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 |
|---|---|---|---|---|---|---|
rgdcopTq p := 'let sp <- sizeT p; rgdcop_recT sp q p. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgdcopT | |
rgdcopTP(k : polyF -> fF) :
(forall p e, qf_eval e (k p) = qf_eval e (k (lift (eval_poly e p)))) ->
forall p q e, qf_eval e (rgdcopT p q k) =
qf_eval e (k (lift (rgdcop (eval_poly e p) (eval_poly e q)))).
Proof. by move=> *; rewrite sizeTP rgdcop_recTP 1?Pk. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgdcopTP | |
rgdcopT_qf(p : polyF) (q : polyF) :
rpoly p -> rpoly q -> qf_cps rpoly (rgdcopT p q).
Proof.
by move=> rp rq k kP; rewrite sizeT_qf => //*; rewrite rgdcop_recT_qf.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgdcopT_qf | |
ex_elim_seq(ps : seq polyF) (q : polyF) : fF :=
('let g <- rgcdpTs ps; 'let d <- rgdcopT q g;
'let n <- sizeT d; ret (n != 1)) GRing.Bool. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | ex_elim_seq | |
ex_elim_seqP(ps : seq polyF) (q : polyF) (e : seq F) :
let gp := (\big[@rgcdp _/0%:P]_(p <- ps)(eval_poly e p)) in
qf_eval e (ex_elim_seq ps q) = (size (rgdcop (eval_poly e q) gp) != 1).
Proof.
by do ![rewrite (rgcdpTsP,rgdcopTP,sizeTP,eval_lift) //= | move=> * //=].
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | ex_elim_seqP | |
ex_elim_seq_qf(ps : seq polyF) (q : polyF) :
all rpoly ps -> rpoly q -> qf (ex_elim_seq ps q).
Proof.
move=> rps rq; apply: rgcdpTs_qf=> // g rg; apply: rgdcopT_qf=> // d rd.
exact : sizeT_qf.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | ex_elim_seq_qf | |
abstrX(i : nat) (t : tF) :=
match t with
| 'X_n => if n == i then [::0; 1] else [::t]
| - x => opppT (abstrX i x)
| x + y => sumpT (abstrX i x) (abstrX i y)
| x * y => mulpT (abstrX i x) (abstrX i y)
| x *+ n => natmulpT n (abstrX i x)
| x ^+ n => let ax := (abstrX i x) in iter n (mulpT ax) [::1]
| _ => [::t]
end%T. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | abstrX | |
abstrXP(i : nat) (t : tF) (e : seq F) (x : F) :
rterm t -> (eval_poly e (abstrX i t)).[x] = eval (set_nth 0 e i x) t.
Proof.
elim: t => [n | r | n | t tP s sP | t tP | t tP n | t tP s sP | t tP | t tP n] h.
- move=> /=; case ni: (_ == _);
rewrite //= ?(mul0r,add0r,addr0,polyC1,mul1r,hornerX,hornerC);
by rewrite // nth_set_nth /= ni.
- by rewrite /= mul0r add0r hornerC.
- by rewrite /= mul0r add0r hornerC.
- by case/andP: h => *; rewrite /= eval_sumpT hornerD tP ?sP.
- by rewrite /= eval_opppT hornerN tP.
- by rewrite /= eval_natmulpT hornerMn tP.
- by case/andP: h => *; rewrite /= eval_mulpT hornerM tP ?sP.
- by [].
- elim: n h => [|n ihn] rt; first by rewrite /= expr0 mul0r add0r hornerC.
by rewrite /= eval_mulpT exprSr hornerM ihn // mulrC tP.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | abstrXP | |
rabstrX(i : nat) (t : tF) : rterm t -> rpoly (abstrX i t).
Proof.
elim: t; do ?[ by move=> * //=; do ?case: (_ == _)].
- move=> t irt s irs /=; case/andP=> rt rs.
by apply: rsumpT; rewrite ?irt ?irs //.
- by move=> t irt /= rt; rewrite rpoly_map_mul ?irt //.
- by move=> t irt /= n rt; rewrite rpoly_map_mul ?irt //.
- move=> t irt s irs /=; case/andP=> rt rs.
by apply: rmulpT; rewrite ?irt ?irs //.
- move=> t irt /= n rt; move: (irt rt) => {}rt; elim: n => [|n ihn] //=.
exact: rmulpT.
Qed.
Implicit Types tx ty : tF. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rabstrX | |
abstrX_mulM(i : nat) : {morph abstrX i : x y / x * y >-> mulpT x y}%T.
Proof. by []. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | abstrX_mulM | |
abstrX1(i : nat) : abstrX i 1%T = [::1%T].
Proof. done. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | abstrX1 | |
eval_poly_mulMe : {morph eval_poly e : x y / mulpT x y >-> x * y}.
Proof. by move=> x y; rewrite eval_mulpT. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_poly_mulM | |
eval_poly1e : eval_poly e [::1%T] = 1.
Proof. by rewrite /= mul0r add0r. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_poly1 | |
abstrX_bigmul:= (big_morph _ (abstrX_mulM _) (abstrX1 _)). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | abstrX_bigmul | |
eval_bigmul:= (big_morph _ (eval_poly_mulM _) (eval_poly1 _)). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_bigmul | |
bigmap_id:= (big_map _ (fun _ => true) id). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | bigmap_id | |
rseq_poly_map(x : nat) (ts : seq tF) :
all (@rterm _) ts -> all rpoly (map (abstrX x) ts).
