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 |
|---|---|---|---|---|---|---|
Definition_ := Finite.on {poly %/ h}.
Hypothesis hI : monic_irreducible_poly h.
HB.instance Definition _ := Finite.on {poly %/ h with hI}. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | Definition | |
card_qfpoly: #|{poly %/ h with hI}| = #|R| ^ (size h).-1.
Proof. by rewrite card_monic_qpoly ?hI. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | card_qfpoly | |
card_qfpoly_gt1: 1 < #|{poly %/ h with hI}|.
Proof. by have := card_finNzRing_gt1 {poly %/ h with hI}. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | card_qfpoly_gt1 | |
in_qpoly_comp_horner(p q : {poly R}) :
in_qpoly h (p \Po q) =
(map_poly (qpolyC h) p).[in_qpoly h q].
Proof.
have hQM := monic_mk_monic h.
rewrite comp_polyE /map_poly poly_def horner_sum /=.
apply: val_inj.
rewrite /= rmodp_sum // poly_of_qpoly_sum.
apply: eq_bigr => i _.
rewrite !hornerE /in_qpoly /=.
rewrite ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | in_qpoly_comp_horner | |
map_poly_div_inj: injective (map_poly (qpolyC h)).
Proof.
apply: map_inj_poly => [x y /val_eqP /eqP /polyC_inj //|].
by rewrite qpolyC0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | map_poly_div_inj | |
qfpoly_const(R : idomainType) (h : {poly R})
(hMI : monic_irreducible_poly h) : R -> {poly %/ h with hMI} :=
qpolyC h. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qfpoly_const | |
map_fpoly_div_inj(R : idomainType) (h : {poly R})
(hMI : monic_irreducible_poly h) :
injective (map_poly (qfpoly_const hMI)).
Proof. by apply: (@map_poly_div_inj R h). Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | map_fpoly_div_inj | |
qfpoly_splitting_field_type:=
FinSplittingFieldType F {poly %/ h with hI}. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qfpoly_splitting_field_type | |
primitive_poly(p: {poly F}) :=
let v := #|{poly %/ p}|.-1 in
[&& p \is monic,
irreducibleb p,
p %| 'X^v - 1 &
[forall n : 'I_v, (p %| 'X^n - 1) ==> (n == 0%N :> nat)]]. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | primitive_poly | |
primitive_polyP(p : {poly F}) :
reflect
(let v := #|{poly %/ p}|.-1 in
[/\ monic_irreducible_poly p,
p %| 'X^v - 1 &
forall n, 0 < n < v -> ~~ (p %| 'X^n - 1)])
(primitive_poly p).
Proof.
apply: (iffP and4P) => [[H1 H2 H3 /forallP H4] v|[[H1 H2] H3 H4]]; split => //.
- by split => //... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | primitive_polyP | |
primitive_mi: monic_irreducible_poly h.
Proof. by case/primitive_polyP: Hh. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | primitive_mi | |
primitive_poly_in_qpoly_eq0p : (in_qpoly h p == 0) = (h %| p).
Proof.
have hM : h \is monic by case/and4P:Hh.
have hMi : monic_irreducible_poly h by apply: primitive_mi.
apply/eqP/idP => [/val_eqP /= | hDp].
by rewrite -Pdiv.IdomainMonic.modpE mk_monicE.
by apply/val_eqP; rewrite /= -Pdiv.IdomainMonic.modpE mk_monicE... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | primitive_poly_in_qpoly_eq0 | |
card_primitive_qpoly: #|{poly %/ h}|= #|F| ^ (size h).-1.
Proof. by rewrite card_monic_qpoly ?primitive_mi. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | card_primitive_qpoly | |
qX_neq0: 'qX != 0 :> qT.
Proof.
apply/eqP => /val_eqP/=.
by rewrite [rmodp _ _]qpolyXE ?polyX_eq0 //; case: primitive_mi.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qX_neq0 | |
qX_in_unit: ('qX : qT) \in GRing.unit.
Proof. by rewrite unitfE /= qX_neq0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qX_in_unit | |
gX: {unit qT} := FinRing.unit _ qX_in_unit. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | gX | |
dvdp_ordern : (h %| 'X^n - 1) = (gX ^+ n == 1)%g.
