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 |
|---|---|---|---|---|---|---|
expvSU V n : (U <= V -> U ^+ n <= V ^+ n)%VS.
Proof.
move=> sUV; elim: n => [|n IHn]; first by rewrite !expv0 subvv.
by rewrite !expvSl prodvS.
Qed. | Lemma | 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 | expvS | |
expv_lineu n : (<[u]> ^+ n = <[u ^+ n]>)%VS.
Proof.
elim: n => [|n IH]; first by rewrite expr0 expv0.
by rewrite exprS expvSl IH prodv_line.
Qed. | Lemma | 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 | expv_line | |
centraliser1_vspaceu := lker (amulr u - amull u).
Local Notation "'C [ u ]" := (centraliser1_vspace u) : vspace_scope. | 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 | centraliser1_vspace | |
centraliser_vspaceV := (\bigcap_i 'C[tnth (vbasis V) i])%VS.
Local Notation "'C ( V )" := (centraliser_vspace V) : vspace_scope. | 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 | centraliser_vspace | |
center_vspaceV := (V :&: 'C(V))%VS.
Local Notation "'Z ( V )" := (center_vspace V) : vspace_scope. | 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 | center_vspace | |
cent1vPu v : reflect (u * v = v * u) (u \in 'C[v]%VS).
Proof. by rewrite (sameP eqlfunP eqP) !lfunE /=; apply: eqP. 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 | cent1vP | |
cent1v1u : 1 \in 'C[u]%VS. Proof. by apply/cent1vP; rewrite commr1. 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 | cent1v1 | |
cent1v_idu : u \in 'C[u]%VS. Proof. exact/cent1vP. 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 | cent1v_id | |
cent1vXu n : u ^+ n \in 'C[u]%VS. Proof. exact/cent1vP/esym/commrX. 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 | cent1vX | |
cent1vCu v : (u \in 'C[v])%VS = (v \in 'C[u])%VS.
Proof. exact/cent1vP/cent1vP. 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 | cent1vC | |
centvPu V : reflect {in V, forall v, u * v = v * u} (u \in 'C(V))%VS.
Proof.
apply: (iffP subv_bigcapP) => [cVu y /coord_vbasis-> | cVu i _].
apply/esym/cent1vP/rpred_sum=> i _; apply: rpredZ.
by rewrite -tnth_nth cent1vC memvE cVu.
exact/cent1vP/cVu/vbasis_mem/mem_tnth.
Qed. | Lemma | 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 | centvP | |
centvsPU V : reflect {in U & V, commutative *%R} (U <= 'C(V))%VS.
Proof. by apply: (iffP subvP) => [cUV u v | cUV u] /cUV-/centvP; apply. Qed. | Lemma | 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 | centvsP | |
subv_cent1U v : (U <= 'C[v])%VS = (v \in 'C(U)%VS).
Proof.
by apply/subvP/centvP=> cUv u Uu; apply/cent1vP; rewrite 1?cent1vC cUv.
Qed. | Lemma | 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 | subv_cent1 | |
centv1V : 1 \in 'C(V)%VS.
Proof. by apply/centvP=> v _; rewrite commr1. 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 | centv1 | |
centvXV u n : u \in 'C(V)%VS -> u ^+ n \in 'C(V)%VS.
Proof. by move/centvP=> cVu; apply/centvP=> v /cVu/esym/commrX->. Qed. | Lemma | 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 | centvX | |
centvCU V : (U <= 'C(V))%VS = (V <= 'C(U))%VS.
Proof. by apply/centvsP/centvsP=> cUV u v UVu /cUV->. 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 | centvC | |
centerv_subV : ('Z(V) <= V)%VS. Proof. exact: capvSl. 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 | centerv_sub | |
cent_centervV : (V <= 'C('Z(V)))%VS.
Proof. by rewrite centvC capvSr. 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 | cent_centerv | |
is_algide U :=
[/\ e \in U, e != 0 & {in U, forall u, e * u = u /\ u * e = u}].