Proof.
by elim: ts => //= t ts iht; case/andP=> rt rts; rewrite rabstrX // iht.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rseq_poly_map | |
ex_elim(x : nat) (pqs : seq tF * seq tF) :=
ex_elim_seq (map (abstrX x) pqs.1)
(abstrX x (\big[GRing.Mul/1%T]_(q <- pqs.2) q)). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | ex_elim | |
ex_elim_qf(x : nat) (pqs : seq tF * seq tF) :
GRing.dnf_rterm pqs -> qf (ex_elim x pqs).
case: pqs => ps qs; case/andP=> /= rps rqs.
apply: ex_elim_seq_qf; first exact: rseq_poly_map.
apply: rabstrX=> /=.
elim: qs rqs=> [|t ts iht] //=; first by rewrite big_nil.
by case/andP=> rt rts; rewrite big_cons /= rt /= iht.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | ex_elim_qf | |
holds_conj: forall e i x ps, all (@rterm _) ps ->
(GRing.holds (set_nth 0 e i x)
(foldr (fun t : tF => GRing.And (t == 0)) GRing.True%T ps)
<-> all ((@root _)^~ x) (map (eval_poly e \o abstrX i) ps)).
Proof.
move=> e i x; elim=> [|p ps ihps] //=.
case/andP=> rp rps; rewrite rootE abstrXP //.
constructor; first by case=> -> hps; rewrite eqxx /=; apply/ihps.
by case/andP; move/eqP=> -> psr; split=> //; apply/ihps.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | holds_conj | |
holds_conjn(e : seq F) (i : nat) (x : F) (ps : seq tF) :
all (@rterm _) ps ->
(GRing.holds (set_nth 0 e i x)
(foldr (fun t : tF => GRing.And (t != 0)) GRing.True ps) <->
all (fun p => ~~root p x) (map (eval_poly e \o abstrX i) ps)).
Proof.
elim: ps => [|p ps ihps] //=.
case/andP=> rp rps; rewrite rootE abstrXP //.
constructor; first by case=> /eqP-> hps /=; apply/ihps.
by case/andP=> pr psr; split; first apply/eqP=> //; apply/ihps.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | holds_conjn | |
holds_ex_elim: GRing.valid_QE_proj ex_elim.
Proof.
move=> i [ps qs] /= e; case/andP=> /= rps rqs.
rewrite ex_elim_seqP big_map.
have -> : \big[@rgcdp _/0%:P]_(j <- ps) eval_poly e (abstrX i j) =
\big[@rgcdp _/0%:P]_(j <- (map (eval_poly e) (map (abstrX i) (ps)))) j.
by rewrite !big_map.
rewrite -!map_comp.
have aux I (l : seq I) (P : I -> {poly F}) :
\big[(@gcdp F)/0]_(j <- l) P j %= \big[(@rgcdp F)/0]_(j <- l) P j.
elim: l => [| u l ihl] /=; first by rewrite !big_nil eqpxx.
rewrite !big_cons; move: ihl; move/(eqp_gcdr (P u)) => h.
by apply: eqp_trans h _; rewrite eqp_sym; apply: eqp_rgcd_gcd.
case g0: (\big[(@rgcdp F)/0%:P]_(j <- map (eval_poly e \o abstrX i) ps) j == 0).
rewrite (eqP g0) rgdcop0.
case m0 : (_ == 0)=> //=; rewrite ?(size_poly1,size_poly0) //=.
rewrite abstrX_bigmul eval_bigmul -bigmap_id in m0.
constructor=> [[x] // []] //.
case=> _; move/holds_conjn=> hc; move/hc:rqs.
by rewrite -root_bigmul //= (eqP m0) root0.
constructor; move/negP:m0; move/negP=>m0.
case: (closed_nonrootP F_closed _ m0) => x {m0}.
rewrite abstrX_bigmul eval_bigmul -bigmap_id root_bigmul=> m0.
exists x; do 2?constructor=> //; last by apply/holds_conjn.
apply/holds_conj; rewrite //= -root_biggcd.
by rewrite (eqp_root (aux _ _ _ )) (eqP g0) root0.
apply: (iffP (closed_rootP F_closed _)) => -[x Px]; exists x; move: Px => //=.
rewrite (eqp_root (@eqp_rgdco_gdco F _ _)) root_gdco ?g0 //.
rewrite -(eqp_root (aux _ _ _ )) root_biggcd abstrX_b
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | holds_ex_elim | |
wf_ex_elim: GRing.wf_QE_proj ex_elim.
Proof. by move=> i bc /= rbc; apply: ex_elim_qf. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | wf_ex_elim | |
RecordField_isAlgClosed F of GRing.Field F := {
solve_monicpoly : GRing.closed_field_axiom F;
}.
HB.builders Context F of Field_isAlgClosed F.
HB.instance Definition _ := GRing.Field_QE_isDecField.Build F
(@ClosedFieldQE.wf_ex_elim F)
(ClosedFieldQE.holds_ex_elim solve_monicpoly).
HB.instance Definition _ := GRing.DecField_isAlgClosed.Build F
solve_monicpoly.
HB.end.
