# math-comp/math-comp

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 (* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *) (* Distributed under the terms of CeCILL-B. *) Require Import mathcomp.ssreflect.ssreflect. From mathcomp Require Import ssrfun ssrbool eqtype ssrnat seq choice fintype. From mathcomp Require Import bigop ssralg poly. (******************************************************************************) (* This file provides a library for the basic theory of Euclidean and pseudo- *) (* Euclidean division for polynomials over ring structures. *) (* The library defines two versions of the pseudo-euclidean division: one for *) (* coefficients in a (not necessarily commutative) ring structure and one for *) (* coefficients equipped with a structure of integral domain. From the latter *) (* we derive the definition of the usual Euclidean division for coefficients *) (* in a field. Only the definition of the pseudo-division for coefficients in *) (* an integral domain is exported by default and benefits from notations. *) (* Also, the only theory exported by default is the one of division for *) (* polynomials with coefficients in a field. *) (* Other definitions and facts are qualified using name spaces indicating the *) (* hypotheses made on the structure of coefficients and the properties of the *) (* polynomial one divides with. *) (* *) (* Pdiv.Field (exported by the present library): *) (* edivp p q == pseudo-division of p by q with p q : {poly R} where *) (* R is an idomainType. *) (* Computes (k, quo, rem) : nat * {poly r} * {poly R}, *) (* such that size rem < size q and: *) (* + if lead_coef q is not a unit, then: *) (* (lead_coef q ^+ k) *: p = q * quo + rem *) (* + else if lead_coef q is a unit, then: *) (* p = q * quo + rem and k = 0 *) (* p %/ q == quotient (second component) computed by (edivp p q). *) (* p %% q == remainder (third component) computed by (edivp p q). *) (* scalp p q == exponent (first component) computed by (edivp p q). *) (* p %| q == tests the nullity of the remainder of the *) (* pseudo-division of p by q. *) (* rgcdp p q == Pseudo-greater common divisor obtained by performing *) (* the Euclidean algorithm on p and q using redivp as *) (* Euclidean division. *) (* p %= q == p and q are associate polynomials, i.e., p %| q and *) (* q %| p, or equivalently, p = c *: q for some nonzero *) (* constant c. *) (* gcdp p q == Pseudo-greater common divisor obtained by performing *) (* the Euclidean algorithm on p and q using edivp as *) (* Euclidean division. *) (* egcdp p q == The pair of Bezout coefficients: if e := egcdp p q, *) (* then size e.1 <= size q, size e.2 <= size p, and *) (* gcdp p q %= e.1 * p + e.2 * q *) (* coprimep p q == p and q are coprime, i.e., (gcdp p q) is a nonzero *) (* constant. *) (* gdcop q p == greatest divisor of p which is coprime to q. *) (* irreducible_poly p <-> p has only trivial (constant) divisors. *) (* *) (* Pdiv.Idomain: theory available for edivp and the related operation under *) (* the sole assumption that the ring of coefficients is canonically an *) (* integral domain (R : idomainType). *) (* *) (* Pdiv.IdomainMonic: theory available for edivp and the related operations *) (* under the assumption that the ring of coefficients is canonically *) (* and integral domain (R : idomainType) an the divisor is monic. *) (* *) (* Pdiv.IdomainUnit: theory available for edivp and the related operations *) (* under the assumption that the ring of coefficients is canonically an *) (* integral domain (R : idomainType) and the leading coefficient of the *) (* divisor is a unit. *) (* *) (* Pdiv.ClosedField: theory available for edivp and the related operation *) (* under the sole assumption that the ring of coefficients is canonically *) (* an algebraically closed field (R : closedField). *) (* *) (* Pdiv.Ring : *) (* redivp p q == pseudo-division of p by q with p q : {poly R} where R is *) (* a ringType. *) (* Computes (k, quo, rem) : nat * {poly r} * {poly R}, *) (* such that if rem = 0 then quo * q = p * (lead_coef q ^+ k) *) (* *) (* rdivp p q == quotient (second component) computed by (redivp p q). *) (* rmodp p q == remainder (third component) computed by (redivp p q). *) (* rscalp p q == exponent (first component) computed by (redivp p q). *) (* rdvdp p q == tests the nullity of the remainder of the pseudo-division *) (* of p by q. *) (* rgcdp p q == analogue of gcdp for coefficients in a ringType. *) (* rgdcop p q == analogue of gdcop for coefficients in a ringType. *) (*rcoprimep p q == analogue of coprimep p q for coefficients in a ringType. *) (* *) (* Pdiv.RingComRreg : theory of the operations defined in Pdiv.Ring, when the *) (* ring of coefficients is canonically commutative (R : comRingType) and *) (* the leading coefficient of the divisor is both right regular and *) (* commutes as a constant polynomial with the divisor itself *) (* *) (* Pdiv.RingMonic : theory of the operations defined in Pdiv.Ring, under the *) (* assumption that the divisor is monic. *) (* *) (* Pdiv.UnitRing: theory of the operations defined in Pdiv.Ring, when the *) (* ring R of coefficients is canonically with units (R : unitRingType). *) (* *) (******************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Import GRing.Theory. Local Open Scope ring_scope. Reserved Notation "p %= q" (at level 70, no associativity). Local Notation simp := Monoid.simpm. Module Pdiv. Module CommonRing. Section RingPseudoDivision. Variable R : ringType. Implicit Types d p q r : {poly R}. (* Pseudo division, defined on an arbitrary ring *) Definition redivp_rec (q : {poly R}) := let sq := size q in let cq := lead_coef q in fix loop (k : nat) (qq r : {poly R})(n : nat) {struct n} := if size r < sq then (k, qq, r) else let m := (lead_coef r) *: 'X^(size r - sq) in let qq1 := qq * cq%:P + m in let r1 := r * cq%:P - m * q in if n is n1.+1 then loop k.+1 qq1 r1 n1 else (k.+1, qq1, r1). Definition redivp_expanded_def p q := if q == 0 then (0%N, 0, p) else redivp_rec q 0 0 p (size p). Fact redivp_key : unit. Proof. by []. Qed. Definition redivp : {poly R} -> {poly R} -> nat * {poly R} * {poly R} := locked_with redivp_key redivp_expanded_def. Canonical redivp_unlockable := [unlockable fun redivp]. Definition rdivp p q := ((redivp p q).1).2. Definition rmodp p q := (redivp p q).2. Definition rscalp p q := ((redivp p q).1).1. Definition rdvdp p q := rmodp q p == 0. (*Definition rmultp := [rel m d | rdvdp d m].*) Lemma redivp_def p q : redivp p q = (rscalp p q, rdivp p q, rmodp p q). Proof. by rewrite /rscalp /rdivp /rmodp; case: (redivp p q) => [[]] /=. Qed. Lemma rdiv0p p : rdivp 0 p = 0. Proof. rewrite /rdivp unlock; case: ifP => // Hp; rewrite /redivp_rec !size_poly0. by rewrite polySpred ?Hp. Qed. Lemma rdivp0 p : rdivp p 0 = 0. Proof. by rewrite /rdivp unlock eqxx. Qed. Lemma rdivp_small p q : size p < size q -> rdivp p q = 0. Proof. rewrite /rdivp unlock; have [-> | _ ltpq] := eqP; first by rewrite size_poly0. by case: (size p) => [|s]; rewrite /= ltpq. Qed. Lemma leq_rdivp p q : size (rdivp p q) <= size p. Proof. have [/rdivp_small->|] := ltnP (size p) (size q); first by rewrite size_poly0. rewrite /rdivp /rmodp /rscalp unlock. case q0: (q == 0) => /=; first by rewrite size_poly0. have: size (0 : {poly R}) <= size p by rewrite size_poly0. move: (leqnn (size p)); move: {2 3 4 6}(size p) => A. elim: (size p) 0%N (0 : {poly R}) {1 3 4}p (leqnn (size p)) => [|n ihn] k q1 r. by move/size_poly_leq0P->; rewrite /= size_poly0 lt0n size_poly_eq0 q0. move=> /= hrn hr hq1 hq; case: ltnP => //= hqr. have sq: 0 < size q by rewrite size_poly_gt0 q0. have sr: 0 < size r by apply: leq_trans sq hqr. apply: ihn => //. - apply/leq_sizeP => j hnj. rewrite coefB -scalerAl coefZ coefXnM ltn_subRL ltnNge. have hj : (size r).