Fact algid_decidable U : decidable (exists e, is_algid e U).
Proof.
have [-> | nzU] := eqVneq U 0%VS.
by right=> [[e []]]; rewrite memv0 => ->.
pose X := vbasis U; pose feq f1 f2 := [tuple of map f1 X ++ map f2 X].
have feqL f i: tnth (feq _ f _) (lshift _ i) = f X`_i.
set v := f _; rewrite (tnth_nth v) /= nth_cat size_map size_tuple.
by rewrite ltn_ord (nth_map 0) ?size_tuple.
have feqR f i: tnth (feq _ _ f) (rshift _ i) = f X`_i.
set v := f _; rewrite (tnth_nth v) /= nth_cat size_map size_tuple.
by rewrite ltnNge leq_addr addKn /= (nth_map 0) ?size_tuple.
apply: decP (vsolve_eq (feq _ amulr amull) (feq _ id id) U) _.
apply: (iffP (vsolve_eqP _ _ _)) => [[e Ue id_e] | [e [Ue _ id_e]]].
suffices idUe: {in U, forall u, e * u = u /\ u * e = u}.
exists e; split=> //; apply: contraNneq nzU => e0; rewrite -subv0.
by apply/subvP=> u /idUe[<- _]; rewrite e0 mul0r mem0v.
move=> u /coord_vbasis->; rewrite mulr_sumr mulr_suml.
split; apply/eq_bigr=> i _; rewrite -(scalerAr, scalerAl); congr (_ *: _).
by have:= id_e (lshift _ i); rewrite !feqL lfunE.
by have:= id_e (rshift _ i); rewrite !feqR lfunE.
have{id_e} /all_and2[ideX idXe]:= id_e _ (vbasis_mem (mem_tnth _ X)).
exists e => // k; rewrite -[k]splitK.
by case: (split k) => i; rewrite !(feqL, feqR) lfunE /= -tnth_nth.
Qed. | 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 | is_algid | |
has_algid: pred {vspace aT} := algid_decidable. | 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 | has_algid | |
has_algidP{U} : reflect (exists e, is_algid e U) (has_algid U).
Proof. exact: sumboolP. 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 | has_algidP | |
has_algid1U : 1 \in U -> has_algid U.
Proof.
move=> U1; apply/has_algidP; exists 1; split; rewrite ?oner_eq0 // => u _.
by rewrite mulr1 mul1r.
Qed. | Lemma | 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 | has_algid1 | |
is_aspaceU := has_algid U && (U * U <= U)%VS. | 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 | is_aspace | |
aspace:= ASpace {asval :> {vspace aT}; _ : is_aspace asval}.
HB.instance Definition _ := [isSub for asval].
HB.instance Definition _ := [Choice of aspace by <:]. | 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 | aspace | |
clone_aspaceU (A : aspace) :=
fun algU & phant_id algU (valP A) => @ASpace U algU : aspace.
Fact aspace1_subproof : is_aspace 1.
Proof. by rewrite /is_aspace prod1v -memvE has_algid1 memv_line. Qed. | 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 | clone_aspace | |
aspace1: aspace := ASpace aspace1_subproof. | 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 | aspace1 | |
aspacef_subproof: is_aspace fullv.
Proof. by rewrite /is_aspace subvf has_algid1 ?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 | aspacef_subproof | |
aspacef: aspace := ASpace aspacef_subproof. | 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 | aspacef | |
polyOver1Pp :
reflect (exists q, p = map_poly (in_alg aT) q) (p \is a polyOver 1%VS).
Proof.
apply: (iffP idP) => [/allP/=Qp | [q ->]]; last first.
by apply/polyOverP=> j; rewrite coef_map rpredZ ?memv_line.
exists (map_poly (coord [tuple 1] 0) p).
rewrite -map_poly_comp map_poly_id // => _ /Qp/vlineP[a ->] /=.
by rewrite linearZ /= (coord_free 0) ?mulr1 // seq1_free ?oner_eq0.