Import CodeSeq. | HB.factory | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | Record | |
countable_field_extension(F : countFieldType) (p : {poly F}) :
size p > 1 ->
{E : countFieldType & {FtoE : {rmorphism F -> E} &
{w : E | root (map_poly FtoE p) w
& forall u : E, exists q, u = (map_poly FtoE q).[w]}}}.
Proof.
pose fix d i :=
if i is i1.+1 then
let d1 := oapp (gcdp (d i1)) 0 (unpickle i1) in
if size d1 > 1 then d1 else d i1
else p.
move=> p_gt1; have sz_d i: size (d i) > 1 by elim: i => //= i IHi; case: ifP.
have dv_d i j: i <= j -> d j %| d i.
move/subnK <-; elim: {j}(j - i)%N => //= j IHj; case: ifP => //=.
case: (unpickle _) => /= [q _|]; last by rewrite size_poly0.
exact: dvdp_trans (dvdp_gcdl _ _) IHj.
pose I : pred {poly F} := [pred q | d (pickle q).+1 %| q].
have I'co q i: q \notin I -> i > pickle q -> coprimep q (d i).
rewrite inE => I'q /dv_d/coprimep_dvdl-> //; apply: contraR I'q.
rewrite coprimep_sym /coprimep /= pickleK /= neq_ltn.
case: ifP => [_ _| ->]; first exact: dvdp_gcdr.
rewrite orbF ltnS leqn0 size_poly_eq0 gcdp_eq0 -size_poly_eq0.
by rewrite -leqn0 leqNgt ltnW //.
have memI q: reflect (exists i, d i %| q) (q \in I).
apply: (iffP idP) => [|[i dv_di_q]]; first by exists (pickle q).+1.
have [le_i_q | /I'co i_co_q] := leqP i (pickle q).
rewrite inE /= pickleK /=; case: ifP => _; first exact: dvdp_gcdr.
exact: dvdp_trans (dv_d _ _ le_i_q) dv_di_q.
apply: contraR i_co_q _.
by rewrite /coprimep (eqp_size (dvdp_gcd_idr dv_di_q)) neq_ltn sz_d orbT.
have I_ideal : idealr_closed I.
split=> [||a q1 q2 Iq1 Iq2]; fi
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | countable_field_extension | |
countable_algebraic_closure(F : countFieldType) :
{K : countClosedFieldType & {FtoK : {rmorphism F -> K} | integralRange FtoK}}.
Proof.
pose minXp (R : nzRingType) (p : {poly R}) := if size p > 1 then p else 'X.
have minXp_gt1 R p: size (minXp R p) > 1.
by rewrite /minXp; case: ifP => // _; rewrite size_polyX.
have minXpE (R : nzRingType) (p : {poly R}) : size p > 1 -> minXp R p = p.
by rewrite /minXp => ->.
have ext1 p := countable_field_extension (minXp_gt1 _ p).
pose ext1fT E p := tag (ext1 E p).
pose ext1to E p : {rmorphism _ -> ext1fT E p} := tag (tagged (ext1 E p)).
pose ext1w E p : ext1fT E p := s2val (tagged (tagged (ext1 E p))).
have ext1root E p: root (map_poly (ext1to E p) (minXp E p)) (ext1w E p).
by rewrite /ext1w; case: (tagged (tagged (ext1 E p))).
have ext1gen E p u: {q | u = (map_poly (ext1to E p) q).[ext1w E p]}.
by apply: sig_eqW; rewrite /ext1w; case: (tagged (tagged (ext1 E p))) u.
pose pExtEnum (E : countFieldType) := nat -> {poly E}.
pose Ext := {E : countFieldType & pExtEnum E}; pose MkExt : Ext := Tagged _ _.
pose EtoInc (E : Ext) i := ext1to (tag E) (tagged E i).
pose incEp E i j :=
let v := map_poly (EtoInc E i) (tagged E j) in
if decode j is [:: i1; k] then
if i1 == i then odflt v (unpickle k) else v
else v.
pose fix E_ i := if i is i1.+1 then MkExt _ (incEp (E_ i1) i1) else MkExt F \0.
pose E i := tag (E_ i); pose Krep := {i : nat & E i}.
pose fix toEadd i k : {rmorphism E i -> E (k + i)%N} :=
if k isn't k1.+1 then idfun else EtoInc _ (k1 + i)%N \o toE
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | countable_algebraic_closure | |
cyclotomic(z : R) n :=
\prod_(k < n | coprime k n) ('X - (z ^+ k)%:P). | Definition | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | cyclotomic | |
cyclotomic_monicz n : cyclotomic z n \is monic.
Proof. exact: monic_prod_XsubC. Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | cyclotomic_monic | |
size_cyclotomicz n : size (cyclotomic z n) = (totient n).+1.
Proof.
rewrite /cyclotomic -big_filter size_prod_XsubC; congr _.+1.
case: big_enumP => _ _ _ [_ ->].
rewrite totient_count_coprime -big_mkcond big_mkord -sum1_card.
by apply: eq_bigl => k; rewrite coprime_sym.
Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | size_cyclotomic | |
separable_Xn_sub_1(R : idomainType) n :
n%:R != 0 :> R -> @separable_poly R ('X^n - 1).