Proof.
have [hM hI] := primitive_mi.
have eqr_add2r (r : nzRingType) (a b c : r) : (a + c == b + c) = (a == b).
by apply/eqP/eqP => [H|->//]; rewrite -(addrK c a) H addrK.
rewrite -val_eqE /= val_unitX /= -val_eqE /=.
rewrite (poly_of_qpolyX) qpolyXE // mk_monicE //.
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | dvdp_order | |
gX_order: #[gX]%g = (#|qT|).-1.
Proof.
have /primitive_polyP[Hp1 Hp2 Hp3] := Hh.
set n := _.-1 in Hp2 Hp3 *.
have n_gt0 : 0 < n by rewrite ltn_predRL card_qfpoly_gt1.
have [hM hI] := primitive_mi.
have gX_neq1 : gX != 1%g.
apply/eqP/val_eqP/eqP/val_eqP=> /=.
rewrite [X in X != _]qpolyXE /= //.
by apply/eqP=> Hx1... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | gX_order | |
gX_all: <[gX]>%g = [set: {unit qT}]%G.
Proof.
apply/eqP; rewrite eqEcard; apply/andP; split.
by apply/subsetP=> i; rewrite inE.
rewrite leq_eqVlt; apply/orP; left; apply/eqP.
rewrite -orderE gX_order card_qfpoly -[in RHS](mk_monicE primitive_mi).
rewrite -card_qpoly -(cardC1 (0 : {poly %/ h with primitive_mi})).
rewr... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | gX_all | |
qlogp(p : qT) : nat :=
odflt (Ordinal pred_card_qT_gt0) (pick [pred i in 'I_ _ | ('qX ^+ i == p)]). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qlogp | |
qlogp_ltp : qlogp p < #|qT|.-1.
Proof. by rewrite /qlogp; case: pickP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qlogp_lt | |
qlogp_qX(p : qT) : p != 0 -> 'qX ^+ (qlogp p) = p.
Proof.
move=> p_neq0.
have Up : p \in GRing.unit by rewrite unitfE.
pose gp : {unit qT}:= FinRing.unit _ Up.
have /cyclePmin[i iLc iX] : gp \in <[gX]>%g by rewrite gX_all inE.
rewrite gX_order in iLc.
rewrite /qlogp; case: pickP => [j /eqP//|/(_ (Ordinal iLc))] /eqP[].... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qlogp_qX | |
qX_order_card: 'qX ^+ (#|qT|).-1 = 1 :> qT.
Proof.
have /primitive_polyP [_ Hd _] := Hh.
rewrite dvdp_order in Hd.
have -> : 1 = val (1%g : {unit qT}) by [].
by rewrite -(eqP Hd) val_unitX.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qX_order_card | |
qX_order_dvd(i : nat) : 'qX ^+ i = 1 :> qT -> (#|qT|.-1 %| i)%N.
Proof.
rewrite -gX_order cyclic.order_dvdn => Hd.
by apply/eqP/val_inj; rewrite /= -Hd val_unitX.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qX_order_dvd | |
qlogp0: qlogp 0 = 0%N.
Proof.
rewrite /qlogp; case: pickP => //= x.
by rewrite (expf_eq0 ('qX : qT)) (negPf qX_neq0) andbF.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qlogp0 | |
qlogp1: qlogp 1 = 0%N.
Proof.
case: (qlogp 1 =P 0%N) => // /eqP log1_neq0.
have := qlogp_lt 1; rewrite ltnNge => /negP[].
apply: dvdn_leq; first by rewrite lt0n.
by rewrite qX_order_dvd // qlogp_qX ?oner_eq0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qlogp1 | |
qlogp_eq0(q : qT) : (qlogp q == 0%N) = (q == 0) || (q == 1).
Proof.
case: (q =P 0) => [->|/eqP q_neq0]/=; first by rewrite qlogp0.
case: (q =P 1) => [->|/eqP q_neq1]/=; first by rewrite qlogp1.
rewrite /qlogp; case: pickP => [x|/(_ (Ordinal (qlogp_lt q)))] /=.
by case: ((x : nat) =P 0%N) => // ->; rewrite expr0 eq_sy... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qlogp_eq0 | |
qX_exp_neq0i : 'qX ^+ i != 0 :> qT.