Qed. | Lemma | 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 | polyOver1P | |
algid_subproofU :
{e | e \in U
& has_algid U ==> (U <= lker (amull e - 1) :&: lker (amulr e - 1))%VS}.
Proof.
apply: sig2W; case: has_algidP => [[e]|]; last by exists 0; rewrite ?mem0v.
case=> Ae _ idAe; exists e => //; apply/subvP=> u /idAe[eu_u ue_u].
by rewrite memv_cap !memv_ker !lfun_simp /= eu_u ue_u subrr eqxx.
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 | algid_subproof | |
algidU := s2val (algid_subproof U). | 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 | algid | |
memv_algidU : algid U \in U.
Proof. by rewrite /algid; case: algid_subproof. 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 | memv_algid | |
algidlA : {in A, left_id (algid A) *%R}.
Proof.
rewrite /algid; case: algid_subproof => e _ /=; have /andP[-> _] := valP A.
move/subvP=> idAe u /idAe/memv_capP[].
by rewrite memv_ker !lfun_simp /= subr_eq0 => /eqP.
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 | algidl | |
algidrA : {in A, right_id (algid A) *%R}.
Proof.
rewrite /algid; case: algid_subproof => e _ /=; have /andP[-> _] := valP A.
move/subvP=> idAe u /idAe/memv_capP[_].
by rewrite memv_ker !lfun_simp /= subr_eq0 => /eqP.
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 | algidr | |
unitr_algid1A u : u \in A -> u \is a GRing.unit -> algid A = 1.
Proof. by move=> Eu /mulrI; apply; rewrite mulr1 algidr. 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 | unitr_algid1 | |
algid_eq1A : (algid A == 1) = (1 \in A).
Proof. by apply/eqP/idP=> [<- | /algidr <-]; rewrite ?memv_algid ?mul1r. 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 | algid_eq1 | |
algid_neq0A : algid A != 0.
Proof.
have /andP[/has_algidP[u [Au nz_u _]] _] := valP A.
by apply: contraNneq nz_u => e0; rewrite -(algidr Au) e0 mulr0.
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 | algid_neq0 | |
dim_algidA : \dim <[algid A]> = 1%N.
Proof. by rewrite dim_vline algid_neq0. 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 | dim_algid | |
adim_gt0A : (0 < \dim A)%N.
Proof. by rewrite -(dim_algid A) dimvS // -memvE ?memv_algid. 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 | adim_gt0 | |
not_asubv0A : ~~ (A <= 0)%VS.
Proof. by rewrite subv0 -dimv_eq0 -lt0n adim_gt0. 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 | not_asubv0 | |
adim1P{A} : reflect (A = <[algid A]>%VS :> {vspace aT}) (\dim A == 1%N).
Proof.
rewrite eqn_leq adim_gt0 -(memv_algid A) andbC -(dim_algid A) -eqEdim eq_sym.
exact: eqP.
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 | adim1P | |
asubvA : (A * A <= A)%VS.
Proof. by have /andP[] := valP A. 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 | asubv | |
memvMA : {in A &, forall u v, u * v \in A}.
Proof. exact/prodvP/asubv. 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 | memvM | |
prodv_idA : (A * A)%VS = A.
Proof.
apply/eqP; rewrite eqEsubv asubv; apply/subvP=> u Au.
by rewrite -(algidl Au) memv_mul // memv_algid.
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 | prodv_id | |
prodv_subU V A : (U <= A -> V <= A -> U * V <= A)%VS.
Proof. by move=> sUA sVA; rewrite -prodv_id prodvS. 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 | prodv_sub | |
expv_idA n : (A ^+ n.+1)%VS = A.
Proof. by elim: n => // n IHn; rewrite !expvSl prodvA prodv_id -expvSl. 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 | expv_id | |
limg_amulrU v : (amulr v @: U = U * <[v]>)%VS.
Proof.
rewrite -(span_basis (vbasisP U)) limg_span !span_def big_distrl /= big_map.
by apply: eq_bigr => u; rewrite prodv_line 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 | limg_amulr | |
memv_cosetP{U v w} :
reflect (exists2 u, u\in U & w = u * v) (w \in U * <[v]>)%VS.
