fieldOver_scale (at level 40).
Fact fieldOver_scaleA a b u : a *F: (b *F: u) = (a * b) *F: u.
Proof. exact: mulrA. Qed.
Fact fieldOver_scale1 u : 1 *F: u = u.
Proof. by rewrite /(1 *F: u) /= algid1 mul1r. Qed.
Fact fieldOver_scaleDr a u v : a *F: (u + v) = a *F: u + a *F: v.
Proof. exact: mulrDr. Qed.
Fact fieldOver_scaleDl v a b : (a + b) *F: v = a *F: v + b *F: v.
Proof. exact: mulrDl. Qed.
HB.instance Definition _ := GRing.Zmodule_isLmodule.Build _ L_F
fieldOver_scaleA fieldOver_scale1 fieldOver_scaleDr fieldOver_scaleDl. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | fieldOver_scale | |
fieldOver_scaleEa (u : L) : a *: (u : L_F) = vsval a * u.
Proof. by []. Qed.
Fact fieldOver_scaleAl a u v : a *F: (u * v) = (a *F: u) * v.
Proof. exact: mulrA. Qed.
HB.instance Definition _ := GRing.Lmodule_isLalgebra.Build _ L_F
fieldOver_scaleAl.
Fact fieldOver_scaleAr a u v : a *F: (u * v) = u * (a *F: v).
Proof. exact: mulrCA. Qed.
HB.instance Definition _ := GRing.Lalgebra_isAlgebra.Build _ L_F
fieldOver_scaleAr.
Fact fieldOver_vectMixin : Lmodule_hasFinDim K_F L_F.
Proof.
have [bL [_ nz_bL] [defL dxSbL]] := field_module_semisimple (subvf (F * _)).
do [set n := \dim_F {:L} in bL nz_bL *; set SbL := (\sum_i _)%VS] in defL dxSbL.
have in_bL i (a : K_F) : val a * (bL`_i : L_F) \in (F * <[bL`_i]>)%VS.
by rewrite memv_mul ?(valP a) ?memv_line.
have nz_bLi (i : 'I_n): bL`_i != 0 by rewrite (memPn nz_bL) ?memt_nth.
pose r2v (v : 'rV[K_F]_n) : L_F := \sum_i v 0 i *: (bL`_i : L_F).
have r2v_lin: linear r2v.
move=> a u v; rewrite /r2v scaler_sumr -big_split /=; apply: eq_bigr => i _.
by rewrite scalerA -scalerDl !mxE.
pose r2vlM := GRing.isLinear.Build _ _ _ _ r2v r2v_lin.
pose r2vL : {linear _ -> _} := HB.pack r2v r2vlM.
have v2rP x: {r : 'rV[K_F]_n | x = r2v r}.
apply: sig_eqW; have /memv_sumP[y Fy ->]: x \in SbL by rewrite defL memvf.
have /fin_all_exists[r Dr] i: exists r, y i = r *: (bL`_i : L_F).
by have /memv_cosetP[a Fa ->] := Fy i isT; exists (Subvs Fa).
by exists (\row_i r i); apply: eq_bigr => i _; rewrite mxE.
pose v2r x := sval (v2rP x).
have v2rK: cancel v2r r2vL
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | fieldOver_scaleE | |
trivial_fieldOver: (1%VS : {vspace L_F}) =i F.
Proof.
move=> x; apply/vlineP/idP=> [[{}x ->] | Fx].
by rewrite fieldOver_scaleE mulr1 (valP x).
by exists (vsproj F x); rewrite fieldOver_scaleE mulr1 vsprojK.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | trivial_fieldOver | |
vspaceOverV := <<vbasis V : seq L_F>>%VS. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | vspaceOver | |
mem_vspaceOverV : vspaceOver V =i (F * V)%VS.
Proof.
move=> y; apply/idP/idP; last rewrite unlock; move/coord_span->.
rewrite (@memv_suml F0 L) // => i _.
by rewrite memv_mul ?subvsP // vbasis_mem ?memt_nth.
rewrite memv_suml // => ij _; rewrite -tnth_nth; set x := tnth _ ij.
have/allpairsP[[u z] /= [Fu Vz {x}->]]: x \in _ := mem_tnth ij _.
by rewrite scalerAl (memvZ (Subvs _)) ?memvZ ?memv_span //= vbasis_mem.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | mem_vspaceOver | |
mem_aspaceOverE : (F <= E)%VS -> vspaceOver E =i E.
Proof.
by move=> sFE y; rewrite mem_vspaceOver field_module_eq ?sup_field_module.
Qed.
Fact aspaceOver_suproof E : is_aspace (vspaceOver E).
Proof.
rewrite /is_aspace has_algid1; last by rewrite mem_vspaceOver (@mem1v _ L).
by apply/prodvP=> u v; rewrite !mem_vspaceOver; apply: memvM.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | mem_aspaceOver |
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