Proof. by rewrite expf_eq0 negb_and qX_neq0 orbT. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qX_exp_neq0 | |
qX_exp_inji j :
i < #|qT|.-1 -> j < #|qT|.-1 -> 'qX ^+ i = 'qX ^+ j :> qT -> i = j.
Proof.
wlog iLj : i j / (i <= j)%N => [Hw|] iL jL Hqx.
case: (ltngtP i j)=> // /ltnW iLj; first by apply: Hw.
by apply/sym_equal/Hw.
suff ji_eq0 : (j - i = 0)%N by rewrite -(subnK iLj) ji_eq0.
case: ((j - i)%N =P 0%N) => // /eqP j... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qX_exp_inj | |
powX_eq_modi j : i = j %[mod #|qT|.-1] -> 'qX ^+ i = 'qX ^+ j :> qT.
Proof.
set n := _.-1 => iEj.
rewrite [i](divn_eq i n) [j](divn_eq j n) !exprD ![(_ * n)%N]mulnC.
by rewrite !exprM !qX_order_card !expr1n !mul1r iEj.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | powX_eq_mod | |
qX_expKi : i < #|qT|.-1 -> qlogp ('qX ^+ i) = i.
Proof.
move=> iLF; apply: qX_exp_inj => //; first by apply: qlogp_lt.
by rewrite qlogp_qX // expf_eq0 (negPf qX_neq0) andbF.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qX_expK | |
qlogpD(q1 q2 : qT) :
q1 != 0 -> q2 != 0 ->qlogp (q1 * q2) = ((qlogp q1 + qlogp q2) %% #|qT|.-1)%N.
Proof.
move=> q1_neq0 q2_neq0.
apply: qX_exp_inj; [apply: qlogp_lt => // | rewrite ltn_mod // |].
rewrite -[RHS]mul1r -(expr1n _ ((qlogp q1 + qlogp q2) %/ #|qT|.-1)).
rewrite -qX_order_card -exprM mulnC -exprD -divn_eq ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qlogpD | |
plogp(p q : {poly F}) :=
if boolP (primitive_poly p) is AltTrue Hh then
qlogp ((in_qpoly p q) : {poly %/ p with primitive_mi Hh})
else 0%N. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | plogp | |
plogp_lt(p q : {poly F}) : 2 < size p -> plogp p q < #|{poly %/ p}|.-1.
Proof.
move=> /ltnW size_gt1.
rewrite /plogp.
case (boolP (primitive_poly p)) => // Hh; first by apply: qlogp_lt.
by rewrite ltn_predRL (card_finNzRing_gt1 {poly %/ p}).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | plogp_lt | |
plogp_X(p q : {poly F}) :
2 < size p -> primitive_poly p -> ~~ (p %| q) -> p %| q - 'X ^+ plogp p q.
Proof.
move=> sp_gt2 Hh pNDq.
rewrite /plogp.
case (boolP (primitive_poly p)) => // Hh'; last by case/negP: Hh'.
have pM : p \is monic by case/and4P: Hh'.
have pMi : monic_irreducible_poly p by apply: primitive_mi.
se... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | plogp_X | |
plogp0(p : {poly F}) : 2 < size p -> plogp p 0 = 0%N.
Proof.
move=> sp_gt2; rewrite /plogp; case (boolP (primitive_poly p)) => // i.
by rewrite in_qpoly0 qlogp0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | plogp0 | |
plogp1(p : {poly F}) : 2 < size p -> plogp p 1 = 0%N.
Proof.
move=> sp_gt2; rewrite /plogp; case (boolP (primitive_poly p)) => // i.
suff->: in_qpoly p 1 = 1 by apply: qlogp1.
apply/val_eqP/eqP; apply: in_qpoly_small.
rewrite mk_monicE ?size_poly1 ?(leq_trans _ sp_gt2) //.
by apply: primitive_mi.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | plogp1 | |
plogp_div_eq0(p q : {poly F}) :
2 < size p -> (p %| q) -> plogp p q = 0%N.