Proof.
rewrite -limg_amulr.
by apply: (iffP memv_imgP) => [] [u] Uu ->; exists u; rewrite ?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 | memv_cosetP | |
dim_cosetv_unitV u : u \is a GRing.unit -> \dim (V * <[u]>) = \dim V.
Proof.
by move/lker0_amulr/eqP=> Uu; rewrite -limg_amulr limg_dim_eq // Uu capv0.
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 | dim_cosetv_unit | |
memvVA u : (u^-1 \in A) = (u \in A).
Proof.
suffices{u} invA: invr_closed A by apply/idP/idP=> /invA; rewrite ?invrK.
move=> u Au; have [Uu | /invr_out-> //] := boolP (u \is a GRing.unit).
rewrite memvE -(limg_ker0 _ _ (lker0_amulr Uu)) limg_line lfunE /= mulVr //.
suff ->: (amulr u @: A)%VS = A by rewrite -memvE -algid_eq1 (unitr_algid1 Au).
by apply/eqP; rewrite limg_amulr -dimv_leqif_eq ?prodv_sub ?dim_cosetv_unit.
Qed.
Fact aspace_cap_subproof A B : algid A \in B -> is_aspace (A :&: B).
Proof.
move=> BeA; apply/andP.
split; [apply/has_algidP | by rewrite subv_cap !prodv_sub ?capvSl ?capvSr].
exists (algid A); rewrite /is_algid algid_neq0 memv_cap memv_algid.
by split=> // u /memv_capP[Au _]; rewrite ?algidl ?algidr.
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 | memvV | |
aspace_capA B BeA := ASpace (@aspace_cap_subproof A B BeA).
Fact centraliser1_is_aspace u : is_aspace 'C[u].
Proof.
rewrite /is_aspace has_algid1 ?cent1v1 //=.
apply/prodvP=> v w /cent1vP-cuv /cent1vP-cuw.
by apply/cent1vP; rewrite -mulrA cuw !mulrA cuv.
Qed. | 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 | aspace_cap | |
centraliser1_aspaceu := ASpace (centraliser1_is_aspace u).
Fact centraliser_is_aspace V : is_aspace 'C(V).
Proof.
rewrite /is_aspace has_algid1 ?centv1 //=.
apply/prodvP=> u w /centvP-cVu /centvP-cVw.
by apply/centvP=> v Vv; rewrite /= -mulrA cVw // !mulrA cVu.
Qed. | 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 | centraliser1_aspace | |
centraliser_aspaceV := ASpace (centraliser_is_aspace V). | 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 | centraliser_aspace | |
centv_algidA : algid A \in 'C(A)%VS.
Proof. by apply/centvP=> u Au; rewrite algidl ?algidr. 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 | centv_algid | |
center_aspaceA := [aspace of 'Z(A) for aspace_cap (centv_algid A)]. | 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 | center_aspace | |
algid_centerA : algid 'Z(A) = algid A.
Proof.
rewrite -(algidl (subvP (centerv_sub A) _ (memv_algid _))) algidr //=.
by rewrite memv_cap memv_algid centv_algid.
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 | algid_center | |
Falgebra_FieldMixin:
GRing.integral_domain_axiom aT -> GRing.field_axiom aT.
Proof.
move=> domT u nz_u; apply/unitrP.
have kerMu: lker (amulr u) == 0%VS.
rewrite eqEsubv sub0v andbT; apply/subvP=> v; rewrite memv_ker lfunE /=.
by move/eqP/domT; rewrite (negPf nz_u) orbF memv0.
have /memv_imgP[v _ vu1]: 1 \in limg (amulr u); last rewrite lfunE /= in vu1.
suffices /eqP->: limg (amulr u) == fullv by rewrite memvf.
by rewrite -dimv_leqif_eq ?subvf ?limg_dim_eq // (eqP kerMu) capv0.
exists v; split=> //; apply: (lker0P kerMu).
by rewrite !lfunE /= -mulrA -vu1 mulr1 mul1r.
