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 |
|---|---|---|---|---|---|---|
dvdpPq p :
reflect (exists2 cqq, cqq.1 != 0 & cqq.1 *: p = cqq.2 * q) (q %| p).
Proof.
rewrite dvdp_eq; apply: (iffP eqP) => [e | [[c qq] cn0 e]].
by exists (lead_coef q ^+ scalp p q, p %/ q) => //=.
apply/eqP; rewrite -dvdp_eq dvdpE.
have Ecc: c%:P != 0 by rewrite polyC_eq0.
have [->|nz_p] := eqVneq p 0; first by rewrite rdvdp0.
pose p1 : {poly R} := lead_coef q ^+ rscalp p q *: qq - c *: (rdivp p q).
have E1: c *: rmodp p q = p1 * q.
rewrite mulrDl mulNr -scalerAl -e scalerA mulrC -scalerA -scalerAl.
by rewrite -scalerBr rdivp_eq addrC addKr.
suff: p1 * q == 0 by rewrite -E1 -mul_polyC mulf_eq0 (negPf Ecc).
rewrite mulf_eq0; apply/norP; case=> p1_nz q_nz; have:= ltn_rmodp p q.
by rewrite q_nz -(size_scale _ cn0) E1 size_mul // polySpred // ltnNge leq_addl.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdpP
| |
mulpKp q : q != 0 -> p * q %/ q = lead_coef q ^+ scalp (p * q) q *: p.
Proof.
move=> qn0; apply: (rregP qn0); rewrite -scalerAl divp_eq.
suff -> : (p * q) %% q = 0 by rewrite addr0.
rewrite modpE RingComRreg.rmodp_mull ?scaler0 ?if_same //.
by red; rewrite mulrC.
by apply/rregP; rewrite lead_coef_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
mulpK
| |
mulKpp q : q != 0 -> q * p %/ q = lead_coef q ^+ scalp (p * q) q *: p.
Proof. by move=> nzq; rewrite mulrC; apply: mulpK. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
mulKp
| |
divppp : p != 0 -> p %/ p = (lead_coef p ^+ scalp p p)%:P.
Proof.
move=> np0; have := divp_eq p p.
suff -> : p %% p = 0 by rewrite addr0 -mul_polyC; move/(mulIf np0).
rewrite modpE Ring.rmodpp; last by red; rewrite mulrC.
by rewrite scaler0 if_same.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
divpp
| |
scalp0p : scalp p 0 = 0.
Proof. by rewrite /scalp unlock lead_coef0 unitr0 unlock eqxx. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
scalp0
| |
divp_smallp q : size p < size q -> p %/ q = 0.
Proof.
move=> spq; rewrite /divp unlock redivp_def /=.
by case: ifP; rewrite rdivp_small // scaler0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
divp_small
| |
leq_divpp q : (size (p %/ q) <= size p).
Proof.
rewrite /divp unlock redivp_def /=; case: ifP => ulcq; rewrite ?leq_rdivp //=.
rewrite size_scale ?leq_rdivp // -exprVn expf_neq0 // invr_eq0.
by case: eqP ulcq => // ->; rewrite unitr0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
leq_divp
| |
div0pp : 0 %/ p = 0.
Proof.
by rewrite /divp unlock redivp_def /=; case: ifP; rewrite rdiv0p // scaler0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
div0p
| |
divp0p : p %/ 0 = 0.
Proof.
by rewrite /divp unlock redivp_def /=; case: ifP; rewrite rdivp0 // scaler0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
divp0
| |
divp1m : m %/ 1 = m.
Proof.
by rewrite divpE lead_coefC unitr1 Ring.rdivp1 expr1n invr1 scale1r.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
divp1
| |
modp0p : p %% 0 = p.
Proof.
rewrite /modp unlock redivp_def; case: ifP; rewrite rmodp0 //= lead_coef0.
by rewrite unitr0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modp0
| |
mod0pp : 0 %% p = 0.
Proof.
by rewrite /modp unlock redivp_def /=; case: ifP; rewrite rmod0p // scaler0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
mod0p
| |
modp1p : p %% 1 = 0.
Proof.
by rewrite /modp unlock redivp_def /=; case: ifP; rewrite rmodp1 // scaler0.
Qed.
Hint Resolve divp0 divp1 mod0p modp0 modp1 : core.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modp1
| |
modp_smallp q : size p < size q -> p %% q = p.
Proof.
move=> spq; rewrite /modp unlock redivp_def; case: ifP; rewrite rmodp_small //.
by rewrite /= rscalp_small // expr0 /= invr1 scale1r.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modp_small
| |
modpCp c : c != 0 -> p %% c%:P = 0.