-1 <= j. by apply: leq_trans hnj; move: hrn; rewrite -{1}(prednK sr) ltnS. rewrite polySpred -?size_poly_gt0 // (leq_ltn_trans hj) /=; last first. by rewrite -{1}(add0n j) ltn_add2r. move: (hj); rewrite leq_eqVlt; case/orP. move/eqP<-; rewrite (@polySpred _ q) ?q0 // subSS coefMC. rewrite subKn; first by rewrite lead_coefE subrr. by rewrite -ltnS -!polySpred // ?q0 -?size_poly_gt0. move=> {hj} hj; move: (hj); rewrite prednK // coefMC; move/leq_sizeP=> -> //. suff: size q <= j - (size r - size q). by rewrite mul0r sub0r; move/leq_sizeP=> -> //; rewrite mulr0 oppr0. rewrite subnBA // addnC -(prednK sq) -(prednK sr) addSn subSS. by rewrite -addnBA ?(ltnW hj) // -{1}[_.-1]addn0 ltn_add2l subn_gt0. - apply: leq_trans (size_add _ _) _; rewrite geq_max; apply/andP; split. apply: leq_trans (size_mul_leq _ _) _. by rewrite size_polyC lead_coef_eq0 q0 /= addn1. rewrite size_opp; apply: leq_trans (size_mul_leq _ _) _. apply: leq_trans hr; rewrite -subn1 leq_subLR -{2}(subnK hqr) addnA leq_add2r. by rewrite add1n -(@size_polyXn R) size_scale_leq. apply: leq_trans (size_add _ _) _; rewrite geq_max; apply/andP; split. apply: leq_trans (size_mul_leq _ _) _. by rewrite size_polyC lead_coef_eq0 q0 /= addnS addn0. apply: leq_trans (size_scale_leq _ _) _; rewrite size_polyXn. by rewrite -subSn // leq_subLR -add1n leq_add. Qed. Lemma rmod0p p : rmodp 0 p = 0. Proof. rewrite /rmodp unlock; case: ifP => // Hp; rewrite /redivp_rec !size_poly0. by rewrite polySpred ?Hp. Qed. Lemma rmodp0 p : rmodp p 0 = p. Proof. by rewrite /rmodp unlock eqxx. Qed. Lemma rscalp_small p q : size p < size q -> rscalp p q = 0%N. Proof. rewrite /rscalp unlock; case: eqP => Eq // spq. by case sp: (size p) => [| s] /=; rewrite spq. Qed. Lemma ltn_rmodp p q : (size (rmodp p q) < size q) = (q != 0). Proof. rewrite /rdivp /rmodp /rscalp unlock; case q0 : (q == 0). by rewrite (eqP q0) /= size_poly0 ltn0. elim: (size p) 0%N 0 {1 3}p (leqnn (size p)) => [|n ihn] k q1 r. rewrite leqn0 size_poly_eq0; move/eqP->; rewrite /= size_poly0 /= lt0n. by rewrite size_poly_eq0 q0 /= size_poly0 lt0n size_poly_eq0 q0. move=> hr /=; case: (@ltnP (size r) _) => //= hsrq; rewrite ihn //. apply/leq_sizeP => j hnj; rewrite coefB. have sr: 0 < size r. by apply: leq_trans hsrq; apply: neq0_lt0n; rewrite size_poly_eq0. have sq: 0 < size q by rewrite size_poly_gt0 q0. have hj : (size r).-1 <= j. by apply: leq_trans hnj; move: hr; rewrite -{1}(prednK sr) ltnS. rewrite -scalerAl !coefZ coefXnM ltn_subRL ltnNge; move: (sr). move/prednK => {1}<-. have -> /= : (size r).-1 < size q + j. apply: (@leq_trans ((size q) + (size r).-1)); last by rewrite leq_add2l. by rewrite -{1}[_.-1]add0n ltn_add2r. move: (hj); rewrite leq_eqVlt; case/orP. move/eqP<-; rewrite -{1}(prednK sq) -{3}(prednK sr) subSS. rewrite subKn; first by rewrite coefMC !lead_coefE subrr. by move: hsrq; rewrite -{1}(prednK sq) -{1}(prednK sr) ltnS. move=> {hj} hj; move: (hj); rewrite prednK // coefMC; move/leq_sizeP=> -> //. suff: size q <= j - (size r - size q). by rewrite mul0r sub0r; move/leq_sizeP=> -> //; rewrite mulr0 oppr0. rewrite subnBA // addnC -(prednK sq) -(prednK sr) addSn subSS. by rewrite -addnBA ?(ltnW hj) // -{1}[_.-1]addn0 ltn_add2l subn_gt0. Qed. Lemma ltn_rmodpN0 p q : q != 0 -> size (rmodp p q) < size q. Proof. by rewrite ltn_rmodp. Qed. Lemma rmodp1 p : rmodp p 1 = 0. Proof. case p0: (p == 0); first by rewrite (eqP p0) rmod0p. apply/eqP; rewrite -size_poly_eq0. by have := (ltn_rmodp p 1); rewrite size_polyC !oner_neq0 ltnS leqn0. Qed. Lemma rmodp_small p q : size p < size q -> rmodp p q = p. Proof. rewrite /rmodp unlock; case: eqP => Eq; first by rewrite Eq size_poly0. by case sp: (size p) => [| s] Hs /=; rewrite sp Hs /=. Qed. Lemma leq_rmodp m d : size (rmodp m d) <= size m. Proof. case: (ltnP (size m) (size d)) => [|h]; first by move/rmodp_small->. case d0: (d == 0); first by rewrite (eqP d0) rmodp0. by apply: leq_trans h; apply: ltnW; rewrite ltn_rmodp d0. Qed. Lemma rmodpC p c : c != 0 -> rmodp p c%:P = 0. Proof. move=> Hc; apply/eqP; rewrite -size_poly_eq0 -leqn0 -ltnS. have -> : 1%N = nat_of_bool (c != 0) by rewrite Hc. by rewrite -size_polyC ltn_rmodp polyC_eq0. Qed. Lemma rdvdp0 d : rdvdp d 0. Proof. by rewrite /rdvdp rmod0p. Qed. Lemma rdvd0p n : (rdvdp 0 n) = (n == 0). Proof. by rewrite /rdvdp rmodp0. Qed. Lemma rdvd0pP n : reflect (n = 0) (rdvdp 0 n). Proof. by apply: (iffP idP); rewrite rdvd0p; move/eqP. Qed. Lemma rdvdpN0 p q : rdvdp p q -> q != 0 -> p != 0. Proof. by move=> pq hq; apply: contraL pq => /eqP ->; rewrite rdvd0p. Qed. Lemma rdvdp1 d : (rdvdp d 1) = ((size d) == 1%N). Proof. rewrite /rdvdp; case d0: (d == 0). by rewrite (eqP d0) rmodp0 size_poly0 (negPf (@oner_neq0 _)). have:= (size_poly_eq0 d); rewrite d0; move/negbT; rewrite -lt0n. rewrite leq_eqVlt; case/orP => hd; last first. by rewrite rmodp_small ?size_poly1 // oner_eq0 -(subnKC hd). rewrite eq_sym in hd; rewrite hd; have [c cn0 ->] := size_poly1P _ hd. rewrite /rmodp unlock -size_poly_eq0 size_poly1 /= size_poly1 size_polyC cn0 /=. by rewrite polyC_eq0 (negPf cn0) !lead_coefC !scale1r subrr !size_poly0. Qed. Lemma rdvd1p m : rdvdp 1 m. Proof. by rewrite /rdvdp rmodp1. Qed. Lemma Nrdvdp_small (n d : {poly R}) : n != 0 -> size n < size d -> (rdvdp d n) = false. Proof. by move=> nn0 hs; rewrite /rdvdp; rewrite (rmodp_small hs); apply: negPf. Qed. Lemma rmodp_eq0P p q : reflect (rmodp p q = 0) (rdvdp q p). Proof. exact: (iffP eqP). Qed. Lemma rmodp_eq0 p q : rdvdp q p -> rmodp p q = 0. Proof. by move/rmodp_eq0P. Qed. Lemma rdvdp_leq p q : rdvdp p q -> q != 0 -> size p <= size q. Proof. by move=> dvd_pq; rewrite leqNgt; apply: contra => /rmodp_small <-. Qed. Definition rgcdp p q := let: (p1, q1) := if size p < size q then (q, p) else (p, q) in if p1 == 0 then q1 else let fix loop (n : nat) (pp qq : {poly R}) {struct n} := let rr := rmodp pp qq in if rr == 0 then qq else if n is n1.+1 then loop n1 qq rr else rr in loop (size p1) p1 q1. Lemma rgcd0p : left_id 0 rgcdp. Proof. move=> p; rewrite /rgcdp size_poly0 size_poly_gt0 if_neg. case: ifP => /= [_ | nzp]; first by rewrite eqxx. by rewrite polySpred !(rmodp0, nzp) //; case: _.-1 => [|m]; rewrite rmod0p eqxx. Qed. Lemma rgcdp0 : right_id 0 rgcdp. Proof. move=> p; have:= rgcd0p p; rewrite /rgcdp size_poly0 size_poly_gt0 if_neg. by case: ifP => /= p0; rewrite ?(eqxx, p0) // (eqP p0). Qed. Lemma rgcdpE p q : rgcdp p q = if size p < size q then rgcdp (rmodp q p) p else rgcdp (rmodp p q) q. Proof. pose rgcdp_rec := fix rgcdp_rec (n : nat) (pp qq : {poly R}) {struct n} := let rr := rmodp pp qq in if rr == 0 then qq else if n is n1.+1 then rgcdp_rec n1 qq rr else rr. have Irec: forall m n p q, size q <= m -> size q <= n -> size q < size p -> rgcdp_rec m p q = rgcdp_rec n p q. + elim=> [|m Hrec] [|n] //= p1 q1. - rewrite leqn0 size_poly_eq0; move/eqP=> -> _. rewrite size_poly0 size_poly_gt0 rmodp0 => nzp. by rewrite (negPf nzp); case: n => [|n] /=; rewrite rmod0p eqxx. - rewrite leqn0 size_poly_eq0 => _; move/eqP=> ->. rewrite size_poly0 size_poly_gt0 rmodp0 => nzp. by rewrite (negPf nzp); case: m {Hrec} => [|m] /=; rewrite rmod0p eqxx. case: ifP => Epq Sm Sn Sq //; rewrite ?Epq //. case: (eqVneq q1 0) => [->|nzq]. by case: n m {Sm Sn Hrec} => [|m] [|n] //=; rewrite rmod0p eqxx. apply: Hrec; last by rewrite ltn_rmodp. by rewrite -ltnS (leq_trans _ Sm) // ltn_rmodp. by rewrite -ltnS (leq_trans _ Sn) // ltn_rmodp. case: (eqVneq p 0) => [-> | nzp]. by rewrite rmod0p rmodp0 rgcd0p rgcdp0 if_same. case: (eqVneq q 0) => [-> | nzq]. by rewrite rmod0p rmodp0 rgcd0p rgcdp0 if_same. rewrite /rgcdp -/rgcdp_rec. case: ltnP; rewrite (negPf nzp, negPf nzq) //=. move=> ltpq; rewrite ltn_rmodp (negPf nzp) //=. rewrite -(ltn_predK ltpq) /=; case: eqP => [->|]. by case: (size p) => [|[|s]]; rewrite /= rmodp0 (negPf nzp) // rmod0p eqxx. move/eqP=> nzqp; rewrite (negPf nzp). apply: Irec => //; last by rewrite ltn_rmodp. by rewrite -ltnS (ltn_predK ltpq) (leq_trans _ ltpq) ?leqW // ltn_rmodp. by rewrite ltnW // ltn_rmodp. move=> leqp; rewrite ltn_rmodp (negPf nzq) //=. have p_gt0: size p > 0 by rewrite size_poly_gt0. rewrite -(prednK p_gt0) /=; case: eqP => [->|]. by case: (size q) => [|[|s]]; rewrite /= rmodp0 (negPf nzq) // rmod0p eqxx. move/eqP=> nzpq; rewrite (negPf nzq). apply: Irec => //; last by rewrite ltn_rmodp. by rewrite -ltnS (prednK p_gt0) (leq_trans _ leqp) // ltn_rmodp. by rewrite ltnW // ltn_rmodp. Qed. Variant comm_redivp_spec m d : nat * {poly R} * {poly R} -> Type := ComEdivnSpec k (q r : {poly R}) of (GRing.comm d (lead_coef d)%:P -> m * (lead_coef d ^+ k)%:P = q * d + r) & (d != 0 -> size r < size d) : comm_redivp_spec m d (k, q, r). Lemma comm_redivpP m d : comm_redivp_spec m d (redivp m d). Proof. rewrite unlock; case: (altP (d =P 0))=> [->| Hd]. by constructor; rewrite !