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 | Falgebra_FieldMixin | |
skew_field_algid1A : algid A = 1.
Proof. by rewrite (unitr_algid1 (memv_algid A)) ?fieldT ?algid_neq0. 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 | skew_field_algid1 | |
skew_field_module_semisimpleA M :
let sumA X := (\sum_(x <- X) A * <[x]>)%VS in
(A * M <= M)%VS -> {X | [/\ sumA X = M, directv (sumA X) & 0 \notin X]}.
Proof.
move=> sumA sAM_M; pose X := Nil aT; pose k := (\dim (A * M) - \dim (sumA X))%N.
have: (\dim (A * M) - \dim (sumA X) < k.+1)%N by [].
have: [/\ (sumA X <= A * M)%VS, directv (sumA X) & 0 \notin X].
by rewrite /sumA directvE /= !big_nil sub0v dimv0.
elim: {X k}k.+1 (X) => // k IHk X [sAX_AM dxAX nzX]; rewrite ltnS => leAXk.
have [sM_AX | /subvPn/sig2W[y My notAXy]] := boolP (M <= sumA X)%VS.
by exists X; split=> //; apply/eqP; rewrite eqEsubv (subv_trans sAX_AM).
have nz_y: y != 0 by rewrite (memPnC notAXy) ?mem0v.
pose AY := sumA (y :: X).
have sAY_AM: (AY <= A * M)%VS by rewrite [AY]big_cons subv_add ?prodvSr.
have dxAY: directv AY.
rewrite directvE /= !big_cons [_ == _]directv_addE dxAX directvE eqxx /=.
rewrite -/(sumA X) eqEsubv sub0v andbT -limg_amulr.
apply/subvP=> _ /memv_capP[/memv_imgP[a Aa ->]]/[!lfunE]/= AXay.
rewrite memv0 (mulIr_eq0 a (mulIr _)) ?fieldT //.
apply: contraR notAXy => /fieldT-Ua; rewrite -[y](mulKr Ua) /sumA.
by rewrite -big_distrr -(prodv_id A) /= -prodvA big_distrr memv_mul ?memvV.
apply: (IHk (y :: X)); first by rewrite !inE eq_sym negb_or nz_y.
rewrite -subSn ?dimvS // (directvP dxAY) /= big_cons -(directvP dxAX) /=.
rewrite subnDA (leq_trans _ leAXk) ?leq_sub2r // leq_subLR -add1n leq_add2r.
by rewrite dim_cosetv_unit ?fieldT ?adim_gt0.
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 | skew_field_module_semisimple | |
skew_field_module_dimSA M : (A * M <= M)%VS -> \dim A %| \dim M.
Proof.
case/skew_field_module_semisimple=> X [<- /directvP-> nzX] /=.
rewrite big_seq prime.dvdn_sum // => x /(memPn nzX)nz_x.
by rewrite dim_cosetv_unit ?fieldT.
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 | skew_field_module_dimS | |
skew_field_dimSA B : (A <= B)%VS -> \dim A %| \dim B.
Proof. by move=> sAB; rewrite skew_field_module_dimS ?prodv_sub. 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 | skew_field_dimS | |
agenvU := (\sum_(i < \dim {:aT}) U ^+ i)%VS.
Local Notation "<< U & vs >>" := (agenv (U + <<vs>>)) : vspace_scope.
Local Notation "<< U ; x >>" := (agenv (U + <[x]>)) : vspace_scope. | 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 | agenv | |
agenvElU : agenv U = (1 + U * agenv U)%VS.