Proof.
move=> sp_gt2; rewrite /plogp; case (boolP (primitive_poly p)) => // i pDq.
suff-> : in_qpoly p q = 0 by apply: qlogp0.
by apply/eqP; rewrite primitive_poly_in_qpoly_eq0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | plogp_div_eq0 | |
plogpD(p q1 q2 : {poly F}) :
2 < size p -> primitive_poly p -> ~~ (p %| q1) -> ~~ (p %| q2) ->
plogp p (q1 * q2) = ((plogp p q1 + plogp p q2) %% #|{poly %/ p}|.-1)%N.
Proof.
move=> sp_gt2 Pp pNDq1 pNDq2.
rewrite /plogp; case (boolP (primitive_poly p)) => [|/negP//] i /=.
have pmi := primitive_mi i.
by rewrite rmorp... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | plogpD | |
separable_poly{R : idomainType} (p : {poly R}) := coprimep p p^`(). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_poly | |
separable_poly_unlockable:= Unlockable separable_poly.unlock. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_poly_unlockable | |
separable_poly_neq0p : separable p -> p != 0.
Proof.
by apply: contraTneq => ->; rewrite unlock deriv0 coprime0p eqp01.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_poly_neq0 | |
poly_square_freePp :
(forall u v, u * v %| p -> coprimep u v)
<-> (forall u, size u != 1 -> ~~ (u ^+ 2 %| p)).
Proof.
split=> [sq'p u | sq'p u v dvd_uv_p].
by apply: contra => /sq'p; rewrite coprimepp.
rewrite coprimep_def (contraLR (sq'p _)) // (dvdp_trans _ dvd_uv_p) //.
by rewrite dvdp_mul ?dvdp_gcdl ?dvdp_gcd... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | poly_square_freeP | |
separable_polyP{p} :
reflect [/\ forall u v, u * v %| p -> coprimep u v
& forall u, u %| p -> 1 < size u -> u^`() != 0]
(separable p).
Proof.
apply: (iffP idP) => [sep_p | [sq'p nz_der1p]].
split=> [u v | u u_dv_p]; last first.
apply: contraTneq => u'0; rewrite unlock in sep_p; rewrite -le... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_polyP | |
separable_coprimep u v : separable p -> u * v %| p -> coprimep u v.
Proof. by move=> /separable_polyP[sq'p _] /sq'p. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_coprime | |
separable_nosquarep u k :
separable p -> 1 < k -> size u != 1 -> (u ^+ k %| p) = false.
Proof.
move=> /separable_polyP[/poly_square_freeP sq'p _] /subnKC <- /sq'p.
by apply: contraNF; apply: dvdp_trans; rewrite exprD dvdp_mulr.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_nosquare | |
separable_deriv_eq0p u :
separable p -> u %| p -> 1 < size u -> (u^`() == 0) = false.
Proof. by move=> /separable_polyP[_ nz_der1p] u_p /nz_der1p/negPf->. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_deriv_eq0 | |
dvdp_separablep q : q %| p -> separable p -> separable q.
Proof.
move=> /(dvdp_trans _)q_dv_p /separable_polyP[sq'p nz_der1p].
by apply/separable_polyP; split=> [u v /q_dv_p/sq'p | u /q_dv_p/nz_der1p].
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | dvdp_separable | |
separable_mulp q :
separable (p * q) = [&& separable p, separable q & coprimep p q].
Proof.
apply/idP/and3P => [sep_pq | [sep_p sep_q co_pq]].
rewrite !(dvdp_separable _ sep_pq) ?dvdp_mulIr ?dvdp_mulIl //.
by rewrite (separable_coprime sep_pq).
rewrite unlock in sep_p sep_q *.
rewrite derivM coprimepMl {1}addrC m... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_mul | |
eqp_separablep q : p %= q -> separable p = separable q.
Proof. by case/andP=> p_q q_p; apply/idP/idP=> /dvdp_separable->. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | eqp_separable | |
separable_rootp x :
separable (p * ('X - x%:P)) = separable p && ~~ root p x.