Proof.
case: n => [/eqP// | n nz_n]; rewrite unlock linearB /= derivC subr0.
rewrite derivXn -scaler_nat coprimepZr //= exprS -scaleN1r coprimep_sym.
by rewrite coprimep_addl_mul coprimepZr ?coprimep1 // (signr_eq0 _ 1).
Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | separable_Xn_sub_1 | |
root_cyclotomicx : root (cyclotomic z n) x = n.-primitive_root x.
Proof.
transitivity (x \in [seq z ^+ i | i : 'I_n in [pred i : 'I_n | coprime i n]]).
by rewrite -root_prod_XsubC big_image.
apply/imageP/idP=> [[k co_k_n ->] | prim_x].
by rewrite prim_root_exp_coprime.
have [k Dx] := prim_rootP prim_z (prim_expr_order prim_x).
exists (Ordinal (ltn_pmod k n_gt0)) => /=; last by rewrite prim_expr_mod.
by rewrite inE coprime_modl -(prim_root_exp_coprime k prim_z) -Dx.
Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | root_cyclotomic | |
prod_cyclotomic:
'X^n - 1 = \prod_(d <- divisors n) cyclotomic (z ^+ (n %/ d)) d.
Proof.
have in_d d: (d %| n)%N -> val (@inord n d) = d by move/dvdn_leq/inordK=> /= ->.
have dv_n k: (n %/ gcdn k n %| n)%N.
by rewrite -{3}(divnK (dvdn_gcdr k n)) dvdn_mulr.
have [uDn _ inDn] := divisors_correct n_gt0.
have defDn: divisors n = map val (map (@inord n) (divisors n)).
by rewrite -map_comp map_id_in // => d; rewrite inDn => /in_d.
rewrite defDn big_map big_uniq /=; last first.
by rewrite -(map_inj_uniq val_inj) -defDn.
pose h (k : 'I_n) : 'I_n.+1 := inord (n %/ gcdn k n).
rewrite -(factor_Xn_sub_1 prim_z) big_mkord.
rewrite (partition_big h (dvdn^~ n)) /= => [|k _]; last by rewrite in_d ?dv_n.
apply: eq_big => d; first by rewrite -(mem_map val_inj) -defDn inDn.
set q := (n %/ d)%N => d_dv_n.
have [q_gt0 d_gt0]: (0 < q /\ 0 < d)%N by apply/andP; rewrite -muln_gt0 divnK.
have fP (k : 'I_d): (q * k < n)%N by rewrite divn_mulAC ?ltn_divLR ?ltn_pmul2l.
rewrite (reindex (fun k => Ordinal (fP k))); last first.
have f'P (k : 'I_n): (k %/ q < d)%N by rewrite ltn_divLR // mulnC divnK.
exists (fun k => Ordinal (f'P k)) => [k _ | k /eqnP/=].
by apply: val_inj; rewrite /= mulKn.
rewrite in_d // => Dd; apply: val_inj; rewrite /= mulnC divnK // /q -Dd.
by rewrite divnA ?mulKn ?dvdn_gcdl ?dvdn_gcdr.
apply: eq_big => k; rewrite ?exprM // -val_eqE in_d //=.
rewrite -eqn_mul ?dvdn_gcdr ?gcdn_gt0 ?n_gt0 ?orbT //.
rewrite -[n in gcdn _ n](divnK d_dv_n) -muln_gcdr mulnCA mulnA divnK //.
by rewrite
... | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | prod_cyclotomic | |
C_prim_root_existsn : (n > 0)%N -> {z : algC | n.-primitive_root z}.
Proof.
pose p : {poly algC} := 'X^n - 1; have [r Dp] := closed_field_poly_normal p.
move=> n_gt0; apply/sigW; rewrite (monicP _) ?monicXnsubC // scale1r in Dp.
have rn1: all n.-unity_root r by apply/allP=> z; rewrite -root_prod_XsubC -Dp.
have sz_r: (n < (size r).+1)%N.
by rewrite -(size_prod_XsubC r id) -Dp size_XnsubC.
have [|z] := hasP (has_prim_root n_gt0 rn1 _ sz_r); last by exists z.
by rewrite -separable_prod_XsubC -Dp separable_Xn_sub_1 // pnatr_eq0 -lt0n.
Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | C_prim_root_exists | |
Cyclotomicn : {poly int} :=
let: exist z _ := C_prim_root_exists (ltn0Sn n.-1) in
map_poly Num.floor (cyclotomic z n). | Definition | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | Cyclotomic | |
Cyclotomic_monicn : 'Phi_n \is monic.
Proof.
rewrite /'Phi_n; case: (C_prim_root_exists _) => z /= _.
rewrite monicE lead_coefE coef_map_id0 ?(int_algC_K 0) ?floor0 //.
by rewrite size_poly_eq -lead_coefE (monicP (cyclotomic_monic _ _)) (intCK 1).
Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | Cyclotomic_monic | |
Cintr_Cyclotomicn z :
n.-primitive_root z -> pZtoC 'Phi_n = cyclotomic z n.