Proof.
move=> cn0; rewrite /modp unlock redivp_def /=; case: ifP; rewrite ?rmodpC //.
by rewrite scaler0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modpC
| |
modp_mullp q : (p * q) %% q = 0.
Proof.
have [-> | nq0] := eqVneq q 0; first by rewrite modp0 mulr0.
have rlcq : GRing.rreg (lead_coef q) by apply/rregP; rewrite lead_coef_eq0.
have hC : GRing.comm q (lead_coef q)%:P by red; rewrite mulrC.
rewrite modpE; case: ifP => ulcq; rewrite RingComRreg.rmodp_mull //.
exact: scaler0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modp_mull
| |
modp_mulrd p : (d * p) %% d = 0. Proof. by rewrite mulrC modp_mull. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modp_mulr
| |
modppd : d %% d = 0.
Proof. by rewrite -[d in d %% _]mul1r modp_mull. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modpp
| |
ltn_modpp q : (size (p %% q) < size q) = (q != 0).
Proof.
rewrite /modp unlock redivp_def /=; case: ifP=> ulcq; rewrite ?ltn_rmodp //=.
rewrite size_scale ?ltn_rmodp // -exprVn expf_neq0 // invr_eq0.
by case: eqP ulcq => // ->; rewrite unitr0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
ltn_modp
| |
ltn_divpld q p : d != 0 ->
(size (q %/ d) < size p) = (size q < size (p * d)).
Proof.
move=> dn0.
have: (lead_coef d) ^+ (scalp q d) != 0 by apply: lc_expn_scalp_neq0.
move/(size_scale q)<-; rewrite divp_eq; have [->|quo0] := eqVneq (q %/ d) 0.
rewrite mul0r add0r size_poly0 size_poly_gt0.
have [->|pn0] := eqVneq p 0; first by rewrite mul0r size_poly0 ltn0.
by rewrite size_mul // (polySpred pn0) addSn ltn_addl // ltn_modp.
rewrite size_polyDl; last first.
by rewrite size_mul // (polySpred quo0) addSn /= ltn_addl // ltn_modp.
have [->|pn0] := eqVneq p 0; first by rewrite mul0r size_poly0 !ltn0.
by rewrite !size_mul ?quo0 // (polySpred dn0) !addnS ltn_add2r.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
ltn_divpl
| |
leq_divprd p q : d != 0 ->
(size p <= size (q %/ d)) = (size (p * d) <= size q).
Proof. by move=> dn0; rewrite leqNgt ltn_divpl // -leqNgt. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
leq_divpr
| |
divpN0d p : d != 0 -> (p %/ d != 0) = (size d <= size p).
Proof.
move=> dn0.
by rewrite -[d in RHS]mul1r -leq_divpr // size_polyC oner_eq0 size_poly_gt0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
divpN0
| |
size_divpp q : q != 0 -> size (p %/ q) = (size p - (size q).-1)%N.
Proof.
move=> nq0; case: (leqP (size q) (size p)) => sqp; last first.
move: (sqp); rewrite -{1}(ltn_predK sqp) ltnS -subn_eq0 divp_small //.
by move/eqP->; rewrite size_poly0.
have np0 : p != 0.
by rewrite -size_poly_gt0; apply: leq_trans sqp; rewrite size_poly_gt0.
have /= := congr1 (size \o @polyseq R) (divp_eq p q).
rewrite size_scale; last by rewrite expf_eq0 lead_coef_eq0 (negPf nq0) andbF.
have [->|qq0] := eqVneq (p %/ q) 0.
by rewrite mul0r add0r=> es; move: nq0; rewrite -(ltn_modp p) -es ltnNge sqp.
rewrite size_polyDl.
by move->; apply/eqP; rewrite size_mul // (polySpred nq0) addnS /= addnK.
rewrite size_mul ?qq0 //.
move: nq0; rewrite -(ltn_modp p); move/leq_trans; apply.
by rewrite (polySpred qq0) addSn /= leq_addl.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
size_divp
| |
ltn_modpN0p q : q != 0 -> size (p %% q) < size q.
Proof. by rewrite ltn_modp. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
ltn_modpN0
| |
modp_idp q : (p %% q) %% q = p %% q.
Proof.
by have [->|qn0] := eqVneq q 0; rewrite ?modp0 // modp_small ?ltn_modp.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modp_id
| |
leq_modpm d : size (m %% d) <= size m.
Proof.
rewrite /modp unlock redivp_def /=; case: ifP; rewrite ?leq_rmodp //.
move=> ud; rewrite size_scale ?leq_rmodp // invr_eq0 expf_neq0 //.
by apply: contraTneq ud => ->; rewrite unitr0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
leq_modp
| |
dvdp0d : d %| 0. Proof. by rewrite /dvdp mod0p. Qed.