Proof.
rewrite separable_mul; apply: andb_id2l => seq_p.
by rewrite unlock derivXsubC coprimep1 coprimep_XsubC.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_root | |
separable_prod_XsubC(r : seq R) :
separable (\prod_(x <- r) ('X - x%:P)) = uniq r.
Proof.
elim: r => [|x r IH]; first by rewrite big_nil unlock /separable_poly coprime1p.
by rewrite big_cons mulrC separable_root IH root_prod_XsubC andbC.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_prod_XsubC | |
make_separablep : p != 0 -> separable (p %/ gcdp p p^`()).
Proof.
set g := gcdp p p^`() => nz_p; apply/separable_polyP.
have max_dvd_u (u : {poly R}): 1 < size u -> exists k, ~~ (u ^+ k %| p).
move=> u_gt1; exists (size p); rewrite gtNdvdp // polySpred //.
by rewrite -(ltn_subRL 1) subn1 size_exp leq_pmull // -(sub... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | make_separable | |
separable_map(F : fieldType) (R : idomainType)
(f : {rmorphism F -> R}) (p : {poly F}) :
separable_poly (map_poly f p) = separable_poly p.
Proof.
by rewrite unlock deriv_map /coprimep -gcdp_map size_map_poly.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_map | |
large_field_PETq :
root (q ^ iota) y -> separable_poly q ->
exists2 r, r != 0
& forall t (z := iota t * y - x), ~~ root r (iota t) -> inFz z x /\ inFz z y.
Proof.
move=> qy_0 sep_q; have nz_q := separable_poly_neq0 sep_q.
have /factor_theorem[q0 Dq] := qy_0.
set p1 := p ^ iota \Po ('X + x%:P); set q1 := q0 \Po ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | large_field_PET | |
pchar0_PET(q : {poly F}) :
q != 0 -> root (q ^ iota) y -> [pchar F] =i pred0 ->
exists n, let z := y *+ n - x in inFz z x /\ inFz z y.
Proof.
move=> nz_q qy_0 /pcharf0P pcharF0.
without loss{nz_q} sep_q: q qy_0 / separable_poly q.
move=> IHq; apply: IHq (make_separable nz_q).
have /dvdpP[q1 Dq] := dvdp_gcdl q... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | pchar0_PET | |
char0_PET:= (pchar0_PET) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | char0_PET | |
Derivation: bool :=
all2rel (fun u v => D (u * v) == D u * v + u * D v) (vbasis K).
Hypothesis derD : Derivation. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation | |
Derivation_mul: {in K &, forall u v, D (u * v) = D u * v + u * D v}.
Proof.
move=> u v /coord_vbasis-> /coord_vbasis->.
rewrite !(mulr_sumr, linear_sum) -big_split; apply: eq_bigr => /= j _.
rewrite !mulr_suml linear_sum -big_split; apply: eq_bigr => /= i _.
rewrite !(=^~ scalerAl, linearZZ) -!scalerAr linearZZ -!scale... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation_mul | |
Derivation_mul_poly(Dp := map_poly D) :
{in polyOver K &, forall p q, Dp (p * q) = Dp p * q + p * Dp q}.
Proof.
move=> p q Kp Kq; apply/polyP=> i; rewrite {}/Dp coefD coef_map /= !coefM.
rewrite linear_sum -big_split; apply: eq_bigr => /= j _.
by rewrite !{1}coef_map Derivation_mul ?(polyOverP _).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation_mul_poly | |
DerivationSE K D : (K <= E)%VS -> Derivation E D -> Derivation K D.
Proof.
move/subvP=> sKE derD; apply/allrelP=> x y Kx Ky; apply/eqP.
by rewrite (Derivation_mul derD) ?sKE // vbasis_mem.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | DerivationS | |
Derivation1: D 1 = 0.
Proof.
apply: (addIr (D (1 * 1))); rewrite add0r {1}mul1r.
by rewrite (Derivation_mul derD) ?mem1v // mulr1 mul1r.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation1 | |
Derivation_scalarx : x \in 1%VS -> D x = 0.