Proof.
elim/ltn_ind: n z => n IHn z0 prim_z0.
rewrite /'Phi_n; case: (C_prim_root_exists _) => z /=.
have n_gt0 := prim_order_gt0 prim_z0; rewrite prednK // => prim_z.
have [uDn _ inDn] := divisors_correct n_gt0.
pose q := \prod_(d <- rem n (divisors n)) 'Phi_d.
have mon_q: q \is monic by apply: monic_prod => d _; apply: Cyclotomic_monic.
have defXn1: cyclotomic z n * pZtoC q = 'X^n - 1.
rewrite (prod_cyclotomic prim_z) (big_rem n) ?inDn //=.
rewrite divnn n_gt0 rmorph_prod /=; congr (_ * _).
apply: eq_big_seq => d; rewrite mem_rem_uniq ?inE //= inDn => /andP[n'd ddvn].
by rewrite -IHn ?dvdn_prim_root // ltn_neqAle n'd dvdn_leq.
have mapXn1 (R1 R2 : nzRingType) (f : {rmorphism R1 -> R2}):
map_poly f ('X^n - 1) = 'X^n - 1.
- by rewrite rmorphB /= rmorph1 map_polyXn.
have nz_q: pZtoC q != 0.
by rewrite -size_poly_eq0 size_map_inj_poly // size_poly_eq0 monic_neq0.
have [r def_zn]: exists r, cyclotomic z n = pZtoC r.
have defZtoC: ZtoC =1 QtoC \o ZtoQ by move=> a; rewrite /= rmorph_int.
have /dvdpP[r0 Dr0]: map_poly ZtoQ q %| 'X^n - 1.
rewrite -(dvdp_map (@ratr algC)) mapXn1 -map_poly_comp.
by rewrite -(eq_map_poly defZtoC) -defXn1 dvdp_mull.
have [r [a nz_a Dr]] := rat_poly_scale r0.
exists (zprimitive r); apply: (mulIf nz_q); rewrite defXn1.
rewrite -rmorphM -(zprimitive_monic mon_q) -zprimitiveM /=.
have ->: r * q = a *: ('X^n - 1).
apply: (map_inj_poly (intr_inj : injective ZtoQ)
... | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | Cintr_Cyclotomic | |
prod_Cyclotomicn :
(n > 0)%N -> \prod_(d <- divisors n) 'Phi_d = 'X^n - 1.
Proof.
move=> n_gt0; have [z prim_z] := C_prim_root_exists n_gt0.
apply: (map_inj_poly (intr_inj : injective ZtoC)) => //.
rewrite rmorphB rmorph1 rmorph_prod /= map_polyXn (prod_cyclotomic prim_z).
apply: eq_big_seq => d; rewrite -dvdn_divisors // => d_dv_n.
by rewrite -Cintr_Cyclotomic ?dvdn_prim_root.
Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | prod_Cyclotomic | |
Cyclotomic0: 'Phi_0 = 1.
Proof.
rewrite /'Phi_0; case: (C_prim_root_exists _) => z /= _.
by rewrite -[1]polyseqK /cyclotomic big_ord0 map_polyE !polyseq1 /= (intCK 1).
Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | Cyclotomic0 | |
size_Cyclotomicn : size 'Phi_n = (totient n).+1.
Proof.
have [-> | n_gt0] := posnP n; first by rewrite Cyclotomic0 polyseq1.
have [z prim_z] := C_prim_root_exists n_gt0.
rewrite -(size_map_inj_poly (can_inj intCK)) //.
by rewrite (Cintr_Cyclotomic prim_z) size_cyclotomic.
Qed. | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | size_Cyclotomic | |
minCpoly_cyclotomicn z :
n.-primitive_root z -> minCpoly z = cyclotomic z n.
Proof.
move=> prim_z; have n_gt0 := prim_order_gt0 prim_z.
have Dpz := Cintr_Cyclotomic prim_z; set pz := cyclotomic z n in Dpz *.
have mon_pz: pz \is monic by apply: cyclotomic_monic.
have pz0: root pz z by rewrite root_cyclotomic.
have [pf [Dpf mon_pf] dv_pf] := minCpolyP z.
have /dvdpP_rat_int[f [af nz_af Df] [g /esym Dfg]]: pf %| pZtoQ 'Phi_n.
rewrite -dv_pf; congr (root _ z): pz0; rewrite -Dpz -map_poly_comp.
by apply: eq_map_poly => b; rewrite /= rmorph_int.
without loss{nz_af} [mon_f mon_g]: af f g Df Dfg / f \is monic /\ g \is monic.
move=> IH; pose cf := lead_coef f; pose cg := lead_coef g.
have cfg1: cf * cg = 1.
by rewrite -lead_coefM Dfg (monicP (Cyclotomic_monic n)).
apply: (IH (af *~ cf) (f *~ cg) (g *~ cf)).
- by rewrite rmorphMz -scalerMzr scalerMzl -mulrzA cfg1.
- by rewrite mulrzAl mulrzAr -mulrzA cfg1.
by rewrite !(intz, =^~ scaler_int) !monicE !lead_coefZ mulrC cfg1.
have{af} Df: pQtoC pf = pZtoC f.
have:= congr1 lead_coef Df.
rewrite lead_coefZ lead_coef_map_inj //; last exact: intr_inj.
rewrite !(monicP _) // mulr1 Df => <-; rewrite scale1r -map_poly_comp.
by apply: eq_map_poly => b; rewrite /= rmorph_int.
have [/size1_polyC Dg | g_gt1] := leqP (size g) 1.
rewrite monicE Dg lead_coefC in mon_g.
by rewrite -Dpz -Dfg Dg (eqP mon_g) mulr1 Dpf.
have [zk gzk0]: exists zk, root (pZtoC g) zk.
have [rg] := closed_field_poly_normal (pZtoC g).
rewrite lead_coef_map_inj
... | Lemma | field | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly finset fingroup finalg zmodp cyclic",
"From mathcomp Require Import ssrnum ssrint archimedean polydiv intdiv mxp... | field/cyclotomic.v | minCpoly_cyclotomic | |
DefinitionFalgebra (R : nzRingType) :=
{ A of Vector R A & GRing.UnitAlgebra R A }.