Hint Resolve dvdp0 : core.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp0
| |
dvd0pp : (0 %| p) = (p == 0). Proof. by rewrite /dvdp modp0. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvd0p
| |
dvd0pPp : reflect (p = 0) (0 %| p).
Proof. by apply: (iffP idP); rewrite dvd0p; move/eqP. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvd0pP
| |
dvdpN0p q : p %| q -> q != 0 -> p != 0.
Proof. by move=> pq hq; apply: contraTneq pq => ->; rewrite dvd0p. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdpN0
| |
dvdp1d : (d %| 1) = (size d == 1).
Proof.
rewrite /dvdp modpE; case ud: (lead_coef d \in GRing.unit); last exact: rdvdp1.
rewrite -size_poly_eq0 size_scale; first by rewrite size_poly_eq0 -rdvdp1.
by rewrite invr_eq0 expf_neq0 //; apply: contraTneq ud => ->; rewrite unitr0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp1
| |
dvd1pm : 1 %| m. Proof. by rewrite /dvdp modp1. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvd1p
| |
gtNdvdpp q : p != 0 -> size p < size q -> (q %| p) = false.
Proof.
by move=> nn0 hs; rewrite /dvdp; rewrite (modp_small hs); apply: negPf.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
gtNdvdp
| |
modp_eq0Pp q : reflect (p %% q = 0) (q %| p).
Proof. exact: (iffP eqP). Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modp_eq0P
| |
modp_eq0p q : (q %| p) -> p %% q = 0. Proof. exact: modp_eq0P. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
modp_eq0
| |
leq_divpld p q :
d %| p -> (size (p %/ d) <= size q) = (size p <= size (q * d)).
Proof.
case: (eqVneq d 0) => [-> /dvd0pP -> | nd0 hd].
by rewrite divp0 size_poly0 !leq0n.
rewrite leq_eqVlt ltn_divpl // (leq_eqVlt (size p)).
case lhs: (size p < size (q * d)); rewrite ?orbT ?orbF //.
have: (lead_coef d) ^+ (scalp p d) != 0 by rewrite expf_neq0 // lead_coef_eq0.
move/(size_scale p)<-; rewrite divp_eq; move/modp_eq0P: hd->; rewrite addr0.
have [-> | quon0] := eqVneq (p %/ d) 0.
rewrite mul0r size_poly0 2!(eq_sym 0) !size_poly_eq0.
by rewrite mulf_eq0 (negPf nd0) orbF.
have [-> | nq0] := eqVneq q 0.
by rewrite mul0r size_poly0 !size_poly_eq0 mulf_eq0 (negPf nd0) orbF.
by rewrite !size_mul // (polySpred nd0) !addnS /= eqn_add2r.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
leq_divpl
| |
dvdp_leqp q : q != 0 -> p %| q -> size p <= size q.
Proof.
move=> nq0 /modp_eq0P.
by case: leqP => // /modp_small -> /eqP; rewrite (negPf nq0).
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_leq
| |
eq_dvdpc quo q p : c != 0 -> c *: p = quo * q -> q %| p.
Proof.
move=> cn0; case: (eqVneq p 0) => [->|nz_quo def_quo] //.
pose p1 : {poly R} := lead_coef q ^+ scalp p q *: quo - c *: (p %/ q).
have E1: c *: (p %% q) = p1 * q.
rewrite mulrDl mulNr -scalerAl -def_quo scalerA mulrC -scalerA.
by rewrite -scalerAl -scalerBr divp_eq addrAC subrr add0r.
rewrite /dvdp; apply/idPn=> m_nz.
have: p1 * q != 0 by rewrite -E1 -mul_polyC mulf_neq0 // polyC_eq0.
rewrite mulf_eq0; case/norP=> p1_nz q_nz.
have := ltn_modp p q; rewrite q_nz -(size_scale (p %% q) cn0) E1.
by rewrite size_mul // polySpred // ltnNge leq_addl.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eq_dvdp
| |
dvdppd : d %| d. Proof. by rewrite /dvdp modpp. Qed.
Hint Resolve dvdpp : core.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdpp
| |
divp_dvdp q : p %| q -> (q %/ p) %| q.
Proof.
have [-> | np0] := eqVneq p 0; first by rewrite divp0.
rewrite dvdp_eq => /eqP h.
apply: (@eq_dvdp ((lead_coef p)^+ (scalp q p)) p); last by rewrite mulrC.
by rewrite expf_neq0 // lead_coef_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
divp_dvd
| |
dvdp_mullm d n : d %| n -> d %| m * n.