Proof. by case/vlineP=> y ->; rewrite linearZ /= Derivation1 scaler0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation_scalar | |
Derivation_expx m : x \in E -> D (x ^+ m) = x ^+ m.-1 *+ m * D x.
Proof.
move=> Ex; case: m; first by rewrite expr0 mulr0n mul0r Derivation1.
elim=> [|m IHm]; first by rewrite mul1r.
rewrite exprS (Derivation_mul derD) //; last by apply: rpredX.
by rewrite mulrC IHm mulrA mulrnAr -exprS -mulrDl.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation_exp | |
Derivation_hornerp x :
p \is a polyOver E -> x \in E ->
D p.[x] = (map_poly D p).[x] + p^`().[x] * D x.
Proof.
move=> Ep Ex; elim/poly_ind: p Ep => [|p c IHp] /polyOverP EpXc.
by rewrite !(raddf0, horner0) mul0r add0r.
have Ep: p \is a polyOver E.
by apply/polyOverP=> i; have:= EpXc i.+1; rewrite coefD coefMX... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation_horner | |
separable_elementU x := separable_poly (minPoly U x). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_element | |
separable_elementP:
reflect (exists f, [/\ f \is a polyOver K, root f x & separable_poly f])
(separable_element K x).
Proof.
apply: (iffP idP) => [sep_x | [f [Kf /(minPoly_dvdp Kf)/dvdpP[g ->]]]].
by exists (minPoly K x); rewrite minPolyOver root_minPoly.
by rewrite separable_mul => /and3P[].
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_elementP | |
base_separable: x \in K -> separable_element K x.
Proof.
move=> Kx; apply/separable_elementP; exists ('X - x%:P).
by rewrite polyOverXsubC root_XsubC unlock !derivCE coprimep1.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | base_separable | |
separable_nz_der: separable_element K x = ((minPoly K x)^`() != 0).
Proof.
rewrite /separable_element unlock.
apply/idP/idP=> [|nzPx'].
by apply: contraTneq => ->; rewrite coprimep0 -size_poly_eq1 size_minPoly.
have gcdK : gcdp (minPoly K x) (minPoly K x)^`() \in polyOver K.
by rewrite gcdp_polyOver ?polyOver_deriv... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_nz_der | |
separablePn_pchar:
reflect (exists2 p, p \in [pchar L] &
exists2 g, g \is a polyOver K & minPoly K x = g \Po 'X^p)
(~~ separable_element K x).
Proof.
rewrite separable_nz_der negbK; set f := minPoly K x.
apply: (iffP eqP) => [f'0 | [p Hp [g _ ->]]]; last first.
by rewrite deriv_comp derivXn -s... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separablePn_pchar | |
separable_root_der: separable_element K x (+) root (minPoly K x)^`() x.
Proof.
have KpKx': _^`() \is a polyOver K := polyOver_deriv (minPolyOver K x).
rewrite separable_nz_der addNb (root_small_adjoin_poly KpKx') ?addbb //.
by rewrite (leq_trans (size_poly _ _)) ?size_minPoly.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_root_der | |
Derivation_separableD :
Derivation <<K; x>> D -> separable_element K x ->
D x = - (map_poly D (minPoly K x)).[x] / (minPoly K x)^`().[x].
Proof.
move=> derD sepKx; have:= separable_root_der; rewrite {}sepKx -sub0r => nzKx'x.
apply: canRL (mulfK nzKx'x) (canRL (addrK _) _); rewrite mulrC addrC.
rewrite -(Derivatio... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation_separable | |
Definition_ := @GRing.isZmodMorphism.Build _ _ body
(extendDerivation_zmod_morphism_subproof E).
HB.instance Definition _ := @GRing.isScalable.Build _ _ _ _ body
(extendDerivation_scalable_subproof E).
Let extendDerivationLinear := Eval hnf in (body : {linear _ -> _}). | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Definition | |
extendDerivation: 'End(L) := linfun extendDerivationLinear. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | extendDerivation | |
extendDerivation_idy : y \in K -> extendDerivation K y = D y.