(simp, eqxx). have: GRing.comm d (lead_coef d)%:P -> m * (lead_coef d ^+ 0)%:P = 0 * d + m. by rewrite !simp. elim: (size m) 0%N 0 {1 4 6}m (leqnn (size m))=> [|n IHn] k q r Hr /=. have{Hr} ->: r = 0 by apply/eqP; rewrite -size_poly_eq0; move: Hr; case: size. suff hsd: size (0: {poly R}) < size d by rewrite hsd => /= ?; constructor. by rewrite size_polyC eqxx (polySpred Hd). case: ltP=> Hlt Heq; first by constructor=> // _; apply/ltP. apply: IHn=> [|Cda]; last first. rewrite mulrDl addrAC -addrA subrK exprSr polyC_mul mulrA Heq //. by rewrite mulrDl -mulrA Cda mulrA. apply/leq_sizeP => j Hj. rewrite coefD coefN coefMC -scalerAl coefZ coefXnM. move/ltP: Hlt; rewrite -leqNgt=> Hlt. move: Hj; rewrite leq_eqVlt; case/predU1P => [<-{j} | Hj]; last first. rewrite nth_default ?(leq_trans Hqq) // ?simp; last by apply: (leq_trans Hr). rewrite nth_default; first by rewrite if_same !simp oppr0. by rewrite -{1}(subKn Hlt) leq_sub2r // (leq_trans Hr). move: Hr; rewrite leq_eqVlt ltnS; case/predU1P=> Hqq; last first. rewrite !nth_default ?if_same ?simp ?oppr0 //. by rewrite -{1}(subKn Hlt) leq_sub2r // (leq_trans Hqq). rewrite {2}/lead_coef Hqq polySpred // subSS ltnNge leq_subr /=. by rewrite subKn ?addrN // -subn1 leq_subLR add1n -Hqq. Qed. Lemma rmodpp p : GRing.comm p (lead_coef p)%:P -> rmodp p p = 0. Proof. move=> hC; rewrite /rmodp unlock; case: ifP => hp /=; first by rewrite (eqP hp). move: (hp); rewrite -size_poly_eq0 /redivp_rec; case sp: (size p)=> [|n] // _. rewrite mul0r sp ltnn add0r subnn expr0 hC alg_polyC subrr. by case: n sp => [|n] sp; rewrite size_polyC /= eqxx. Qed. Definition rcoprimep (p q : {poly R}) := size (rgcdp p q) == 1%N. Fixpoint rgdcop_rec q p n := if n is m.+1 then if rcoprimep p q then p else rgdcop_rec q (rdivp p (rgcdp p q)) m else (q == 0)%:R. Definition rgdcop q p := rgdcop_rec q p (size p). Lemma rgdcop0 q : rgdcop q 0 = (q == 0)%:R. Proof. by rewrite /rgdcop size_poly0. Qed. End RingPseudoDivision. End CommonRing. Module RingComRreg. Import CommonRing. Section ComRegDivisor. Variable R : ringType. Variable d : {poly R}. Hypothesis Cdl : GRing.comm d (lead_coef d)%:P. Hypothesis Rreg : GRing.rreg (lead_coef d). Implicit Types p q r : {poly R}. Lemma redivp_eq q r : size r < size d -> let k := (redivp (q * d + r) d).1.1 in let c := (lead_coef d ^+ k)%:P in redivp (q * d + r) d = (k, q * c, r * c). Proof. move=> lt_rd; case: comm_redivpP=> k q1 r1; move/(_ Cdl)=> Heq. have: d != 0 by case: (size d) lt_rd (size_poly_eq0 d) => // n _ <-. move=> dn0; move/(_ dn0)=> Hs. have eC : q * d * (lead_coef d ^+ k)%:P = q * (lead_coef d ^+ k)%:P * d. by rewrite -mulrA polyC_exp (GRing.commrX k Cdl) mulrA. suff e1 : q1 = q * (lead_coef d ^+ k)%:P. congr (_, _, _) => //=; move/eqP: Heq; rewrite [_ + r1]addrC. rewrite -subr_eq; move/eqP<-; rewrite e1 mulrDl addrAC -{2}(add0r (r * _)). by rewrite eC subrr add0r. have : (q1 - q * (lead_coef d ^+ k)%:P) * d = r * (lead_coef d ^+ k)%:P - r1. apply: (@addIr _ r1); rewrite subrK. apply: (@addrI _ ((q * (lead_coef d ^+ k)%:P) * d)). by rewrite mulrDl mulNr !addrA [_ + (q1 * d)]addrC addrK -eC -mulrDl. move/eqP; rewrite -[_ == _ - _]subr_eq0 rreg_div0 //. by case/andP; rewrite subr_eq0; move/eqP. rewrite size_opp; apply: (leq_ltn_trans (size_add _ _)); rewrite size_opp. rewrite gtn_max Hs (leq_ltn_trans (size_mul_leq _ _)) //. rewrite size_polyC; case: (_ == _); last by rewrite addnS addn0. by rewrite addn0; apply: leq_ltn_trans lt_rd; case: size. Qed. (* this is a bad name *) Lemma rdivp_eq p : p * (lead_coef d ^+ (rscalp p d))%:P = (rdivp p d) * d + (rmodp p d). Proof. by rewrite /rdivp /rmodp /rscalp; case: comm_redivpP=> k q1 r1 Hc _; apply: Hc. Qed. (* section variables impose an inconvenient order on parameters *) Lemma eq_rdvdp k q1 p: p * ((lead_coef d)^+ k)%:P = q1 * d -> rdvdp d p. Proof. move=> he. have Hnq0 := rreg_lead0 Rreg; set lq := lead_coef d. pose v := rscalp p d; pose m := maxn v k. rewrite /rdvdp -(rreg_polyMC_eq0 _ (@rregX _ _ (m - v) Rreg)). suff: ((rdivp p d) * (lq ^+ (m - v))%:P - q1 * (lq ^+ (m - k))%:P) * d + (rmodp p d) * (lq ^+ (m - v))%:P == 0. rewrite rreg_div0 //; first by case/andP. by rewrite rreg_size ?ltn_rmodp //; apply rregX. rewrite mulrDl addrAC mulNr -!mulrA polyC_exp -(GRing.commrX (m-v) Cdl). rewrite -polyC_exp mulrA -mulrDl -rdivp_eq // [(_ ^+ (m - k))%:P]polyC_exp. rewrite -(GRing.commrX (m-k) Cdl) -polyC_exp mulrA -he -!mulrA -!polyC_mul. rewrite -/v -!exprD addnC subnK ?leq_maxl //. by rewrite addnC subnK ?subrr ?leq_maxr. Qed. Variant rdvdp_spec p q : {poly R} -> bool -> Type := | Rdvdp k q1 & p * ((lead_coef q)^+ k)%:P = q1 * q : rdvdp_spec p q 0 true | RdvdpN & rmodp p q != 0 : rdvdp_spec p q (rmodp p q) false. (* Is that version useable ? *) Lemma rdvdp_eqP p : rdvdp_spec p d (rmodp p d) (rdvdp d p). Proof. case hdvd: (rdvdp d p); last by apply: RdvdpN; move/rmodp_eq0P/eqP: hdvd. move/rmodp_eq0P: (hdvd)->; apply: (@Rdvdp _ _ (rscalp p d) (rdivp p d)). by rewrite rdivp_eq //; move/rmodp_eq0P: (hdvd)->; rewrite addr0. Qed. Lemma rdvdp_mull p : rdvdp d (p * d). Proof. by apply: (@eq_rdvdp 0%N p); rewrite expr0 mulr1. Qed. Lemma rmodp_mull p : rmodp (p * d) d = 0. Proof. case: (d =P 0)=> Hd; first by rewrite Hd simp rmod0p. by apply/eqP; apply: rdvdp_mull. Qed. Lemma rmodpp : rmodp d d = 0. Proof. by rewrite -{1}(mul1r d) rmodp_mull. Qed. Lemma rdivpp : rdivp d d = (lead_coef d ^+ rscalp d d)%:P. have dn0 : d != 0 by rewrite -lead_coef_eq0 rreg_neq0. move: (rdivp_eq d); rewrite rmodpp addr0. suff ->: GRing.comm d (lead_coef d ^+ rscalp d d)%:P by move/(rreg_lead Rreg)->. by rewrite polyC_exp; apply: commrX. Qed. Lemma rdvdpp : rdvdp d d. Proof. by apply/eqP; apply: rmodpp. Qed. Lemma rdivpK p : rdvdp d p -> (rdivp p d) * d = p * (lead_coef d ^+ rscalp p d)%:P. Proof. by rewrite rdivp_eq /rdvdp; move/eqP->; rewrite addr0. Qed. End ComRegDivisor. End RingComRreg. Module RingMonic. Import CommonRing. Import RingComRreg. Section MonicDivisor. Variable R : ringType. Implicit Types p q r : {poly R}. Variable d : {poly R}. Hypothesis mond : d \is monic. Lemma redivp_eq q r : size r < size d -> let k := (redivp (q * d + r) d).1.1 in redivp (q * d + r) d = (k, q, r). Proof. case: (monic_comreg mond)=> Hc Hr; move/(redivp_eq Hc Hr q). by rewrite (eqP mond) => -> /=; rewrite expr1n !mulr1. Qed. Lemma rdivp_eq p : p = (rdivp p d) * d + (rmodp p d). Proof. rewrite -rdivp_eq; rewrite (eqP mond); last exact: commr1. by rewrite expr1n mulr1. Qed. Lemma rdivpp : rdivp d d = 1. Proof. by case: (monic_comreg mond) => hc hr; rewrite rdivpp // (eqP mond) expr1n. Qed. Lemma rdivp_addl_mul_small q r : size r < size d -> rdivp (q * d + r) d = q. Proof. by move=> Hd; case: (monic_comreg mond)=> Hc Hr; rewrite /rdivp redivp_eq. Qed. Lemma rdivp_addl_mul q r : rdivp (q * d + r) d = q + rdivp r d. Proof. case: (monic_comreg mond)=> Hc Hr; rewrite {1}(rdivp_eq r) addrA. by rewrite -mulrDl rdivp_addl_mul_small // ltn_rmodp monic_neq0. Qed. Lemma rdivp_addl q r : rdvdp d q -> rdivp (q + r) d = rdivp q d + rdivp r d. Proof. case: (monic_comreg mond)=> Hc Hr; rewrite {1}(rdivp_eq r) addrA. rewrite {2}(rdivp_eq q); move/rmodp_eq0P->; rewrite addr0. by rewrite -mulrDl rdivp_addl_mul_small // ltn_rmodp monic_neq0. Qed. Lemma rdivp_addr q r : rdvdp d r -> rdivp (q + r) d = rdivp q d + rdivp r d. Proof. by rewrite addrC; move/rdivp_addl->; rewrite addrC. Qed. Lemma rdivp_mull p : rdivp (p * d) d = p. Proof. by rewrite -[p * d]addr0 rdivp_addl_mul rdiv0p addr0. Qed. Lemma rmodp_mull p : rmodp (p * d) d = 0. Proof. by apply: rmodp_mull; rewrite (eqP mond); [apply: commr1 | apply: rreg1]. Qed. Lemma rmodpp : rmodp d d = 0. Proof. by apply: rmodpp; rewrite (eqP mond); [apply: commr1 | apply: rreg1]. Qed. Lemma rmodp_addl_mul_small q r : size r < size d -> rmodp (q * d + r) d = r. Proof. by move=> Hd; case: (monic_comreg mond)=> Hc Hr; rewrite /rmodp redivp_eq. Qed. Lemma rmodp_add p q : rmodp (p + q) d = rmodp p d + rmodp q d. Proof. rewrite {1}(rdivp_eq p) {1}(rdivp_eq q). rewrite addrCA 2!addrA -mulrDl (addrC (rdivp q d)) -addrA. rewrite rmodp_addl_mul_small //; apply: (leq_ltn_trans (size_add _ _)). by rewrite gtn_max !ltn_rmodp // monic_neq0. Qed. Lemma rmodp_mulmr p q : rmodp (p * (rmodp q d)) d = rmodp (p * q) d. Proof. have -> : rmodp q d = q - (rdivp q d) * d. by rewrite {2}(rdivp_eq q) addrAC subrr add0r. rewrite mulrDr rmodp_add -mulNr mulrA. by rewrite -{2}[rmodp _ _]addr0; congr (_ + _); apply: rmodp_mull. Qed. Lemma rdvdpp : rdvdp d d. Proof. by apply: rdvdpp; rewrite (eqP mond); [apply: commr1 | apply: rreg1]. Qed. (* section variables impose an inconvenient order on parameters *) Lemma eq_rdvdp q1 p : p = q1 * d -> rdvdp d p. Proof. (* this probably means I need to specify impl args for comm_rref_rdvdp *) move=> h; apply: (@eq_rdvdp _ _ _ _ 1%N q1); rewrite (eqP mond). - exact: commr1. - exact: rreg1. by rewrite expr1n mulr1. Qed. Lemma rdvdp_mull p : rdvdp d (p * d). Proof. by apply: rdvdp_mull; rewrite (eqP mond) //; [apply: commr1 | apply: rreg1]. Qed. Lemma rdvdpP p : reflect (exists qq, p = qq * d) (rdvdp d p). Proof. case: (monic_comreg mond)=> Hc Hr; apply: (iffP idP). case: rdvdp_eqP=> // k qq; rewrite (eqP mond) expr1n mulr1 => -> _. by exists qq. by case=> [qq]; move/eq_rdvdp. Qed. Lemma rdivpK p : rdvdp d p -> (rdivp p d) * d = p. Proof. by move=> dvddp; rewrite {2}[p]rdivp_eq rmodp_eq0 ?addr0. Qed. End MonicDivisor. End RingMonic. Module Ring. Include CommonRing. Import RingMonic. Section ExtraMonicDivisor. Variable R : ringType. Implicit Types d p q r : {poly R}. Lemma rdivp1 p : rdivp p 1 = p. Proof. by rewrite -{1}(mulr1 p) rdivp_mull // monic1. Qed. Lemma rdvdp_XsubCl p x : rdvdp ('X - x%:P) p = root p x. Proof. have [HcX Hr] := (monic_comreg (monicXsubC x)). apply/rmodp_eq0P/factor_theorem; last first. by case=> p1 ->; apply: rmodp_mull; apply: monicXsubC. move=> e0; exists (rdivp p ('X - x%:P)). by rewrite {1}(rdivp_eq (monicXsubC x) p) e0 addr0. Qed. Lemma polyXsubCP p x : reflect (p.[x] = 0) (rdvdp ('X - x%:P) p). Proof. by apply: (iffP idP); rewrite rdvdp_XsubCl; move/rootP. Qed. Lemma root_factor_theorem p x : root p x = (rdvdp ('X - x%:P) p). Proof. by rewrite rdvdp_XsubCl. Qed. End ExtraMonicDivisor. End Ring. Module ComRing. Import Ring. Import RingComRreg. Section CommutativeRingPseudoDivision. Variable R : comRingType. Implicit Types d p q m n r : {poly R}. Variant redivp_spec (m d : {poly R}) : nat * {poly R} * {poly R} -> Type := EdivnSpec k (q r: {poly R}) of (lead_coef d ^+ k) *: m = q * d + r & (d != 0 -> size r < size d) : redivp_spec m d (k, q, r). Lemma redivpP m d : redivp_spec m d (redivp m d). Proof. rewrite redivp_def; constructor; last by move=> dn0; rewrite ltn_rmodp. by rewrite -mul_polyC mulrC rdivp_eq //= /GRing.comm mulrC. Qed. Lemma rdivp_eq d p : (lead_coef d ^+ (rscalp p d)) *: p = (rdivp p d) * d + (rmodp p d). Proof. by rewrite /rdivp /rmodp /rscalp; case: redivpP=> k q1 r1 Hc _; apply: Hc. Qed. Lemma rdvdp_eqP d p : rdvdp_spec p d (rmodp p d) (rdvdp d p). Proof. case hdvd: (rdvdp d p); last by apply: RdvdpN; move/rmodp_eq0P/eqP: hdvd. move/rmodp_eq0P: (hdvd)->; apply: (@Rdvdp _ _ _ (rscalp p d) (rdivp p d)). by rewrite mulrC mul_polyC rdivp_eq; move/rmodp_eq0P: (hdvd)->; rewrite addr0. Qed. Lemma rdvdp_eq q p : (rdvdp q p) = ((lead_coef q) ^+ (rscalp p q) *: p == (rdivp p q) * q). apply/rmodp_eq0P/eqP; rewrite rdivp_eq; first by move->; rewrite addr0. by move/eqP; rewrite eq_sym addrC -subr_eq subrr; move/eqP->. Qed. End CommutativeRingPseudoDivision. End ComRing. Module UnitRing. Import Ring. Section UnitRingPseudoDivision. Variable R : unitRingType. Implicit Type p q r d : {poly R}. Lemma uniq_roots_rdvdp p rs : all (root p) rs -> uniq_roots rs -> rdvdp (\prod_(z <- rs) ('X - z%:P)) p. Proof. move=> rrs; case/(uniq_roots_prod_XsubC rrs)=> q ->. exact/RingMonic.rdvdp_mull/monic_prod_XsubC. Qed. End UnitRingPseudoDivision. End UnitRing. Module IdomainDefs. Import Ring. Section IDomainPseudoDivisionDefs. Variable R : idomainType. Implicit Type p q r d : {poly R}. Definition edivp_expanded_def p q := let: (k, d, r) as edvpq := redivp p q in if lead_coef q \in GRing.unit then (0%N, (lead_coef q)^-k *: d, (lead_coef q)^-k *: r) else edvpq. Fact edivp_key : unit. Proof. by []. Qed. Definition edivp := locked_with edivp_key edivp_expanded_def. Canonical edivp_unlockable := [unlockable fun edivp]. Definition divp p q := ((edivp p q).1).2. Definition modp p q := (edivp p q).2. Definition scalp p q := ((edivp p q).1).1. Definition dvdp p q := modp q p == 0. Definition eqp p q := (dvdp p q) && (dvdp q p). End IDomainPseudoDivisionDefs. Notation "m %/ d" := (divp m d) : ring_scope. Notation "m %% d" := (modp m d) : ring_scope. Notation "p %| q" := (dvdp p q) : ring_scope. Notation "p %= q" := (eqp p q) : ring_scope. End IdomainDefs. Module WeakIdomain. Import Ring ComRing UnitRing IdomainDefs. Section WeakTheoryForIDomainPseudoDivision. Variable R : idomainType. Implicit Type p q r d : {poly R}. Lemma edivp_def p q : edivp p q = (scalp p q, divp p q, modp p q). Proof. by rewrite /scalp /divp /modp; case: (edivp p q) => [[]] /=. Qed. Lemma edivp_redivp p q : (lead_coef q \in GRing.unit) = false -> edivp p q = redivp p q. Proof. by move=> hu; rewrite unlock hu; case: (redivp p q) => [[? ?] ?]. Qed. Lemma divpE p q : p %/ q = if lead_coef q \in GRing.unit then (lead_coef q)^-(rscalp p q) *: (rdivp p q) else rdivp p q. Proof. by case ulcq: (lead_coef q \in GRing.unit); rewrite /divp unlock redivp_def ulcq. Qed. Lemma modpE p q : p %% q = if lead_coef q \in GRing.unit then (lead_coef q)^-(rscalp p q) *: (rmodp p q) else rmodp p q. Proof. by case ulcq: (lead_coef q \in GRing.unit); rewrite /modp unlock redivp_def ulcq. Qed. Lemma scalpE p q : scalp p q = if lead_coef q \in GRing.unit then 0%N else rscalp p q. Proof. by case h: (lead_coef q \in GRing.unit); rewrite /scalp unlock redivp_def h. Qed. Lemma dvdpE p q : p %| q = rdvdp p q. Proof. rewrite /dvdp modpE /rdvdp; case ulcq: (lead_coef p \in GRing.unit)=> //. rewrite -[_ *: _ == 0]size_poly_eq0 size_scale ?size_poly_eq0 //. by rewrite invr_eq0 expf_neq0 //; apply: contraTneq ulcq => ->; rewrite unitr0. Qed. Lemma lc_expn_scalp_neq0 p q : lead_coef q ^+ scalp p q != 0. Proof. case: (eqVneq q 0) => [->|nzq]; last by rewrite expf_neq0 ?lead_coef_eq0. by rewrite /scalp 2!unlock /= eqxx lead_coef0 unitr0 /= oner_neq0. Qed. Hint Resolve lc_expn_scalp_neq0. Variant edivp_spec (m d : {poly R}) : nat * {poly R} * {poly R} -> bool -> Type := |Redivp_spec k (q r: {poly R}) of (lead_coef d ^+ k) *: m = q * d + r & lead_coef d \notin GRing.unit & (d != 0 -> size r < size d) : edivp_spec m d (k, q, r) false |Fedivp_spec (q r: {poly R}) of m = q * d + r & (lead_coef d \in GRing.unit) & (d != 0 -> size r < size d) : edivp_spec m d (0%N, q, r) true. (* There are several ways to state this fact. The most appropriate statement*) (* might be polished in light of usage. *) Lemma edivpP m d : edivp_spec m d (edivp m d) (lead_coef d \in GRing.unit). Proof. have hC : GRing.comm d (lead_coef d)%:P by rewrite /GRing.comm mulrC. case ud: (lead_coef d \in GRing.unit); last first. rewrite edivp_redivp // redivp_def; constructor; rewrite ?ltn_rmodp // ?ud //. by rewrite rdivp_eq. have cdn0: lead_coef d != 0 by apply: contraTneq ud => ->; rewrite unitr0. rewrite unlock ud redivp_def; constructor => //. rewrite -scalerAl -scalerDr -mul_polyC. have hn0 : (lead_coef d ^+ rscalp m d)%:P != 0. by rewrite polyC_eq0; apply: expf_neq0. apply: (mulfI hn0); rewrite !mulrA -exprVn !polyC_exp -exprMn -polyC_mul. by rewrite divrr // expr1n mul1r -polyC_exp mul_polyC rdivp_eq. move=> dn0; rewrite