Proof.
case: (eqVneq d 0) => [-> /dvd0pP -> | dn0]; first by rewrite mulr0 dvdpp.
rewrite dvdp_eq => /eqP e.
apply: (@eq_dvdp (lead_coef d ^+ scalp n d) (m * (n %/ d))).
by rewrite expf_neq0 // lead_coef_eq0.
by rewrite scalerAr e mulrA.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_mull
| |
dvdp_mulrn d m : d %| m -> d %| m * n.
Proof. by move=> hdm; rewrite mulrC dvdp_mull. Qed.
Hint Resolve dvdp_mull dvdp_mulr : core.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_mulr
| |
dvdp_muld1 d2 m1 m2 : d1 %| m1 -> d2 %| m2 -> d1 * d2 %| m1 * m2.
Proof.
case: (eqVneq d1 0) => [-> /dvd0pP -> | d1n0]; first by rewrite !mul0r dvdpp.
case: (eqVneq d2 0) => [-> _ /dvd0pP -> | d2n0]; first by rewrite !mulr0.
rewrite dvdp_eq; set c1 := _ ^+ _; set q1 := _ %/ _; move/eqP=> Hq1.
rewrite dvdp_eq; set c2 := _ ^+ _; set q2 := _ %/ _; move/eqP=> Hq2.
apply: (@eq_dvdp (c1 * c2) (q1 * q2)).
by rewrite mulf_neq0 // expf_neq0 // lead_coef_eq0.
rewrite -scalerA scalerAr scalerAl Hq1 Hq2 -!mulrA.
by rewrite [d1 * (q2 * _)]mulrCA.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_mul
| |
dvdp_addrm d n : d %| m -> (d %| m + n) = (d %| n).
Proof.
case: (eqVneq d 0) => [-> /dvd0pP -> | dn0]; first by rewrite add0r.
rewrite dvdp_eq; set c1 := _ ^+ _; set q1 := _ %/ _; move/eqP=> Eq1.
apply/idP/idP; rewrite dvdp_eq; set c2 := _ ^+ _; set q2 := _ %/ _.
have sn0 : c1 * c2 != 0.
by rewrite !mulf_neq0 // expf_eq0 lead_coef_eq0 (negPf dn0) andbF.
move/eqP=> Eq2; apply: (@eq_dvdp _ (c1 *: q2 - c2 *: q1) _ _ sn0).
rewrite mulrDl -scaleNr -!scalerAl -Eq1 -Eq2 !scalerA.
by rewrite mulNr mulrC scaleNr -scalerBr addrC addKr.
have sn0 : c1 * c2 != 0.
by rewrite !mulf_neq0 // expf_eq0 lead_coef_eq0 (negPf dn0) andbF.
move/eqP=> Eq2; apply: (@eq_dvdp _ (c1 *: q2 + c2 *: q1) _ _ sn0).
by rewrite mulrDl -!scalerAl -Eq1 -Eq2 !scalerA mulrC addrC scalerDr.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_addr
| |
dvdp_addln d m : d %| n -> (d %| m + n) = (d %| m).
Proof. by rewrite addrC; apply: dvdp_addr. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_addl
| |
dvdp_addd m n : d %| m -> d %| n -> d %| m + n.
Proof. by move/dvdp_addr->. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_add
| |
dvdp_add_eqd m n : d %| m + n -> (d %| m) = (d %| n).
Proof. by move=> ?; apply/idP/idP; [move/dvdp_addr <-| move/dvdp_addl <-]. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_add_eq
| |
dvdp_subrd m n : d %| m -> (d %| m - n) = (d %| n).
Proof. by move=> ?; apply: dvdp_add_eq; rewrite -addrA addNr simp. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_subr
| |
dvdp_subld m n : d %| n -> (d %| m - n) = (d %| m).
Proof. by move/dvdp_addl<-; rewrite subrK. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_subl
| |
dvdp_subd m n : d %| m -> d %| n -> d %| m - n.
Proof. by move=> *; rewrite dvdp_subl. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_sub
| |
dvdp_modd n m : d %| n -> (d %| m) = (d %| m %% n).