#[deprecated(since="mathcomp 2.0.0", note="Use falgType instead.")] | HB.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 | Definition | |
FalgType:= falgType. | Notation | 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 | FalgType | |
RecordAlgebra_isFalgebra (K : fieldType) A
of Vector K A & GRing.Algebra K A := {}.
HB.builders Context K A of Algebra_isFalgebra K A.
Let vA : Vector.type K := A.
Let am u := linfun (u \o* idfun : vA -> vA).
Let uam := [pred u | lker (am u) == 0%VS].
Let vam := [fun u => if u \in uam then (am u)^-1%VF 1 else u]. | HB.factory | 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 | Record | |
amEu v : am u v = v * u. Proof. by 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 | amE | |
mulVr: {in uam, left_inverse 1 vam *%R}.
Proof. by move=> u Uu; rewrite /= Uu -amE lker0_lfunVK. 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 | mulVr | |
divrr: {in uam, right_inverse 1 vam *%R}.
Proof.
by move=> u Uu; apply/(lker0P Uu); rewrite !amE -mulrA mulVr // mul1r mulr1.
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 | divrr | |
unitrP: forall x y, y * x = 1 /\ x * y = 1 -> uam x.
Proof.
move=> u v [_ uv1].
by apply/lker0P=> w1 w2 /(congr1 (am v)); rewrite !amE -!mulrA uv1 !mulr1.
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 | unitrP | |
invr_out: {in [predC uam], vam =1 id}.
Proof. by move=> u /negbTE/= ->. Qed.
HB.instance Definition _ := GRing.NzRing_hasMulInverse.Build A
mulVr divrr unitrP invr_out.
HB.end. | 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 | invr_out | |
Definition_ (K : fieldType) n :=
Algebra_isFalgebra.Build K 'M[K]_n.+1.
HB.instance Definition _ (R : comUnitRingType) := GRing.UnitAlgebra.on R^o. | 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 | |
regular_fullv(K : fieldType) : (fullv = 1 :> {vspace K^o})%VS.
Proof. by apply/esym/eqP; rewrite eqEdim subvf dim_vline oner_eq0 dimvf. Qed. | Lemma | 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 | regular_fullv | |
FalgType_proper: dim aT > 0.
Proof.
rewrite lt0n; apply: contraNneq (oner_neq0 aT) => aT0.
by apply/eqP/v2r_inj; do 2!move: (v2r _); rewrite aT0 => u v; rewrite !thinmx0.
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 | FalgType_proper | |
Definition_ := GRing.Algebra.copy 'End(aT)
(lfun_algType (FalgType_proper aT)). | 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 | |
lfun_mulEf g u : (f * g) u = g (f u). Proof. exact: 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 | lfun_mulE | |
lfun_compEf g : (g \o f)%VF = f * g. 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 | lfun_compE | |
lfun_invrf := if lker f == 0%VS then f^-1%VF else 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 | lfun_invr | |
lfun_mulVrf : lker f == 0%VS -> f^-1%VF * f = 1.
Proof. exact: lker0_compfV. 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 | lfun_mulVr | |
lfun_mulrVf : lker f == 0%VS -> f * f^-1%VF = 1.
Proof. exact: lker0_compVf. Qed.
Fact lfun_mulRVr f : lker f == 0%VS -> lfun_invr f * f = 1.
Proof. by move=> Uf; rewrite /lfun_invr Uf lfun_mulVr. Qed.
Fact lfun_mulrRV f : lker f == 0%VS -> f * lfun_invr f = 1.
Proof. by move=> Uf; rewrite /lfun_invr Uf lfun_mulrV. Qed.
Fact lfun_unitrP f g : g * f = 1 /\ f * g = 1 -> lker f == 0%VS.
Proof.
case=> _ fK; apply/lker0P; apply: can_inj (g) _ => u.
by rewrite -lfun_mulE fK lfunE.
Qed. | 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 | lfun_mulrV | |
lfun_invr_outf : lker f != 0%VS -> lfun_invr f = f.
Proof. by rewrite /lfun_invr => /negPf->. Qed.
HB.instance Definition _ := GRing.NzRing_hasMulInverse.Build 'End(aT)
lfun_mulRVr lfun_mulrRV lfun_unitrP lfun_invr_out. | 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 | lfun_invr_out | |
lfun_invEf : lker f == 0%VS -> f^-1%VF = f^-1.
Proof. by rewrite /f^-1 /= /lfun_invr => ->. 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 | lfun_invE | |
amullu : 'End(aT) := linfun (u \*o @idfun aT). | 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 | amull | |
amulru : 'End(aT) := linfun (u \o* @idfun aT). | 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 | amulr | |
amull_inj: injective amull.
Proof. by move=> u v /lfunP/(_ 1); rewrite !lfunE /= !mulr1. 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 | amull_inj | |
amulr_inj: injective amulr.
Proof. by move=> u v /lfunP/(_ 1); rewrite !lfunE /= !mul1r. Qed.
Fact amull_is_linear : linear amull.
Proof.
move=> a u v; apply/lfunP => w.
by rewrite !lfunE /= scale_lfunE !lfunE /= mulrDl scalerAl.
Qed.
#[hnf]
HB.instance Definition _ := GRing.isSemilinear.Build K aT (hom aT aT) _ amull
(GRing.semilinear_linear amull_is_linear). | 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 | amulr_inj | |
amull1: amull 1 = \1%VF.
Proof. by apply/lfunP => z; rewrite id_lfunE lfunE /= 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 | amull1 | |
amullMu v : (amull (u * v) = amull v * amull u)%VF.
Proof. by apply/lfunP => w; rewrite comp_lfunE !lfunE /= mulrA. 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 | amullM | |
amulr_is_linear: linear amulr.
Proof.
move=> a u v; apply/lfunP => w.
by rewrite !lfunE /= !lfunE /= lfunE mulrDr /= scalerAr.
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 | amulr_is_linear | |
amulr_is_monoid_morphism: monoid_morphism amulr.
Proof.
split=> [|x y]; first by apply/lfunP => w; rewrite id_lfunE !lfunE /= mulr1.
by apply/lfunP=> w; rewrite comp_lfunE !lfunE /= mulrA.
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `amulr_is_monoid_morphism` 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 | amulr_is_monoid_morphism | |
amulr_is_multiplicative:=
(fun p => (p.2, p.1)) amulr_is_monoid_morphism.
#[hnf]
HB.instance Definition _ := GRing.isSemilinear.Build K aT (hom aT aT) _ amulr
(GRing.semilinear_linear amulr_is_linear).
#[hnf]
HB.instance Definition _ := GRing.isMonoidMorphism.Build aT (hom aT aT) amulr
amulr_is_monoid_morphism. | 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 | amulr_is_multiplicative | |
lker0_amullu : u \is a GRing.unit -> lker (amull u) == 0%VS.
Proof. by move=> Uu; apply/lker0P=> v w; rewrite !lfunE; apply: mulrI. 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 | lker0_amull | |
lker0_amulru : u \is a GRing.unit -> lker (amulr u) == 0%VS.
Proof. by move=> Uu; apply/lker0P=> v w; rewrite !lfunE; apply: mulIr. 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 | lker0_amulr | |
lfun1_poly(p : {poly aT}) : map_poly \1%VF p = p.
Proof. by apply: map_poly_id => u _; apply: id_lfunE. Qed.
Fact prodv_key : unit. 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 | lfun1_poly | |
prodv:=
locked_with prodv_key (fun U V => <<allpairs *%R (vbasis U) (vbasis V)>>%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 | prodv | |
prodv_unlockable:= [unlockable fun prodv].
Local Notation "A * B" := (prodv A B) : vspace_scope. | 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 | prodv_unlockable | |
memv_mulU V : {in U & V, forall u v, u * v \in (U * V)%VS}.
Proof.
move=> u v /coord_vbasis-> /coord_vbasis->.
rewrite mulr_suml; apply: memv_suml => i _.
rewrite mulr_sumr; apply: memv_suml => j _.
rewrite -scalerAl -scalerAr !memvZ // [prodv]unlock memv_span //.
by apply/allpairsP; exists ((vbasis U)`_i, (vbasis V)`_j); rewrite !memt_nth.
Qed. | Lemma | 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_mul | |
prodvP{U V W} :
reflect {in U & V, forall u v, u * v \in W} (U * V <= W)%VS.
Proof.
apply: (iffP idP) => [sUVW u v Uu Vv | sUVW].
by rewrite (subvP sUVW) ?memv_mul.
rewrite [prodv]unlock; apply/span_subvP=> _ /allpairsP[[u v] /= [Uu Vv ->]].
by rewrite sUVW ?vbasis_mem.
Qed. | Lemma | 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 | prodvP | |
prodv_lineu v : (<[u]> * <[v]> = <[u * v]>)%VS.
Proof.
apply: subv_anti; rewrite -memvE memv_mul ?memv_line // andbT.
apply/prodvP=> _ _ /vlineP[a ->] /vlineP[b ->].
by rewrite -scalerAr -scalerAl !memvZ ?memv_line.
Qed. | Lemma | 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_line | |
dimv1: \dim (1%VS : {vspace aT}) = 1.
Proof. by rewrite dim_vline oner_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 | dimv1 | |
dim_prodvU V : \dim (U * V) <= \dim U * \dim V.
Proof. by rewrite unlock (leq_trans (dim_span _)) ?size_tuple. 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_prodv | |
vspace1_neq0: (1 != 0 :> {vspace aT})%VS.
Proof. by rewrite -dimv_eq0 dimv1. 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 | vspace1_neq0 | |
vbasis1: exists2 k, k != 0 & vbasis 1 = [:: k%:A] :> seq aT.
Proof.
move: (vbasis 1) (@vbasisP K aT 1); rewrite dim_vline oner_neq0.
case/tupleP=> x X0; rewrite {X0}tuple0 => defX; have Xx := mem_head x nil.
have /vlineP[k def_x] := basis_mem defX Xx; exists k; last by rewrite def_x.
by have:= basis_not0 defX Xx; rewrite def_x scaler_eq0 oner_eq0 orbF.