Proof.
move=> yK; rewrite lfunE /= Fadjoin_polyC // derivC map_polyC hornerC.
by rewrite horner0 mul0r addr0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | extendDerivation_id | |
extendDerivation_hornerp :
p \is a polyOver K -> separable_element K x ->
extendDerivation K p.[x] = (map_poly D p).[x] + p^`().[x] * Dx K.
Proof.
move=> Kp sepKx; have:= separable_root_der; rewrite {}sepKx /= => nz_pKx'x.
rewrite [in RHS](divp_eq p (minPoly K x)) lfunE /= Fadjoin_poly_mod ?raddfD //=.
rewrite (D... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | extendDerivation_horner | |
extendDerivationP:
separable_element K x -> Derivation <<K; x>> (extendDerivation K).
Proof.
move=> sep; apply/allrelP=> u v /vbasis_mem Hu /vbasis_mem Hv; apply/eqP.
rewrite -(Fadjoin_poly_eq Hu) -(Fadjoin_poly_eq Hv) -hornerM.
rewrite !{1}extendDerivation_horner ?{1}rpredM ?Fadjoin_polyOver //.
rewrite (Derivation_... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | extendDerivationP | |
Derivation_separableP:
reflect
(forall D, Derivation <<K; x>> D -> K <= lker D -> <<K; x>> <= lker D)%VS
(separable_element K x).
Proof.
apply: (iffP idP) => [sepKx D derD /subvP DK_0 | derKx_0].
have{} DK_0 q: q \is a polyOver K -> map_poly D q = 0.
move=> /polyOverP Kq; apply/polyP=> i; apply/eqP.
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Derivation_separableP | |
separablePn:= (separablePn_pchar) (only parsing).
Arguments separable_elementP {K x}. | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separablePn | |
separable_elementSK E x :
(K <= E)%VS -> separable_element K x -> separable_element E x.
Proof.
move=> sKE /separable_elementP[f [fK rootf sepf]]; apply/separable_elementP.
by exists f; rewrite (polyOverSv sKE).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_elementS | |
adjoin_separableP{K x} :
reflect (forall y, y \in <<K; x>>%VS -> separable_element K y)
(separable_element K x).
Proof.
apply: (iffP idP) => [sepKx | -> //]; last exact: memv_adjoin.
move=> _ /Fadjoin_polyP[q Kq ->]; apply/Derivation_separableP=> D derD DK_0.
apply/subvP=> _ /Fadjoin_polyP[p Kp ->].
rewrite... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | adjoin_separableP | |
separable_exponent_pcharK x :
exists n, [pchar L].-nat n && separable_element K (x ^+ n).
Proof.
pose d := adjoin_degree K x; move: {2}d.+1 (ltnSn d) => n.
elim: n => // n IHn in x @d *; rewrite ltnS => le_d_n.
have [[p pcharLp]|] := altP (separablePn_pchar K x); last by rewrite negbK; exists 1.
case=> g Kg defKx; ha... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_exponent_pchar | |
separable_exponent:= (separable_exponent_pchar) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_exponent | |
pcharf0_separableK : [pchar L] =i pred0 -> forall x, separable_element K x.
Proof.
move=> pcharL0 x; have [n /andP[pcharLn]] := separable_exponent_pchar K x.
by rewrite (pnat_1 pcharLn (sub_in_pnat _ pcharLn)) // => p _; rewrite pcharL0.
Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use pcharf0_separable instead.")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | pcharf0_separable | |
charf0_separable:= (pcharf0_separable) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | charf0_separable | |
pcharf_p_separableK x e p :
p \in [pchar L] -> separable_element K x = (x \in <<K; x ^+ (p ^ e.+1)>>%VS).
Proof.
move=> pcharLp; apply/idP/idP=> [sepKx | /Fadjoin_poly_eq]; last first.
set m := p ^ _; set f := Fadjoin_poly K _ x => Dx; apply/separable_elementP.
have mL0: m%:R = 0 :> L by apply/eqP; rewrite -(dvdn... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | pcharf_p_separable | |
charf_p_separable:= (pcharf_p_separable) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | charf_p_separable | |
pcharf_n_separableK x n :
[pchar L].-nat n -> 1 < n -> separable_element K x = (x \in <<K; x ^+ n>>%VS).