size_scale ?ltn_rmodp // -exprVn expf_eq0 negb_and. by rewrite invr_eq0 cdn0 orbT. Qed. Lemma edivp_eq d q r : size r < size d -> lead_coef d \in GRing.unit -> edivp (q * d + r) d = (0%N, q, r). Proof. have hC : GRing.comm d (lead_coef d)%:P by apply: mulrC. move=> hsrd hu; rewrite unlock hu; case et: (redivp _ _) => [[s qq] rr]. have cdn0 : lead_coef d != 0. by move: hu; case d0: (lead_coef d == 0) => //; rewrite (eqP d0) unitr0. move: (et); rewrite RingComRreg.redivp_eq //; last by apply/rregP. rewrite et /=; case=> e1 e2; rewrite -!mul_polyC -!exprVn !polyC_exp. suff h x y: x * (lead_coef d ^+ s)%:P = y -> ((lead_coef d)^-1)%:P ^+ s * y = x. by congr (_, _, _); apply: h. have hn0 : (lead_coef d)%:P ^+ s != 0 by apply: expf_neq0; rewrite polyC_eq0. move=> hh; apply: (mulfI hn0); rewrite mulrA -exprMn -polyC_mul divrr //. by rewrite expr1n mul1r -polyC_exp mulrC; apply: sym_eq. Qed. Lemma divp_eq p q : (lead_coef q ^+ (scalp p q)) *: p = (p %/ q) * q + (p %% q). Proof. rewrite divpE modpE scalpE. case uq: (lead_coef q \in GRing.unit); last by rewrite rdivp_eq. rewrite expr0 scale1r; case: (altP (q =P 0)) => [-> | qn0]. rewrite mulr0 add0r lead_coef0 rmodp0 /rscalp unlock eqxx expr0 invr1. by rewrite scale1r. have hn0 : (lead_coef q ^+ rscalp p q)%:P != 0. by rewrite polyC_eq0 expf_neq0 // lead_coef_eq0. apply: (mulfI hn0). rewrite -scalerAl -scalerDr !mul_polyC scalerA mulrV ?unitrX //. by rewrite scale1r rdivp_eq. Qed. Lemma dvdp_eq q p : (q %| p) = ((lead_coef q) ^+ (scalp p q) *: p == (p %/ q) * q). Proof. rewrite dvdpE rdvdp_eq scalpE divpE; case: ifP => ulcq //. rewrite expr0 scale1r; apply/eqP/eqP. by rewrite -scalerAl; move<-; rewrite scalerA mulVr ?scale1r // unitrX. by move=> {2}->; rewrite scalerAl scalerA mulrV ?scale1r // unitrX. Qed. Lemma divpK d p : d %| p -> p %/ d * d = ((lead_coef d) ^+ (scalp p d)) *: p. Proof. by rewrite dvdp_eq; move/eqP->. Qed. Lemma divpKC d p : d %| p -> d * (p %/ d) = ((lead_coef d) ^+ (scalp p d)) *: p. Proof. by move=> ?; rewrite mulrC divpK. Qed. Lemma dvdpP q 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. case: (eqVneq p 0) => [->|nz_p]; 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 {1}mulNr -scalerAl -e scalerA mulrC -scalerA -scalerAl. by rewrite -scalerBr rdivp_eq addrC addKr. rewrite /dvdp; apply/idPn=> m_nz. have: p1 * q != 0 by rewrite -E1 -mul_polyC mulf_neq0 // -/(dvdp q p) dvdpE. rewrite mulf_eq0; case/norP=> p1_nz q_nz; have:= ltn_rmodp p q. rewrite q_nz -(size_scale _ cn0) E1 size_mul //. by rewrite polySpred // ltnNge leq_addl. Qed. Lemma mulpK p q : q != 0 -> p * q %/ q = lead_coef q ^+ scalp (p * q) q *: p. Proof. move=> qn0; move/rregP: (qn0); apply; 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 mulKp p q : q != 0 -> q * p %/ q = lead_coef q ^+ scalp (p * q) q *: p. Proof. by move=> nzq; rewrite mulrC; apply: mulpK. Qed. Lemma divpp p : 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; move/eqP; rewrite -mul_polyC (inj_eq (mulIf np0)); move/eqP. rewrite modpE Ring.rmodpp; last by red; rewrite mulrC. by rewrite scaler0 if_same. Qed. End WeakTheoryForIDomainPseudoDivision. Hint Resolve lc_expn_scalp_neq0. End WeakIdomain. Module CommonIdomain. Import Ring ComRing UnitRing IdomainDefs WeakIdomain. Section IDomainPseudoDivision. Variable R : idomainType. Implicit Type p q r d m n : {poly R}. Lemma scalp0 p : scalp p 0 = 0%N. Proof. by rewrite /scalp unlock lead_coef0 unitr0 unlock eqxx. Qed. Lemma divp_small p 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 leq_divp p q : (size (p %/ q) <= size p). Proof. rewrite /divp unlock redivp_def /=; case: ifP=> /=; rewrite ?leq_rdivp //. move=> ulcq; rewrite size_scale ?leq_rdivp //. rewrite -exprVn expf_neq0 // invr_eq0. by move: ulcq; case lcq0: (lead_coef q == 0) => //; rewrite (eqP lcq0) unitr0. Qed. Lemma div0p p : 0 %/ p = 0. Proof. by rewrite /divp unlock redivp_def /=; case: ifP; rewrite rdiv0p // scaler0. Qed. Lemma divp0 p : p %/ 0 = 0. Proof. by rewrite /divp unlock redivp_def /=; case: ifP; rewrite rdivp0 // scaler0. Qed. Lemma divp1 m : m %/ 1 = m. Proof. by rewrite divpE lead_coefC unitr1 Ring.rdivp1 expr1n invr1 scale1r. Qed. Lemma modp0 p : p %% 0 = p. Proof. rewrite /modp unlock redivp_def; case: ifP; rewrite rmodp0 //= lead_coef0. by rewrite unitr0. Qed. Lemma mod0p p : 0 %% p = 0. Proof. by rewrite /modp unlock redivp_def /=; case: ifP; rewrite rmod0p // scaler0. Qed. Lemma modp1 p : p %% 1 = 0. Proof. by rewrite /modp unlock redivp_def /=; case: ifP; rewrite rmodp1 // scaler0. Qed. Hint Resolve divp0 divp1 mod0p modp0 modp1. Lemma modp_small p 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 modpC p c : c != 0 -> p %% c%:P = 0. Proof. move=> cn0; rewrite /modp unlock redivp_def /=; case: ifP; rewrite ?rmodpC //. by rewrite scaler0. Qed. Lemma modp_mull p q : (p * q) %% q = 0. Proof. case: (altP (q =P 0)) => [-> | nq0]; 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. by rewrite modpE; case: ifP => ulcq; rewrite RingComRreg.rmodp_mull // scaler0. Qed. Lemma modp_mulr d p : (d * p) %% d = 0. Proof. by rewrite mulrC modp_mull. Qed. Lemma modpp d : d %% d = 0. Proof. by rewrite -{1}(mul1r d) modp_mull. Qed. Lemma ltn_modp p q : (size (p %% q) < size q) = (q != 0). Proof. rewrite /modp unlock redivp_def /=; case: ifP=> /=; rewrite ?ltn_rmodp //. move=> ulcq; rewrite size_scale ?ltn_rmodp //. rewrite -exprVn expf_neq0 // invr_eq0. by move: ulcq; case lcq0: (lead_coef q == 0) => //; rewrite (eqP lcq0) unitr0. Qed. Lemma ltn_divpl d q p : d != 0 -> (size (q %/ d) < size p) = (size q < size (p * d)). Proof. move=> dn0; have sd : size d > 0 by rewrite size_poly_gt0 dn0. have: (lead_coef d) ^+ (scalp q d) != 0 by apply: lc_expn_scalp_neq0. move/size_scale; move/(_ q)<-; rewrite divp_eq; case quo0 : (q %/ d == 0). rewrite (eqP quo0) mul0r add0r size_poly0. case p0 : (p == 0); first by rewrite (eqP p0) mul0r size_poly0 ltnn ltn0. have sp : size p > 0 by rewrite size_poly_gt0 p0. rewrite /= size_mul ?p0 // sp; apply: sym_eq; move/prednK:(sp)<-. by rewrite addSn /= ltn_addl // ltn_modp. rewrite size_addl; last first. rewrite size_mul ?quo0 //; move/negbT: quo0; rewrite -size_poly_gt0. by move/prednK<-; rewrite addSn /= ltn_addl // ltn_modp. case: (altP (p =P 0)) => [-> | pn0]; first by rewrite mul0r size_poly0 !ltn0. by rewrite !size_mul ?quo0 //; move/prednK: sd<-; rewrite !addnS ltn_add2r. Qed. Lemma leq_divpr d 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 divpN0 d p : d != 0 -> (p %/ d != 0) = (size d <= size p). Proof. move=> dn0; rewrite -{2}(mul1r d) -leq_divpr // size_polyC oner_eq0 /=. by rewrite size_poly_gt0. Qed. Lemma size_divp p 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. move: (nq0); rewrite -size_poly_gt0 => lt0sq. move: (sqp); move/(leq_trans lt0sq) => lt0sp. move: (lt0sp); rewrite size_poly_gt0=> p0. move: (divp_eq p q); move/(congr1 (size \o (@polyseq R)))=> /=. rewrite size_scale; last by rewrite expf_eq0 lead_coef_eq0 (negPf nq0) andbF. case: (eqVneq (p %/ q) 0) => [-> | qq0]. by rewrite mul0r add0r=> es; move: nq0; rewrite -(ltn_modp p) -es ltnNge sqp. move/negP:(qq0); move/negP; rewrite -size_poly_gt0 => lt0qq. rewrite size_addl. rewrite size_mul ?qq0 // => ->. apply/eqP; rewrite -(eqn_add2r ((size q).-1)). rewrite subnK; first by rewrite -subn1 addnBA // subn1. rewrite /leq -(subnDl 1%N) !add1n prednK // (@ltn_predK (size q)) //. by rewrite addnC subnDA subnn sub0n. by rewrite -[size q]add0n ltn_add2r. rewrite size_mul ?qq0 //. move: nq0; rewrite -(ltn_modp p); move/leq_trans; apply; move/prednK: lt0qq<-. by rewrite addSn /= leq_addl. Qed. Lemma ltn_modpN0 p q : q != 0 -> size (p %% q) < size q. Proof. by rewrite ltn_modp. Qed. Lemma modp_mod p q : (p %% q) %% q = p %% q. Proof. by case: (eqVneq q 0) => [-> | qn0]; rewrite ?modp0 // modp_small ?ltn_modp. Qed. Lemma leq_modp m 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 dvdp0 d : d %| 0. Proof. by rewrite /dvdp mod0p. Qed. Hint Resolve dvdp0. Lemma dvd0p p : (0 %| p) = (p == 0). Proof. by rewrite /dvdp modp0. Qed. Lemma dvd0pP p : reflect (p = 0) (0 %| p). Proof. by apply: (iffP idP); rewrite dvd0p; move/eqP. Qed. Lemma dvdpN0 p q : p %| q -> q != 0 -> p != 0. Proof. by move=> pq hq; apply: contraL pq=> /eqP ->; rewrite dvd0p. Qed. Lemma dvdp1 d : (d %| 1) = ((size d) == 1%N). 