Proof.
have [-> | nn0] := eqVneq n 0; first by rewrite modp0.
case: (eqVneq d 0) => [-> /dvd0pP -> | dn0]; first by rewrite modp0.
rewrite dvdp_eq; set c1 := _ ^+ _; set q1 := _ %/ _; move/eqP=> Eq1.
apply/idP/idP; rewrite dvdp_eq; set c2 := _ ^+ _; set q2 := _ %/ _.
have sn0 : c1 * c2 != 0.
by rewrite !mulf_neq0 // expf_eq0 lead_coef_eq0 (negPf dn0) andbF.
pose quo := (c1 * lead_coef n ^+ scalp m n) *: q2 - c2 *: (m %/ n) * q1.
move/eqP=> Eq2; apply: (@eq_dvdp _ quo _ _ sn0).
rewrite mulrDl mulNr -!scalerAl -!mulrA -Eq1 -Eq2 -scalerAr !scalerA.
rewrite mulrC [_ * c2]mulrC mulrA -[((_ * _) * _) *: _]scalerA -scalerBr.
by rewrite divp_eq addrC addKr.
have sn0 : c1 * c2 * lead_coef n ^+ scalp m n != 0.
rewrite !mulf_neq0 // expf_eq0 lead_coef_eq0 ?(negPf dn0) ?andbF //.
by rewrite (negPf nn0) andbF.
move/eqP=> Eq2; apply: (@eq_dvdp _ (c2 *: (m %/ n) * q1 + c1 *: q2) _ _ sn0).
rewrite -scalerA divp_eq scalerDr -!scalerA Eq2 scalerAl scalerAr Eq1.
by rewrite scalerAl mulrDl mulrA.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_mod
| |
dvdp_trans: transitive (@dvdp R).
Proof.
move=> n d m.
case: (eqVneq d 0) => [-> /dvd0pP -> // | dn0].
case: (eqVneq n 0) => [-> _ /dvd0pP -> // | nn0].
rewrite dvdp_eq; set c1 := _ ^+ _; set q1 := _ %/ _; move/eqP=> Hq1.
rewrite dvdp_eq; set c2 := _ ^+ _; set q2 := _ %/ _; move/eqP=> Hq2.
have sn0 : c1 * c2 != 0 by rewrite mulf_neq0 // expf_neq0 // lead_coef_eq0.
apply: (@eq_dvdp _ (q2 * q1) _ _ sn0).
by rewrite -scalerA Hq2 scalerAr Hq1 mulrA.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_trans
| |
dvdp_mulIlp q : p %| p * q. Proof. exact/dvdp_mulr/dvdpp. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_mulIl
| |
dvdp_mulIrp q : q %| p * q. Proof. exact/dvdp_mull/dvdpp. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_mulIr
| |
dvdp_mul2rr p q : r != 0 -> (p * r %| q * r) = (p %| q).
Proof.
move=> nzr.
have [-> | pn0] := eqVneq p 0.
by rewrite mul0r !dvd0p mulf_eq0 (negPf nzr) orbF.
have [-> | qn0] := eqVneq q 0; first by rewrite mul0r !dvdp0.
apply/idP/idP; last by move=> ?; rewrite dvdp_mul ?dvdpp.
rewrite dvdp_eq; set c := _ ^+ _; set x := _ %/ _; move/eqP=> Hx.
apply: (@eq_dvdp c x); first by rewrite expf_neq0 // lead_coef_eq0 mulf_neq0.
by apply: (mulIf nzr); rewrite -mulrA -scalerAl.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_mul2r
| |
dvdp_mul2lr p q: r != 0 -> (r * p %| r * q) = (p %| q).
Proof. by rewrite ![r * _]mulrC; apply: dvdp_mul2r. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_mul2l
| |
ltn_divprd p q :
d %| q -> (size p < size (q %/ d)) = (size (p * d) < size q).
Proof. by move=> dv_d_q; rewrite !ltnNge leq_divpl. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
ltn_divpr
| |
dvdp_expd k p : 0 < k -> d %| p -> d %| (p ^+ k).
Proof. by case: k => // k _ d_dv_m; rewrite exprS dvdp_mulr. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_exp
| |
dvdp_exp2ld k l : k <= l -> d ^+ k %| d ^+ l.
Proof. by move/subnK <-; rewrite exprD dvdp_mull // ?lead_coef_exp ?unitrX. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_exp2l
| |
dvdp_Pexp2ld k l : 1 < size d -> (d ^+ k %| d ^+ l) = (k <= l).
Proof.
move=> sd; case: leqP => [|gt_n_m]; first exact: dvdp_exp2l.
have dn0 : d != 0 by rewrite -size_poly_gt0; apply: ltn_trans sd.
rewrite gtNdvdp ?expf_neq0 // polySpred ?expf_neq0 // size_exp /=.
rewrite [size (d ^+ k)]polySpred ?expf_neq0 // size_exp ltnS ltn_mul2l.
by move: sd; rewrite -subn_gt0 subn1; move->.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_Pexp2l
| |
dvdp_exp2rp q k : p %| q -> p ^+ k %| q ^+ k.