Proof.
pose f V := (1 + U * V)%VS; rewrite -/(f _); pose n := \dim {:aT}.
have ->: agenv U = iter n f 0%VS.
rewrite /agenv -/n; elim: n => [|n IHn]; first by rewrite big_ord0.
rewrite big_ord_recl /= -{}IHn; congr (1 + _)%VS; rewrite big_distrr /=.
by apply: eq_bigr => i; rewrite expvSl.
have fS i j: i <= j -> (iter i f 0 <= iter j f 0)%VS.
by elim: i j => [|i IHi] [|j] leij; rewrite ?sub0v //= addvS ?prodvSr ?IHi.
suffices /(@trajectP _ f _ n.+1)[i le_i_n Dfi]: looping f 0%VS n.+1.
by apply/eqP; rewrite eqEsubv -iterS fS // Dfi fS.
apply: contraLR (dimvS (subvf (iter n.+1 f 0%VS))); rewrite -/n -ltnNge.
rewrite -looping_uniq; elim: n.+1 => // i IHi; rewrite trajectSr rcons_uniq.
rewrite {1}trajectSr mem_rcons inE negb_or eq_sym eqEdim fS ?leqW // -ltnNge.
by rewrite -andbA => /and3P[lt_fi _ /IHi/leq_ltn_trans->].
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 | agenvEl | |
agenvErU : agenv U = (1 + agenv U * U)%VS.
Proof.
rewrite [lhs in lhs = _]agenvEl big_distrr big_distrl /=; congr (_ + _)%VS.
by apply: eq_bigr => i _ /=; rewrite -expvSr -expvSl.
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 | agenvEr | |
agenv_modlU V : (U * V <= V -> agenv U * V <= V)%VS.
Proof.
rewrite big_distrl /= => idlU_V; apply/subv_sumP=> [[i _] /= _].
elim: i => [|i]; first by rewrite expv0 prod1v.
by apply: subv_trans; rewrite expvSr -prodvA prodvSr.
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 | agenv_modl | |
agenv_modrU V : (V * U <= V -> V * agenv U <= V)%VS.
Proof.
rewrite big_distrr /= => idrU_V; apply/subv_sumP=> [[i _] /= _].
elim: i => [|i]; first by rewrite expv0 prodv1.
by apply: subv_trans; rewrite expvSl prodvA prodvSl.
Qed.
Fact agenv_is_aspace U : is_aspace (agenv U).
Proof.
rewrite /is_aspace has_algid1; last by rewrite memvE agenvEl addvSl.
by rewrite agenv_modl // [V in (_ <= V)%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 | agenv_modr | |
agenv_aspaceU : {aspace aT} := ASpace (agenv_is_aspace U). | 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 | agenv_aspace | |
agenvEU : agenv U = agenv_aspace U. Proof. by []. 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 | agenvE | |
agenvMU : (agenv U * agenv U)%VS = agenv U. Proof. exact: prodv_id. 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 | agenvM | |
agenvXn U : (agenv U ^+ n.+1)%VS = agenv U. Proof. exact: expv_id. 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 | agenvX | |
sub1_agenvU : (1 <= agenv U)%VS. Proof. by rewrite agenvEl addvSl. 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 | sub1_agenv | |
sub_agenvU : (U <= agenv U)%VS.
Proof. by rewrite 2!agenvEl addvC prodvDr prodv1 -addvA addvSl. 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 | sub_agenv | |
subX_agenvU n : (U ^+ n <= agenv U)%VS.
Proof.
by case: n => [|n]; rewrite ?sub1_agenv // -(agenvX n) expvS // sub_agenv.
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 | subX_agenv | |
agenv_sub_modlU V : (1 <= V -> U * V <= V -> agenv U <= V)%VS.
Proof.
move=> s1V /agenv_modl; apply: subv_trans.
by rewrite -[Us in (Us <= _)%VS]prodv1 prodvSr.
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 | agenv_sub_modl | |
agenv_sub_modrU V : (1 <= V -> V * U <= V -> agenv U <= V)%VS.
Proof.
move=> s1V /agenv_modr; apply: subv_trans.
by rewrite -[Us in (Us <= _)%VS]prod1v prodvSl.
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 | agenv_sub_modr | |
agenv_idU : agenv (agenv U) = agenv U.