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 dvd1p m : 1 %| m. Proof. by rewrite /dvdp modp1. Qed. Lemma gtNdvdp p 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 modp_eq0P p q : reflect (p %% q = 0) (q %| p). Proof. exact: (iffP eqP). Qed. Lemma modp_eq0 p q : (q %| p) -> p %% q = 0. Proof. by move/modp_eq0P. Qed. Lemma leq_divpl d p q : d %| p -> (size (p %/ d) <= size q) = (size p <= size (q * d)). Proof. case: (eqVneq d 0) => [-> | nd0]. by move/dvd0pP->; rewrite divp0 size_poly0 !leq0n. move=> hd; 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; move/(_ p)<-; rewrite divp_eq. move/modp_eq0P: hd->; rewrite addr0; case: (altP (p %/ d =P 0))=> [-> | quon0]. rewrite mul0r size_poly0 eq_sym (eq_sym 0%N) size_poly_eq0. case: (altP (q =P 0)) => [-> | nq0]; first by rewrite mul0r size_poly0 eqxx. by rewrite size_poly_eq0 mulf_eq0 (negPf nq0) (negPf nd0). case: (altP (q =P 0)) => [-> | nq0]. by rewrite mul0r size_poly0 !size_poly_eq0 mulf_eq0 (negPf nd0) orbF. rewrite !size_mul //; move: nd0; rewrite -size_poly_gt0; move/prednK<-. by rewrite !addnS /= eqn_add2r. Qed. Lemma dvdp_leq p q : q != 0 -> p %| q -> size p <= size q. move=> nq0 /modp_eq0P => rpq; case: (ltnP (size p) (size q)). by move/ltnW->. rewrite leq_eqVlt; case/orP; first by move/eqP->. by move/modp_small; rewrite rpq => h; move: nq0; rewrite h eqxx. Qed. Lemma eq_dvdp c 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 {1}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 dvdpp d : d %| d. Proof. by rewrite /dvdp modpp. Qed. Hint Resolve dvdpp. Lemma divp_dvd p q : (p %| q) -> ((q %/ p) %| q). Proof. case: (eqVneq p 0) => [-> | np0]; 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 dvdp_mull m d n : d %| n -> d %| m * n. Proof. case: (eqVneq d 0) => [-> |dn0]; first by move/dvd0pP->; 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 dvdp_mulr n d m : d %| m -> d %| m * n. Proof. by move=> hdm; rewrite mulrC dvdp_mull. Qed. Hint Resolve dvdp_mull dvdp_mulr. Lemma dvdp_mul d1 d2 m1 m2 : d1 %| m1 -> d2 %| m2 -> d1 * d2 %| m1 * m2. Proof. case: (eqVneq d1 0) => [-> |d1n0]; first by move/dvd0pP->; rewrite !mul0r dvdpp. case: (eqVneq d2 0) => [-> |d2n0]; first by move=> _ /dvd0pP ->; 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 dvdp_addr m d n : d %| m -> (d %| m + n) = (d %| n). Proof. case: (altP (d =P 0)) => [-> | dn0]; first by move/dvd0pP->; 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 dvdp_addl n d m : d %| n -> (d %| m + n) = (d %| m). Proof. by rewrite addrC; apply: dvdp_addr. Qed. Lemma dvdp_add d m n : d %| m -> d %| n -> d %| m + n. Proof. by move/dvdp_addr->. Qed. Lemma dvdp_add_eq d m n : d %| m + n -> (d %| m) = (d %| n). Proof. by move=> ?; apply/idP/idP; [move/dvdp_addr <-| move/dvdp_addl <-]. Qed. Lemma dvdp_subr d m n : d %| m -> (d %| m - n) = (d %| n). Proof. by move=> ?; apply dvdp_add_eq; rewrite -addrA addNr simp. Qed. Lemma dvdp_subl d m n : d %| n -> (d %| m - n) = (d %| m). Proof. by move/dvdp_addl<-; rewrite subrK. Qed. Lemma dvdp_sub d m n : d %| m -> d %| n -> d %| m - n. Proof. by move=> *; rewrite dvdp_subl. Qed. Lemma dvdp_mod d n m : d %| n -> (d %| m) = (d %| m %% n). Proof. case: (altP (n =P 0)) => [-> | nn0]; first by rewrite modp0. case: (altP (d =P 0)) => [-> | dn0]; first by move/dvd0pP->; 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 dvdp_trans : transitive (@dvdp R). Proof. move=> n d m. case: (altP (d =P 0)) => [-> | dn0]; first by move/dvd0pP->. case: (altP (n =P 0)) => [-> | nn0]; first by move=> _ /dvd0pP ->. 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. by apply: (@eq_dvdp _ (q2 * q1) _ _ sn0); rewrite -scalerA Hq2 scalerAr Hq1 mulrA. Qed. Lemma dvdp_mulIl p q : p %| p * q. Proof. by apply: dvdp_mulr; apply: dvdpp. Qed. Lemma dvdp_mulIr p q : q %| p * q. Proof. by apply: dvdp_mull; apply: dvdpp. Qed. Lemma dvdp_mul2r r p q : r != 0 -> (p * r %| q * r) = (p %| q). Proof. move=> nzr. case: (eqVneq p 0) => [-> | pn0]. by rewrite mul0r !dvd0p mulf_eq0 (negPf nzr) orbF. case: (eqVneq q 0) => [-> | qn0]; 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). by rewrite expf_neq0 // lead_coef_eq0 mulf_neq0. by apply: (GRing.mulIf nzr); rewrite -GRing.mulrA -GRing.scalerAl. Qed. Lemma dvdp_mul2l r p q: r != 0 -> (r * p %| r * q) = (p %| q). Proof. by rewrite ![r * _]GRing.mulrC; apply: dvdp_mul2r. Qed. Lemma ltn_divpr d 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 dvdp_exp d k p : 0 < k -> d %| p -> d %| (p ^+ k). Proof. by case: k => // k _ d_dv_m; rewrite exprS dvdp_mulr. Qed. Lemma dvdp_exp2l d k l : k <= l -> d ^+ k %| d ^+ l. Proof. by move/subnK <-; rewrite exprD dvdp_mull // ?lead_coef_exp ?unitrX. Qed. Lemma dvdp_Pexp2l d 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 dvdp_exp2r p q k : p %| q -> p ^+ k %| q ^+ k. Proof. case: (eqVneq p 0) => [-> | pn0]; first by move/dvd0pP->. 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 dvdp_exp_sub p q k l: p != 0 -> (p ^+ k %| q * p ^+ l) = (p ^+ (k - l) %| q). Proof. move=> pn0; case: (leqP k l)=> hkl. move: (hkl); rewrite -subn_eq0; move/eqP->; rewrite expr0 dvd1p. apply: dvdp_mull; case: (ltnP 1%N (size p)) => sp. by rewrite dvdp_Pexp2l. move: sp; case esp: (size p) => [|sp]. by move/eqP: esp; rewrite size_poly_eq0 (negPf pn0). rewrite ltnS leqn0; move/eqP=> sp0; move/eqP: esp; rewrite sp0. by case/size_poly1P => c cn0 ->; move/subnK: hkl<-; rewrite exprD dvdp_mulIr. rewrite -{1}[k](@subnK l) 1?ltnW// exprD dvdp_mul2r//. elim: l {hkl}=> [|l ihl]; first by rewrite expr0 oner_eq0. by rewrite exprS mulf_neq0. Qed. Lemma dvdp_XsubCl p x : ('X - x%:P) %| p = root p x. Proof. by rewrite dvdpE; apply: Ring.rdvdp_XsubCl. Qed. Lemma polyXsubCP p x : reflect (p.[x] = 0) (('X - x%:P) %| p). Proof. by rewrite dvdpE; apply: Ring.polyXsubCP. Qed. Lemma eqp_div_XsubC p c : (p == (p %/ ('X - c%:P)) * ('X - c%:P)) = ('X - c%:P %| p). Proof. by rewrite dvdp_eq lead_coefXsubC expr1n scale1r. Qed. Lemma root_factor_theorem p x : root p x = (('X - x%:P) %| p). Proof. by rewrite dvdp_XsubCl. Qed. Lemma uniq_roots_dvdp p 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 root_bigmul : forall x (ps : seq {poly R}), ~~root (\big[*%R/1]_(p <- ps) p) x = all (fun p => ~~ root p x) ps. Proof. move=> x; elim; first by rewrite big_nil root1. by move=> p ps ihp; rewrite big_cons /= rootM negb_or ihp. Qed. Lemma eqpP m 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) => [-> | m_nz]. by case/andP => /dvd0pP -> _; exists (1, 1); rewrite ?scaler0 // oner_eq0. case: (eqVneq n 0) => [-> | n_nz]. by case/andP => _ /dvd0pP ->; exists (1, 1); rewrite ?scaler0 // oner_eq0. case/andP; 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%N. have:= size_mul nz_q1 nz_q2; rewrite def_q12 size_polyC mulf_neq0 //=. by rewrite polySpred // => ->; rewrite leq_addl. rewrite leq_eqVlt ltnS leqn0 size_poly_eq0 (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 eqp_eq p 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 //. rewrite !mul_polyC scalerA -eqC mulrC -scalerA eq. by rewrite !scalerA mulrC. Qed. Lemma eqpxx : reflexive (@eqp R). Proof. by move=> p; rewrite /eqp dvdpp. Qed. Hint Resolve eqpxx. Lemma eqp_sym : symmetric (@eqp R). Proof. by move=> p q; rewrite /eqp andbC. Qed. Lemma 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 eqp_ltrans : left_transitive (@eqp R). Proof. move=> p q r pq. by apply/idP/idP=> e; apply: eqp_trans e; rewrite // eqp_sym. Qed. Lemma eqp_rtrans : right_transitive (@eqp R). Proof. by move=> x y xy z; rewrite eqp_sym (eqp_ltrans xy) eqp_sym. Qed. Lemma eqp0 : forall p, (p %= 0) = (p == 0). Proof. move=> p; case: eqP; move/eqP=> Ep; first by rewrite (eqP Ep) eqpxx. by apply/negP; case/andP=> _; rewrite /dvdp modp0 (negPf Ep). Qed. Lemma eqp01 : 0 %= (1 : {poly R}) = false. Proof. case abs : (0 %= 1) => //; case/eqpP: abs=> [[c1 c2]] /andP [c1n0 c2n0] /=. by rewrite scaler0 alg_polyC; move/eqP; rewrite eq_sym polyC_eq0 (negbTE c2n0). Qed. Lemma eqp_scale p 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 eqp_size p q : p %= q -> size p = size q. Proof. case: (q =P 0); move/eqP => Eq; first by rewrite (eqP Eq) eqp0; move/eqP->. rewrite eqp_sym; case: (p =P 0); move/eqP => Ep. by rewrite (eqP Ep) eqp0; move/eqP->. by case/andP => Dp Dq; apply: anti_leq; rewrite !dvdp_leq. Qed. Lemma size_poly_eq1 p : (size p == 1%N) = (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; move/size_poly1P. Qed. Lemma polyXsubC_eqp1 (x : R) : ('X - x%:P %= 1) = false. Proof. by rewrite -size_poly_eq1 size_XsubC. Qed. Lemma dvdp_eqp1 p q : p %| q -> q %= 1 -> p %= 1. Proof. move=> dpq hq. have sizeq : size q == 1%N by rewrite size_poly_eq1. have n0q : q != 0. by case abs: (q == 0) => //; move: hq; rewrite (eqP abs) eqp01. rewrite -size_poly_eq1 eqn_leq -{1}(eqP sizeq) dvdp_leq //=. case p0 : (size p == 0%N); last by rewrite neq0_lt0n. by move: dpq; rewrite size_poly_eq0 in p0; rewrite (eqP p0) dvd0p (negbTE n0q). Qed. Lemma eqp_dvdr q 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 eqp_dvdl d2 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 dvdp_scaler c m n : c != 0 -> m %| c *: n = (m %| n). Proof. by move=> cn0; apply: eqp_dvdr; apply: eqp_scale. Qed. Lemma dvdp_scalel c m n : c != 0 -> (c *: m %| n) = (m %| n). Proof. by move=> cn0; apply: eqp_dvdl; apply: eqp_scale. Qed. Lemma dvdp_opp d p : d %| (- p) = (d %| p). Proof. by apply: eqp_dvdr; rewrite -scaleN1r eqp_scale ?oppr_eq0 ?oner_eq0. Qed. Lemma eqp_mul2r r p q : r != 0 -> (p * r %= q * r) = (p %= q). Proof. by move=> nz_r; rewrite /eqp !dvdp_mul2r. Qed. Lemma eqp_mul2l r p q: r != 0 -> (r * p %= r * q) = (p %= q). Proof. by move=> nz_r; rewrite /eqp !dvdp_mul2l. Qed. Lemma eqp_mull r 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 eqp_mulr q p r : (p %= q) -> (p * r %= q * r). Proof. by move=> epq; rewrite ![_ * r]mulrC eqp_mull. Qed. Lemma eqp_exp p 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 polyC_eqp1 (c : R) : (c%:P %= 1) = (c != 0). Proof. apply/eqpP/idP => [[[x y]] |nc0] /=. case c0: (c == 0); rewrite // alg_polyC (eqP c0) scaler0. by case/andP=> _ /=; move/negbTE<-; move/eqP; rewrite eq_sym polyC_eq0. exists (1, c); first by rewrite nc0 /= oner_neq0. by rewrite alg_polyC scale1r. Qed. Lemma dvdUp d p: d %= 1 -> d %| p. Proof. by move/eqp_dvdl->; rewrite dvd1p. Qed. Lemma dvdp_size_eqp p q : p %| q -> size p == size q = (p %= q). Proof. move=> pq; apply/idP/idP; last by move/eqp_size->. case (q =P 0)=> [->|]; [|move/eqP => Hq]. by rewrite size_poly0 size_poly_eq0; move/eqP->; rewrite eqpxx. case (p =P 0)=> [->|]; [|move/eqP => Hp]. by rewrite size_poly0 eq_sym size_poly_eq0; move/eqP->; rewrite eqpxx. move: pq; rewrite dvdp_eq; set c := _ ^+ _; set x := _ %/ _; move/eqP=> eqpq. move: (eqpq); move/(congr1 (size \o (@polyseq R)))=> /=. have cn0 : c != 0 by rewrite expf_neq0 // lead_coef_eq0. rewrite (@eqp_size _ q); last by apply: 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 eqp_root p 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. rewrite rootE -[_==_]orFb -c0 -mulf_eq0 -hornerZ e hornerZ. by rewrite mulf_eq0 d0. Qed. Lemma eqp_rmod_mod p 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 eqp_rdiv_div p q : rdivp p q %= divp p q. Proof. rewrite divpE eqp_sym; case: ifP=> ulcq //; apply: eqp_scale; rewrite invr_eq0 //. by apply: expf_neq0; apply: contraTneq ulcq => ->; rewrite unitr0. Qed. Lemma dvd_eqp_divl d p q (dvd_dp : d %| q) (eq_pq : p %= q) : p %/ d %= q %/ d. Proof. case: (eqVneq q 0) eq_pq=> [->|q_neq0]; first by rewrite eqp0=> /eqP->. have d_neq0: d != 0 by apply: contraL dvd_dp=> /eqP->; rewrite dvd0p. move=> eq_pq; rewrite -(@eqp_mul2r d) // !divpK // ?(eqp_dvdr _ eq_pq) //. rewrite (eqp_ltrans (eqp_scale _ _)) ?lc_expn_scalp_neq0 //. by rewrite (eqp_rtrans (eqp_scale _ _)) ?lc_expn_scalp_neq0. Qed. Definition gcdp_rec p q := let: (p1, q1) := if size p < size q then (q, p) else (p, q) in if p1 == 0 then q1 else let fix loop (n : nat) (pp qq : {poly R}) {struct n} := let rr := modp pp qq in if rr == 0 then qq else if n is n1.+1 then loop n1 qq rr else rr in loop (size p1) p1 q1. Definition gcdp := nosimpl gcdp_rec. Lemma gcd0p : left_id 0 gcdp. Proof. move=> p; rewrite /gcdp /gcdp_rec size_poly0 size_poly_gt0 if_neg. case: ifP => /= [_ | nzp]; first by rewrite eqxx. by rewrite polySpred !(modp0, nzp) //; case: _.-1 => [|m]; rewrite mod0p eqxx. Qed. Lemma gcdp0 : right_id 0 gcdp. Proof. move=> p; have:= gcd0p p; rewrite /gcdp /gcdp_rec size_poly0 size_poly_gt0. by rewrite if_neg; case: ifP => /= p0; rewrite ?(eqxx, p0) // (eqP p0). Qed. Lemma gcdpE p q : gcdp p q = if size p < size q then gcdp (modp q p) p else gcdp (modp p q) q. Proof. pose gcdpE_rec := fix gcdpE_rec (n : nat) (pp qq : {poly R}) {struct n} := let rr := modp pp qq in if rr == 0 then qq else if n is n1.+1 then gcdpE_rec n1 qq rr else rr. have Irec: forall k l p q, size q <= k -> size q <= l -> size q < size p -> gcdpE_rec k p q = gcdpE_rec l p q. + elim=> [|m Hrec] [|n] //= p1 q1. - rewrite leqn0 size_poly_eq0; move/eqP=> -> _. rewrite size_poly0 size_poly_gt0 modp0 => nzp. by rewrite (negPf nzp); case: n => [|n] /=; rewrite mod0p eqxx. - rewrite leqn0 size_poly_eq0 => _; move/eqP=> ->. rewrite size_poly0 size_poly_gt0 modp0 => nzp. by rewrite (negPf nzp); case: m {Hrec} => [|m] /=; rewrite mod0p eqxx. case: ifP => Epq Sm Sn Sq //; rewrite ?Epq //. case: (eqVneq q1 0) => [->|nzq]. by case: n m {Sm Sn Hrec} => [|m] [|n] //=; rewrite mod0p eqxx. apply: Hrec; last by rewrite ltn_modp. by rewrite -ltnS (leq_trans _ Sm) // ltn_modp. by rewrite -ltnS (leq_trans _ Sn) // ltn_modp. case: (eqVneq p 0) => [-> | nzp]. by rewrite mod0p modp0 gcd0p gcdp0 if_same. case: (eqVneq q 0) => [-> | nzq]. by rewrite mod0p modp0 gcd0p gcdp0 if_same. rewrite /gcdp /gcdp_rec. case: ltnP; rewrite (negPf nzp, negPf nzq) //=. move=> ltpq; rewrite ltn_modp (negPf nzp) //=. rewrite -(ltn_predK ltpq) /=; case: eqP => [->|]. by case: (size p) => [|[|s]]; rewrite /= modp0 (negPf nzp) // mod0p eqxx. move/eqP=> nzqp; rewrite (negPf nzp). apply: Irec => //; last by rewrite ltn_modp. by rewrite -ltnS (ltn_predK ltpq) (leq_trans _ ltpq) ?leqW // ltn_modp. by rewrite ltnW // ltn_modp. move=> leqp; rewrite ltn_modp (negPf nzq) //=. have p_gt0: size p > 0 by rewrite size_poly_gt0. rewrite -(prednK p_gt0) /=; case: eqP => [->|]. by case: (size q) => [|[|s]]; rewrite /= modp0 (negPf nzq) // mod0p eqxx. move/eqP=> nzpq; rewrite (negPf nzq); apply: Irec => //; rewrite ?ltn_modp //. by rewrite -ltnS (prednK p_gt0) (leq_trans _ leqp) // ltn_modp. by rewrite ltnW // ltn_modp. Qed. Lemma size_gcd1p p : size (gcdp 1 p) = 1%N. Proof. rewrite gcdpE size_polyC oner_eq0 /= modp1; case: ltnP. by rewrite gcd0p size_polyC oner_eq0. move/size1_polyC=> e; rewrite e. case p00: (p_0 == 0); first by rewrite (eqP p00) modp0 gcdp0 size_poly1. by rewrite modpC ?p00 // gcd0p size_polyC p00. Qed. Lemma size_gcdp1 p : size (gcdp p 1) = 1%N. rewrite gcdpE size_polyC oner_eq0 /= modp1; case: ltnP; last first. by rewrite gcd0p size_polyC oner_eq0. rewrite ltnS leqn0 size_poly_eq0; move/eqP->; rewrite gcdp0 modp0 size_polyC. by rewrite oner_eq0. Qed. Lemma gcdpp : idempotent gcdp. Proof. by move=> p; rewrite gcdpE ltnn modpp gcd0p. Qed. Lemma dvdp_gcdlr p q : (gcdp p q %| p) && (gcdp p q %| q). Proof. elim: {p q}minn {-2}p {-2}q (leqnn (minn (size q) (size p))) => [|r Hrec] p q. rewrite geq_min !leqn0 !size_poly_eq0. by case/pred2P=> ->; rewrite (gcdp0, gcd0p) dvdpp ?andbT /=. case: (eqVneq p 0) => [-> _|nz_p]; first by rewrite gcd0p dvdpp andbT. case: (eqVneq q 0) => [->|nz_q]; first by rewrite gcdp0 dvdpp /=. rewrite gcdpE minnC /minn; case: ltnP => [lt_pq | le_pq] le_qr. suffices: minn (size p) (size (q %% p)) <= r. by move/Hrec; case/andP => E1 E2; rewrite E2 (dvdp_mod _ E2). by rewrite geq_min orbC -ltnS (leq_trans _ le_qr) ?ltn_modp. suffices: minn (size q) (size (p %% q)) <= r. by move/Hrec; case/andP => E1 E2; rewrite E2 andbT (dvdp_mod _ E2). by rewrite geq_min orbC -ltnS (leq_trans _ le_qr) ?ltn_modp. Qed. Lemma dvdp_gcdl p q : gcdp p q %| p. Proof. by case/andP: (dvdp_gcdlr p q). Qed. Lemma dvdp_gcdr p q :gcdp p q %| q. Proof. by case/andP: (dvdp_gcdlr p q). Qed. Lemma leq_gcdpl p q : p != 0 -> size (gcdp p q) <= size p. Proof. by move=> pn0; move: (dvdp_gcdl p q); apply: dvdp_leq. Qed. Lemma leq_gcdpr p q : q != 0 -> size (gcdp p q) <= size q. Proof. by move=> qn0; move: (dvdp_gcdr p q); apply: dvdp_leq. Qed. Lemma dvdp_gcd p m n : p %| gcdp m n = (p %| m) && (p %| n). Proof. apply/idP/andP=> [dv_pmn | [dv_pm dv_pn]]. by rewrite ?