Proof.
case: (eqVneq p 0) => [-> /dvd0pP -> // | pn0].
rewrite dvdp_eq; set c := _ ^+ _; set t := _ %/ _; move/eqP=> e.
apply: (@eq_dvdp (c ^+ k) (t ^+ k)); first by rewrite !expf_neq0 ?lead_coef_eq0.
by rewrite -exprMn -exprZn; congr (_ ^+ k).
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_exp2r
| |
dvdp_exp_subp q k l: p != 0 ->
(p ^+ k %| q * p ^+ l) = (p ^+ (k - l) %| q).
Proof.
move=> pn0; case: (leqP k l)=> [|/ltnW] hkl.
move: (hkl); rewrite -subn_eq0; move/eqP->; rewrite expr0 dvd1p.
exact/dvdp_mull/dvdp_exp2l.
by rewrite -[in LHS](subnK hkl) exprD dvdp_mul2r // expf_eq0 (negPf pn0) andbF.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_exp_sub
| |
dvdp_XsubClp x : ('X - x%:P) %| p = root p x.
Proof. by rewrite dvdpE; apply: Ring.rdvdp_XsubCl. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_XsubCl
| |
root_dvdpp q x : p %| q -> root p x -> root q x.
Proof. by rewrite -!dvdp_XsubCl => /[swap]; exact: dvdp_trans. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
root_dvdp
| |
polyXsubCPp x : reflect (p.[x] = 0) (('X - x%:P) %| p).
Proof. by rewrite dvdpE; apply: Ring.polyXsubCP. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
polyXsubCP
| |
eqp_div_XsubCp c :
(p == (p %/ ('X - c%:P)) * ('X - c%:P)) = ('X - c%:P %| p).
Proof. by rewrite dvdp_eq lead_coefXsubC expr1n scale1r. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_div_XsubC
| |
root_factor_theoremp x : root p x = (('X - x%:P) %| p).
Proof. by rewrite dvdp_XsubCl. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
root_factor_theorem
| |
uniq_roots_dvdpp rs : all (root p) rs -> uniq_roots rs ->
(\prod_(z <- rs) ('X - z%:P)) %| p.
Proof.
move=> rrs; case/(uniq_roots_prod_XsubC rrs)=> q ->.
by apply: dvdp_mull; rewrite // (eqP (monic_prod_XsubC _)) unitr1.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
uniq_roots_dvdp
| |
root_bigmulx (ps : seq {poly R}) :
~~root (\big[*%R/1]_(p <- ps) p) x = all (fun p => ~~ root p x) ps.
Proof.
elim: ps => [|p ps ihp]; first by rewrite big_nil root1.
by rewrite big_cons /= rootM negb_or ihp.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
root_bigmul
| |
eqpPm n :
reflect (exists2 c12, (c12.1 != 0) && (c12.2 != 0) & c12.1 *: m = c12.2 *: n)
(m %= n).
Proof.
apply: (iffP idP) => [| [[c1 c2]/andP[nz_c1 nz_c2 eq_cmn]]]; last first.
rewrite /eqp (@eq_dvdp c2 c1%:P) -?eq_cmn ?mul_polyC // (@eq_dvdp c1 c2%:P)//.
by rewrite eq_cmn mul_polyC.
case: (eqVneq m 0) => [-> /andP [/dvd0pP -> _] | m_nz].
by exists (1, 1); rewrite ?scaler0 // oner_eq0.
case: (eqVneq n 0) => [-> /andP [_ /dvd0pP ->] | n_nz /andP []].
by exists (1, 1); rewrite ?scaler0 // oner_eq0.
rewrite !dvdp_eq; set c1 := _ ^+ _; set c2 := _ ^+ _.
set q1 := _ %/ _; set q2 := _ %/ _; move/eqP => Hq1 /eqP Hq2;
have Hc1 : c1 != 0 by rewrite expf_eq0 lead_coef_eq0 negb_and m_nz orbT.
have Hc2 : c2 != 0 by rewrite expf_eq0 lead_coef_eq0 negb_and n_nz orbT.
have def_q12: q1 * q2 = (c1 * c2)%:P.
apply: (mulIf m_nz); rewrite mulrAC mulrC -Hq1 -scalerAr -Hq2 scalerA.
by rewrite -mul_polyC.
have: q1 * q2 != 0 by rewrite def_q12 -size_poly_eq0 size_polyC mulf_neq0.
rewrite mulf_eq0; case/norP=> nz_q1 nz_q2.
have: size q2 <= 1.
have:= size_mul nz_q1 nz_q2; rewrite def_q12 size_polyC mulf_neq0 //=.
by rewrite polySpred // => ->; rewrite leq_addl.
rewrite leq_eqVlt ltnS size_poly_leq0 (negPf nz_q2) orbF.
case/size_poly1P=> c cn0 cqe; exists (c2, c); first by rewrite Hc2.
by rewrite Hq2 -mul_polyC -cqe.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqpP
| |
eqp_eqp q: p %= q -> (lead_coef q) *: p = (lead_coef p) *: q.
Proof.
move=> /eqpP [[c1 c2] /= /andP [nz_c1 nz_c2]] eq.
have/(congr1 lead_coef) := eq; rewrite !lead_coefZ.
move=> eqC; apply/(@mulfI _ c2%:P); rewrite ?polyC_eq0 //.
by rewrite !mul_polyC scalerA -eqC mulrC -scalerA eq !scalerA mulrC.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_eq
| |
eqpxx: reflexive (@eqp R). Proof. by move=> p; rewrite /eqp dvdpp. Qed.
Hint Resolve eqpxx : core.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqpxx
| |
eqpWp q : p = q -> p %= q. Proof. by move->; rewrite eqpxx. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqpW
| |
eqp_sym: symmetric (@eqp R).
Proof. by move=> p q; rewrite /eqp andbC. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_sym
| |
eqp_trans: transitive (@eqp R).
Proof.
move=> p q r; case/andP=> Dp pD; case/andP=> Dq qD.
by rewrite /eqp (dvdp_trans Dp) // (dvdp_trans qD).
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_trans
| |
eqp_ltrans: left_transitive (@eqp R).
Proof. exact: sym_left_transitive eqp_sym eqp_trans. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_ltrans
| |
eqp_rtrans: right_transitive (@eqp R).
Proof. exact: sym_right_transitive eqp_sym eqp_trans. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_rtrans
| |
eqp0p : (p %= 0) = (p == 0).
Proof. by apply/idP/eqP => [/andP [_ /dvd0pP] | -> //]. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp0
| |
eqp01: 0 %= (1 : {poly R}) = false.
Proof. by rewrite eqp_sym eqp0 oner_eq0. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp01
| |
eqp_scalep c : c != 0 -> c *: p %= p.
Proof.
move=> c0; apply/eqpP; exists (1, c); first by rewrite c0 oner_eq0.
by rewrite scale1r.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_scale
| |
eqp_sizep q : p %= q -> size p = size q.
Proof.
have [->|Eq] := eqVneq q 0; first by rewrite eqp0; move/eqP->.
rewrite eqp_sym; have [->|Ep] := eqVneq p 0; first by rewrite eqp0; move/eqP->.
by case/andP => Dp Dq; apply: anti_leq; rewrite !dvdp_leq.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_size
| |
size_poly_eq1p : (size p == 1) = (p %= 1).
Proof.
apply/size_poly1P/idP=> [[c cn0 ep] |].
by apply/eqpP; exists (1, c); rewrite ?oner_eq0 // alg_polyC scale1r.
by move/eqp_size; rewrite size_poly1; move/eqP/size_poly1P.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
size_poly_eq1
| |
polyXsubC_eqp1(x : R) : ('X - x%:P %= 1) = false.
Proof. by rewrite -size_poly_eq1 size_XsubC. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
polyXsubC_eqp1
| |
dvdp_eqp1p q : p %| q -> q %= 1 -> p %= 1.
Proof.
move=> dpq hq.
have sizeq : size q == 1 by rewrite size_poly_eq1.
have n0q : q != 0 by case: eqP hq => // ->; rewrite eqp01.
rewrite -size_poly_eq1 eqn_leq -{1}(eqP sizeq) dvdp_leq //= size_poly_gt0.
by apply/eqP => p0; move: dpq n0q; rewrite p0 dvd0p => ->.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_eqp1
| |
eqp_dvdrq p d: p %= q -> d %| p = (d %| q).
Proof.
suff Hmn m n: m %= n -> (d %| m) -> (d %| n).
by move=> mn; apply/idP/idP; apply: Hmn=> //; rewrite eqp_sym.
by rewrite /eqp; case/andP=> pq qp dp; apply: (dvdp_trans dp).
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_dvdr
| |
eqp_dvdld2 d1 p : d1 %= d2 -> d1 %| p = (d2 %| p).
suff Hmn m n: m %= n -> (m %| p) -> (n %| p).
by move=> ?; apply/idP/idP; apply: Hmn; rewrite // eqp_sym.
by rewrite /eqp; case/andP=> dd' d'd dp; apply: (dvdp_trans d'd).
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_dvdl
| |
dvdpZrc m n : c != 0 -> m %| c *: n = (m %| n).