Proof.
apply/eqP; rewrite eqEsubv sub_agenv andbT.
by rewrite agenv_sub_modl ?sub1_agenv ?agenvM.
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 | agenv_id | |
agenvSU V : (U <= V -> agenv U <= agenv V)%VS.
Proof.
move=> sUV; rewrite agenv_sub_modl ?sub1_agenv //.
by rewrite -[Vs in (_ <= Vs)%VS]agenvM prodvSl ?(subv_trans sUV) ?sub_agenv.
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 | agenvS | |
agenv_add_idU V : agenv (agenv U + V) = agenv (U + V).
Proof.
apply/eqP; rewrite eqEsubv andbC agenvS ?addvS ?sub_agenv //=.
rewrite agenv_sub_modl ?sub1_agenv //.
rewrite -[rhs in (_ <= rhs)%VS]agenvM prodvSl // subv_add agenvS ?addvSl //=.
exact: subv_trans (addvSr U V) (sub_agenv _).
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 | agenv_add_id | |
subv_adjoinU x : (U <= <<U; x>>)%VS.
Proof. by rewrite (subv_trans (sub_agenv _)) ?agenvS ?addvSl. 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 | subv_adjoin | |
subv_adjoin_seqU xs : (U <= <<U & xs>>)%VS.
Proof. by rewrite (subv_trans (sub_agenv _)) // ?agenvS ?addvSl. 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 | subv_adjoin_seq | |
memv_adjoinU x : x \in <<U; x>>%VS.
Proof. by rewrite memvE (subv_trans (sub_agenv _)) ?agenvS ?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 | memv_adjoin | |
seqv_sub_adjoinU xs : {subset xs <= <<U & xs>>%VS}.
Proof.
by apply/span_subvP; rewrite (subv_trans (sub_agenv _)) ?agenvS ?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 | seqv_sub_adjoin | |
subvP_adjoinU x y : y \in U -> y \in <<U; x>>%VS.
Proof. exact/subvP/subv_adjoin. 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 | subvP_adjoin | |
adjoin_nilV : <<V & [::]>>%VS = agenv V.
Proof. by rewrite span_nil addv0. 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 | adjoin_nil | |
adjoin_consV x rs : <<V & x :: rs>>%VS = << <<V; x>> & rs>>%VS.
Proof. by rewrite span_cons addvA agenv_add_id. 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 | adjoin_cons | |
adjoin_rconsV rs x : <<V & rcons rs x>>%VS = << <<V & rs>>%VS; x>>%VS.
Proof. by rewrite -cats1 span_cat addvA span_seq1 agenv_add_id. 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 | adjoin_rcons | |
adjoin_seq1V x : <<V & [:: x]>>%VS = <<V; x>>%VS.
Proof. by rewrite adjoin_cons adjoin_nil agenv_id. 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 | adjoin_seq1 | |
adjoinCV x y : << <<V; x>>; y>>%VS = << <<V; y>>; x>>%VS.
Proof. by rewrite !agenv_add_id -!addvA (addvC <[x]>%VS). 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 | adjoinC | |
adjoinSlU V x : (U <= V -> <<U; x>> <= <<V; x>>)%VS.
Proof. by move=> sUV; rewrite agenvS ?addvS. 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 | adjoinSl | |
adjoin_seqSlU V rs : (U <= V -> <<U & rs>> <= <<V & rs>>)%VS.
Proof. by move=> sUV; rewrite agenvS ?addvS. 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 | adjoin_seqSl | |
adjoin_seqSrU rs1 rs2 :
{subset rs1 <= rs2} -> (<<U & rs1>> <= <<U & rs2>>)%VS.
Proof. by move/sub_span=> s_rs12; rewrite agenvS ?addvS. 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 | adjoin_seqSr | |
subvs_one:= Subvs (memv_algid A). | 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 | subvs_one | |
subvs_mul(u v : subvs_of A) :=
Subvs (subv_trans (memv_mul (subvsP u) (subvsP v)) (asubv _)).
Fact subvs_mulA : associative subvs_mul.