(dvdp_trans dv_pmn) ?dvdp_gcdl ?dvdp_gcdr. move: (leqnn (minn (size n) (size m))) dv_pm dv_pn. elim: {m n}minn {-2}m {-2}n => [|r Hrec] m n. rewrite geq_min !leqn0 !size_poly_eq0. by case/pred2P=> ->; rewrite (gcdp0, gcd0p). case: (eqVneq m 0) => [-> _|nz_m]; first by rewrite gcd0p /=. case: (eqVneq n 0) => [->|nz_n]; first by rewrite gcdp0 /=. rewrite gcdpE minnC /minn; case: ltnP => Cnm le_r dv_m dv_n. apply: Hrec => //; last by rewrite -(dvdp_mod _ dv_m). by rewrite geq_min orbC -ltnS (leq_trans _ le_r) ?ltn_modp. apply: Hrec => //; last by rewrite -(dvdp_mod _ dv_n). by rewrite geq_min orbC -ltnS (leq_trans _ le_r) ?ltn_modp. Qed. Lemma gcdpC : forall p q, gcdp p q %= gcdp q p. Proof. by move=> p q; rewrite /eqp !dvdp_gcd !dvdp_gcdl !dvdp_gcdr. Qed. Lemma gcd1p p : gcdp 1 p %= 1. Proof. rewrite -size_poly_eq1 gcdpE size_poly1; case: ltnP. by rewrite modp1 gcd0p size_poly1 eqxx. move/size1_polyC=> e; rewrite e. case p00: (p_0 == 0); first by rewrite (eqP p00) modp0 gcdp0 size_poly1. by rewrite modpC ?p00 // gcd0p size_polyC p00. Qed. Lemma gcdp1 p : gcdp p 1 %= 1. Proof. by rewrite (eqp_ltrans (gcdpC _ _)) gcd1p. Qed. Lemma gcdp_addl_mul p q r: gcdp r (p * r + q) %= gcdp r q. Proof. suff h m n d : gcdp d n %| gcdp d (m * d + n). apply/andP; split => //; rewrite {2}(_: q = (-p) * r + (p * r + q)) ?H //. by rewrite GRing.mulNr GRing.addKr. by rewrite dvdp_gcd dvdp_gcdl /= dvdp_addr ?dvdp_gcdr ?dvdp_mull ?dvdp_gcdl. Qed. Lemma gcdp_addl m n : gcdp m (m + n) %= gcdp m n. Proof. by rewrite -{2}(mul1r m) gcdp_addl_mul. Qed. Lemma gcdp_addr m n : gcdp m (n + m) %= gcdp m n. Proof. by rewrite addrC gcdp_addl. Qed. Lemma gcdp_mull m n : gcdp n (m * n) %= n. Proof. case: (eqVneq n 0) => [-> | nn0]; first by rewrite gcd0p mulr0 eqpxx. case: (eqVneq m 0) => [-> | mn0]; first by rewrite mul0r gcdp0 eqpxx. rewrite gcdpE modp_mull gcd0p size_mul //; case: ifP; first by rewrite eqpxx. rewrite (polySpred mn0) addSn /= -{1}[size n]add0n ltn_add2r; move/negbT. rewrite -ltnNge prednK ?size_poly_gt0 // leq_eqVlt ltnS leqn0 size_poly_eq0. rewrite (negPf mn0) orbF; case/size_poly1P=> c cn0 -> {mn0 m}; rewrite mul_polyC. suff -> : n %% (c *: n) = 0 by rewrite gcd0p; apply: eqp_scale. by apply/modp_eq0P; rewrite dvdp_scalel. Qed. Lemma gcdp_mulr m n : gcdp n (n * m) %= n. Proof. by rewrite mulrC gcdp_mull. Qed. Lemma gcdp_scalel c m n : c != 0 -> gcdp (c *: m) n %= gcdp m n. Proof. move=> cn0; rewrite /eqp dvdp_gcd [gcdp m n %| _]dvdp_gcd !dvdp_gcdr !andbT. apply/andP; split; last first. by apply: dvdp_trans (dvdp_gcdl _ _) _; rewrite dvdp_scaler. by apply: dvdp_trans (dvdp_gcdl _ _) _; rewrite dvdp_scalel. Qed. Lemma gcdp_scaler c m n : c != 0 -> gcdp m (c *: n) %= gcdp m n. Proof. move=> cn0; apply: eqp_trans (gcdpC _ _) _. by apply: eqp_trans (gcdp_scalel _ _ _) _ => //; apply: gcdpC. Qed. Lemma dvdp_gcd_idl m n : m %| n -> gcdp m n %= m. Proof. case: (eqVneq m 0) => [-> | mn0]. by rewrite dvd0p => /eqP ->; rewrite gcdp0 eqpxx. rewrite dvdp_eq; move/eqP; move/(f_equal (gcdp m)) => h. apply: eqp_trans (gcdp_mull (n %/ m) _); rewrite -h eqp_sym gcdp_scaler //. by rewrite expf_neq0 // lead_coef_eq0. Qed. Lemma dvdp_gcd_idr m n : n %| m -> gcdp m n %= n. Proof. by move/dvdp_gcd_idl => h; apply: eqp_trans h; apply: gcdpC. Qed. Lemma gcdp_exp p k l : gcdp (p ^+ k) (p ^+ l) %= p ^+ minn k l. Proof. wlog leqmn: k l / k <= l. move=> hwlog; case: (leqP k l); first exact: hwlog. by move/ltnW; rewrite minnC; move/hwlog=> h; apply: eqp_trans h; apply: gcdpC. rewrite (minn_idPl leqmn); move/subnK: leqmn<-; rewrite exprD. by apply: eqp_trans (gcdp_mull _ _) _; apply: eqpxx. Qed. Lemma gcdp_eq0 p q : gcdp p q == 0 = (p == 0) && (q == 0). Proof. apply/idP/idP; last by case/andP => /eqP -> /eqP ->; rewrite gcdp0. have h m n: gcdp m n == 0 -> (m == 0). by rewrite -(dvd0p m); move/eqP<-; rewrite dvdp_gcdl. by move=> ?; rewrite (h _ q) // (h _ p) // -eqp0 (eqp_ltrans (gcdpC _ _)) eqp0. Qed. Lemma eqp_gcdr p q r : q %= r -> gcdp p q %= gcdp p r. Proof. move=> eqr; rewrite /eqp !(dvdp_gcd, dvdp_gcdl, andbT) /=. by rewrite -(eqp_dvdr _ eqr) dvdp_gcdr (eqp_dvdr _ eqr) dvdp_gcdr. Qed. Lemma eqp_gcdl r p q : p %= q -> gcdp p r %= gcdp q r. move=> eqr; rewrite /eqp !(dvdp_gcd, dvdp_gcdr, andbT) /=. by rewrite -(eqp_dvdr _ eqr) dvdp_gcdl (eqp_dvdr _ eqr) dvdp_gcdl. Qed. Lemma eqp_gcd p1 p2 q1 q2 : p1 %= p2 -> q1 %= q2 -> gcdp p1 q1 %= gcdp p2 q2. Proof. move=> e1 e2. by apply: eqp_trans (eqp_gcdr _ e2); apply: eqp_trans (eqp_gcdl _ e1). Qed. Lemma eqp_rgcd_gcd p q : rgcdp p q %= gcdp p q. Proof. move: (leqnn (minn (size p) (size q))); move: {2}(minn (size p) (size q)) => n. elim: n p q => [p q|n ihn p q hs]. rewrite leqn0 /minn; case: ltnP => _; rewrite size_poly_eq0; move/eqP->. by rewrite gcd0p rgcd0p eqpxx. by rewrite gcdp0 rgcdp0 eqpxx. case: (eqVneq p 0) => [-> | pn0]; first by rewrite gcd0p rgcd0p eqpxx. case: (eqVneq q 0) => [-> | qn0]; first by rewrite gcdp0 rgcdp0 eqpxx. rewrite gcdpE rgcdpE; case: ltnP => sp. have e := (eqp_rmod_mod q p); move: (e); move/(eqp_gcdl p) => h. apply: eqp_trans h; apply: ihn; rewrite (eqp_size e) geq_min. by rewrite -ltnS (leq_trans _ hs) // (minn_idPl (ltnW _)) ?ltn_modp. have e := (eqp_rmod_mod p q); move: (e); move/(eqp_gcdl q) => h. apply: eqp_trans h; apply: ihn; rewrite (eqp_size e) geq_min. by rewrite -ltnS (leq_trans _ hs) // (minn_idPr _) ?ltn_modp. Qed. Lemma gcdp_modr m n : gcdp m (n %% m) %= gcdp m n. Proof. case: (eqVneq m 0) => [-> | mn0]; first by rewrite modp0 eqpxx. have : (lead_coef m) ^+ (scalp n m) != 0 by rewrite expf_neq0 // lead_coef_eq0. move/gcdp_scaler; move/(_ m n) => h; apply: eqp_trans h; rewrite divp_eq. by rewrite eqp_sym gcdp_addl_mul. Qed. Lemma gcdp_modl m n : gcdp (m %% n) n %= gcdp m n. Proof. apply: eqp_trans (gcdpC _ _) _; apply: eqp_trans (gcdp_modr _ _) _. exact: gcdpC. Qed. Lemma gcdp_def d m n : d %| m -> d %| n -> (forall d', d' %| m -> d' %| n -> d' %| d) -> gcdp m n %= d. Proof. move=> dm dn h; rewrite /eqp dvdp_gcd dm dn !andbT. by apply: h; [apply: dvdp_gcdl | apply: dvdp_gcdr]. Qed. Definition coprimep p q := size (gcdp p q) == 1%N. Lemma coprimep_size_gcd p q : coprimep p q -> size (gcdp p q) = 1%N. Proof. by rewrite /coprimep=> /eqP. Qed. Lemma coprimep_def p q : (coprimep p q) = (size (gcdp p q) == 1%N). Proof. done. Qed. Lemma coprimep_scalel c m n : c != 0 -> coprimep (c *: m) n = coprimep m n. Proof. by move=> ?; rewrite !coprimep_def (eqp_size (gcdp_scalel _ _ _)). Qed. Lemma coprimep_scaler c m n: c != 0 -> coprimep m (c *: n) = coprimep m n. Proof. by move=> ?; rewrite !coprimep_def (eqp_size (gcdp_scaler _ _ _)). Qed. Lemma coprimepp p : coprimep p p = (size p == 1%N). Proof. by rewrite coprimep_def gcdpp. Qed. Lemma gcdp_eqp1 p q : gcdp p q %= 1 = (coprimep p q). Proof. by rewrite coprimep_def size_poly_eq1. Qed. Lemma coprimep_sym p q : coprimep p q = coprimep q p. Proof. by rewrite -!gcdp_eqp1; apply: eqp_ltrans; rewrite gcdpC. Qed. Lemma coprime1p p : coprimep 1 p. Proof. rewrite /coprimep -[1%N](size_poly1 R); apply/eqP; apply: eqp_size. exact: gcd1p. Qed. Lemma coprimep1 p : coprimep p 1. Proof. by rewrite coprimep_sym; apply: coprime1p. Qed. Lemma coprimep0 p : coprimep p 0 = (p %= 1). Proof. by rewrite /coprimep gcdp0 size_poly_eq1. Qed. Lemma coprime0p p : coprimep 0 p = (p %= 1). Proof. by rewrite coprimep_sym coprimep0. Qed. (* This is different from coprimeP in div. shall we keep this? *) