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) [:... | 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 rewri... | 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 //.
-... | 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.
Q... | 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 //.
construc... | 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 r... | 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.
... | 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 _... | 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 (unpic... | 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... | 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_or... | 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: di... | 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: (... | 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 (... | 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_diviso... | 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... | 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%... | 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... | 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
... | 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` in... | 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)`_... | 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; rewri... | 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
pro... | 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 ... | 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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