Proof. by move=> x y z; apply/val_inj/mulrA. Qed.
Fact subvs_mu1l : left_id subvs_one subvs_mul.
Proof. by move=> x; apply/val_inj/algidl/(valP x). Qed.
Fact subvs_mul1 : right_id subvs_one subvs_mul.
Proof. by move=> x; apply/val_inj/algidr/(valP x). Qed.
Fact subvs_mulDl : left_distributive subvs_mul +%R.
Proof. move=> x y z; apply/val_inj/mulrDl. Qed.
Fact subvs_mulDr : right_distributive subvs_mul +%R.
Proof. move=> x y z; apply/val_inj/mulrDr. Qed.
HB.instance Definition _ := GRing.Zmodule_isNzRing.Build (subvs_of A)
subvs_mulA subvs_mu1l subvs_mul1 subvs_mulDl subvs_mulDr (algid_neq0 _). | 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 | subvs_mul | |
subvs_scaleAlk (x y : subvs_of A) : k *: (x * y) = (k *: x) * y.
Proof. exact/val_inj/scalerAl. Qed.
HB.instance Definition _ := GRing.Lmodule_isLalgebra.Build K (subvs_of A)
subvs_scaleAl. | 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 | subvs_scaleAl | |
subvs_scaleArk (x y : subvs_of A) : k *: (x * y) = x * (k *: y).
Proof. exact/val_inj/scalerAr. Qed.
HB.instance Definition _ := GRing.Lalgebra_isAlgebra.Build K (subvs_of A)
subvs_scaleAr.
HB.instance Definition _ := Algebra_isFalgebra.Build K (subvs_of A).
Implicit Type w : subvs_of A. | 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 | subvs_scaleAr | |
vsval_unitrw : vsval w \is a GRing.unit -> w \is a GRing.unit.
Proof.
case: w => /= u Au Uu; have Au1: u^-1 \in A by rewrite memvV.
apply/unitrP; exists (Subvs Au1).
by split; apply: val_inj; rewrite /= ?mulrV ?mulVr ?(unitr_algid1 Au).
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 | vsval_unitr | |
vsval_invrw : vsval w \is a GRing.unit -> val w^-1 = (val w)^-1.
Proof.
move=> Uu; have def_w: w / w * w = w by rewrite divrK ?vsval_unitr.
by apply: (mulrI Uu); rewrite -[in u in u / _]def_w ?mulrK.
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 | vsval_invr | |
ahom_in(U : {vspace aT}) (f : 'Hom(aT, rT)) :=
all2rel (fun x y : aT => f (x * y) == f x * f y) (vbasis U) && (f 1 == 1). | 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_in | |
ahom_inP{f : 'Hom(aT, rT)} {U : {vspace aT}} :
reflect ({in U &, {morph f : x y / x * y >-> x * y}} * (f 1 = 1))
(ahom_in U f).
Proof.
apply: (iffP andP) => [[/allrelP fM /eqP f1] | [fM f1]]; last first.
rewrite f1; split=> //; apply/allrelP => x y Ax Ay.
by rewrite fM // vbasis_mem.
split=> // x y /coord_vbasis -> /coord_vbasis ->.
rewrite !mulr_suml ![f _]linear_sum mulr_suml; apply: eq_bigr => i _ /=.
rewrite !mulr_sumr linear_sum; apply: eq_bigr => j _ /=.
rewrite !linearZ -!scalerAr -!scalerAl 2!linearZ /=; congr (_ *: (_ *: _)).
by apply/eqP/fM; apply: memt_nth.
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 | ahom_inP | |
ahomP_tmp{f : 'Hom(aT, rT)} : reflect (monoid_morphism f) (ahom_in {:aT} f).
Proof.
apply: (iffP ahom_inP) => [[fM f1] | fRM_P]; last first.
by split=> [x y|]; [rewrite fRM_P.2|rewrite fRM_P.1].
by split=> // x y; rewrite fM ?memvf.
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `ahomP_tmp` instead")] | 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_tmp |
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