Proof.
rewrite -pi_pdiv; set p := pdiv n => pcharLn pi_n_p.
have pcharLp: p \in [pchar L] := pnatPpi pcharLn pi_n_p.
have <-: (n`_p)%N = n by rewrite -(eq_partn n (pcharf_eq pcharLp)) part_pnat_id.
by rewrite p_pa... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | pcharf_n_separable | |
charf_n_separable:= (pcharf_n_separable) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | charf_n_separable | |
purely_inseparable_elementU x :=
x ^+ ex_minn (separable_exponent_pchar <<U>> x) \in U. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | purely_inseparable_element | |
purely_inseparable_elementP_pchar{K x} :
reflect (exists2 n, [pchar L].-nat n & x ^+ n \in K)
(purely_inseparable_element K x).
Proof.
rewrite /purely_inseparable_element.
case: ex_minnP => n /andP[pcharLn /=]; rewrite subfield_closed => sepKxn min_xn.
apply: (iffP idP) => [Kxn | [m pcharLm Kxm]]; first by ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | purely_inseparable_elementP_pchar | |
purely_inseparable_elementP:= (purely_inseparable_elementP_pchar) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | purely_inseparable_elementP | |
separable_inseparable_elementK x :
separable_element K x && purely_inseparable_element K x = (x \in K).
Proof.
rewrite /purely_inseparable_element; case: ex_minnP => [[|m]] //=.
rewrite subfield_closed; case: m => /= [-> //| m _ /(_ 1)/implyP/= insepKx].
by rewrite (negPf insepKx) (contraNF (@base_separable K x) inse... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable_inseparable_element | |
base_inseparableK x : x \in K -> purely_inseparable_element K x.
Proof. by rewrite -separable_inseparable_element => /andP[]. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | base_inseparable | |
sub_inseparableK E x :
(K <= E)%VS -> purely_inseparable_element K x ->
purely_inseparable_element E x.
Proof.
move/subvP=> sKE /purely_inseparable_elementP_pchar[n pcharLn /sKE Exn].
by apply/purely_inseparable_elementP_pchar; exists n.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | sub_inseparable | |
finite_PET: K_is_large \/ exists z, (<< <<K; y>>; x>> = <<K; z>>)%VS.
Proof.
have [-> | /cyclic_or_large[|[a Dxa]]] := eqVneq x 0; first 2 [by left].
by rewrite addv0 subfield_closed; right; exists y.
have [-> | /cyclic_or_large[|[b Dyb]]] := eqVneq y 0; first 2 [by left].
by rewrite addv0 subfield_closed; right; e... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | finite_PET | |
Primitive_Element_Theorem: exists z, (<< <<K; y>>; x>> = <<K; z>>)%VS.
Proof.
have /polyOver_subvs[p Dp]: minPoly K x \is a polyOver K := minPolyOver K x.
have nz_pKx: minPoly K x != 0 by rewrite monic_neq0 ?monic_minPoly.
have{nz_pKx} nz_p: p != 0 by rewrite Dp map_poly_eq0 in nz_pKx.
have{Dp} px0: root (map_poly vsva... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | Primitive_Element_Theorem | |
adjoin_separable: separable_element <<K; y>> x -> separable_element K x.
Proof.
have /Derivation_separableP derKy := sepKy => /Derivation_separableP derKy_x.
have [z defKz] := Primitive_Element_Theorem.
suffices /adjoin_separableP: separable_element K z.
by apply; rewrite -defKz memv_adjoin.
apply/Derivation_separabl... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | adjoin_separable | |
strong_Primitive_Element_TheoremK x y :
separable_element <<K; x>> y ->
exists2 z : L, (<< <<K; y>>; x>> = <<K; z>>)%VS
& separable_element K x -> separable_element K y.
Proof.
move=> sepKx_y; have [n /andP[pcharLn sepKyn]] := separable_exponent_pchar K y.
have adjK_C z t: (<<<<K; z>>; t>> = <<<<K;... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | strong_Primitive_Element_Theorem | |
separableU W : bool :=
all (separable_element U) (vbasis W). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism... | field/separable.v | separable |
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