Proof. by move=> cn0; exact/eqp_dvdr/eqp_scale. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdpZr
| |
dvdpZlc m n : c != 0 -> (c *: m %| n) = (m %| n).
Proof. by move=> cn0; exact/eqp_dvdl/eqp_scale. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdpZl
| |
dvdpNld p : (- d) %| p = (d %| p).
Proof.
by rewrite -scaleN1r; apply/eqp_dvdl/eqp_scale; rewrite oppr_eq0 oner_neq0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdpNl
| |
dvdpNrd p : d %| (- p) = (d %| p).
Proof. by apply: eqp_dvdr; rewrite -scaleN1r eqp_scale ?oppr_eq0 ?oner_eq0. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdpNr
| |
eqp_mul2rr p q : r != 0 -> (p * r %= q * r) = (p %= q).
Proof. by move=> nz_r; rewrite /eqp !dvdp_mul2r. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_mul2r
| |
eqp_mul2lr p q: r != 0 -> (r * p %= r * q) = (p %= q).
Proof. by move=> nz_r; rewrite /eqp !dvdp_mul2l. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_mul2l
| |
eqp_mullr p q: q %= r -> p * q %= p * r.
Proof.
case/eqpP=> [[c d]] /andP [c0 d0 e]; apply/eqpP; exists (c, d); rewrite ?c0 //.
by rewrite scalerAr e -scalerAr.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_mull
| |
eqp_mulrq p r : p %= q -> p * r %= q * r.
Proof. by move=> epq; rewrite ![_ * r]mulrC eqp_mull. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_mulr
| |
eqp_expp q k : p %= q -> p ^+ k %= q ^+ k.
Proof.
move=> pq; elim: k=> [|k ihk]; first by rewrite !expr0 eqpxx.
by rewrite !exprS (@eqp_trans (q * p ^+ k)) // (eqp_mulr, eqp_mull).
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_exp
| |
polyC_eqp1(c : R) : (c%:P %= 1) = (c != 0).
Proof.
apply/eqpP/idP => [[[x y]] |nc0] /=.
case: (eqVneq c) => [->|] //= /andP [_] /negPf <- /eqP.
by rewrite alg_polyC scaler0 eq_sym polyC_eq0.
exists (1, c); first by rewrite nc0 /= oner_neq0.
by rewrite alg_polyC scale1r.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
polyC_eqp1
| |
dvdUpd p: d %= 1 -> d %| p.
Proof. by move/eqp_dvdl->; rewrite dvd1p. Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdUp
| |
dvdp_size_eqpp q : p %| q -> size p == size q = (p %= q).
Proof.
move=> pq; apply/idP/idP; last by move/eqp_size->.
have [->|Hq] := eqVneq q 0; first by rewrite size_poly0 size_poly_eq0 eqp0.
have [->|Hp] := eqVneq p 0.
by rewrite size_poly0 eq_sym size_poly_eq0 eqp_sym eqp0.
move: pq; rewrite dvdp_eq; set c := _ ^+ _; set x := _ %/ _; move/eqP=> eqpq.
have /= := congr1 (size \o @polyseq R) eqpq.
have cn0 : c != 0 by rewrite expf_neq0 // lead_coef_eq0.
rewrite (@eqp_size _ q); last exact: eqp_scale.
rewrite size_mul ?p0 // => [-> HH|]; last first.
apply/eqP=> HH; move: eqpq; rewrite HH mul0r.
by move/eqP; rewrite scale_poly_eq0 (negPf Hq) (negPf cn0).
suff: size x == 1%N.
case/size_poly1P=> y H1y H2y.
by apply/eqpP; exists (y, c); rewrite ?H1y // eqpq H2y mul_polyC.
case: (size p) HH (size_poly_eq0 p)=> [|n]; first by case: eqP Hp.
by rewrite addnS -add1n eqn_add2r; move/eqP->.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
dvdp_size_eqp
| |
eqp_rootp q : p %= q -> root p =1 root q.
Proof.
move/eqpP=> [[c d]] /andP [c0 d0 e] x; move/negPf:c0=>c0; move/negPf:d0=>d0.
by rewrite rootE -[_==_]orFb -c0 -mulf_eq0 -hornerZ e hornerZ mulf_eq0 d0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_root
| |
eqp_rmod_modp q : rmodp p q %= modp p q.
Proof.
rewrite modpE eqp_sym; case: ifP => ulcq //.
apply: eqp_scale; rewrite invr_eq0 //.
by apply: expf_neq0; apply: contraTneq ulcq => ->; rewrite unitr0.
Qed.
|
Lemma
|
algebra
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] |
algebra/polydiv.v
|
eqp_rmod_mod
|
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