Qed. | Lemma | 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 | vbasis1 | |
prod0v: left_zero 0%VS prodv.
Proof.
move=> U; apply/eqP; rewrite -dimv_eq0 -leqn0 (leq_trans (dim_prodv 0 U)) //.
by rewrite dimv0.
Qed. | Lemma | 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 | prod0v | |
prodv0: right_zero 0%VS prodv.
Proof.
move=> U; apply/eqP; rewrite -dimv_eq0 -leqn0 (leq_trans (dim_prodv U 0)) //.
by rewrite dimv0 muln0.
Qed.
HB.instance Definition _ := Monoid.isMulLaw.Build {vspace aT} 0%VS prodv
prod0v prodv0. | 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 | prodv0 | |
prod1v: left_id 1%VS prodv.
Proof.
move=> U; apply/subv_anti/andP; split.
by apply/prodvP=> _ u /vlineP[a ->] Uu; rewrite mulr_algl memvZ.
by apply/subvP=> u Uu; rewrite -[u]mul1r memv_mul ?memv_line.
Qed. | Lemma | 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 | prod1v | |
prodv1: right_id 1%VS prodv.
Proof.
move=> U; apply/subv_anti/andP; split.
by apply/prodvP=> u _ Uu /vlineP[a ->]; rewrite mulr_algr memvZ.
by apply/subvP=> u Uu; rewrite -[u]mulr1 memv_mul ?memv_line.
Qed. | Lemma | 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 | prodv1 | |
prodvSU1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 * V1 <= U2 * V2)%VS.
Proof.
move/subvP=> sU12 /subvP sV12; apply/prodvP=> u v Uu Vv.
by rewrite memv_mul ?sU12 ?sV12.
Qed. | Lemma | 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 | prodvS | |
prodvSlU1 U2 V : (U1 <= U2 -> U1 * V <= U2 * V)%VS.
Proof. by move/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 | prodvSl | |
prodvSrU V1 V2 : (V1 <= V2 -> U * V1 <= U * V2)%VS.
Proof. exact: 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 | prodvSr | |
prodvDl: left_distributive prodv addv.
Proof.
move=> U1 U2 V; apply/esym/subv_anti/andP; split.
by rewrite subv_add 2?prodvS ?addvSl ?addvSr.
apply/prodvP=> _ v /memv_addP[u1 Uu1 [u2 Uu2 ->]] Vv.
by rewrite mulrDl memv_add ?memv_mul.
Qed. | Lemma | 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 | prodvDl | |
prodvDr: right_distributive prodv addv.
Proof.
move=> U V1 V2; apply/esym/subv_anti/andP; split.
by rewrite subv_add 2?prodvS ?addvSl ?addvSr.
apply/prodvP=> u _ Uu /memv_addP[v1 Vv1 [v2 Vv2 ->]].
by rewrite mulrDr memv_add ?memv_mul.
Qed.
HB.instance Definition _ := Monoid.isAddLaw.Build {vspace aT} prodv addv
prodvDl prodvDr. | 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 | prodvDr | |
prodvA: associative prodv.
Proof.
move=> U V W; rewrite -(span_basis (vbasisP U)) span_def !big_distrl /=.
apply: eq_bigr => u _; rewrite -(span_basis (vbasisP W)) span_def !big_distrr.
apply: eq_bigr => w _; rewrite -(span_basis (vbasisP V)) span_def /=.
rewrite !(big_distrl, big_distrr) /=; apply: eq_bigr => v _.
by rewrite !prodv_line mulrA.
Qed.
HB.instance Definition _ := Monoid.isLaw.Build {vspace aT} 1%VS prodv
prodvA prod1v prodv1. | 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 | prodvA | |
expvU n := iterop n.+1.-1 prodv U 1%VS.
Local Notation "A ^+ n" := (expv A n) : 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 | expv | |
expv0U : (U ^+ 0 = 1)%VS. 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 | expv0 | |
expv1U : (U ^+ 1 = U)%VS. 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 | expv1 | |
expv2U : (U ^+ 2 = U * U)%VS. 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 | expv2 | |
expvSlU n : (U ^+ n.+1 = U * U ^+ n)%VS.
Proof. by case: n => //; rewrite prodv1. 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 | expvSl | |
expv0nn : (0 ^+ n = if n is _.+1 then 0 else 1)%VS.
Proof. by case: n => // n; rewrite expvSl prod0v. 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 | expv0n | |
expv1nn : (1 ^+ n = 1)%VS.
Proof. by elim: n => // n IHn; rewrite expvSl IHn prodv1. 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 | expv1n | |
expvDU m n : (U ^+ (m + n) = U ^+ m * U ^+ n)%VS.
Proof. by elim: m => [|m IHm]; rewrite ?prod1v // !expvSl IHm prodvA. 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 | expvD | |
expvSrU n : (U ^+ n.+1 = U ^+ n * U)%VS.
Proof. by rewrite -addn1 expvD. 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 | expvSr | |
expvMU m n : (U ^+ (m * n) = U ^+ m ^+ n)%VS.
Proof. by elim: n => [|n IHn]; rewrite ?muln0 // mulnS expvD IHn 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 | expvM |
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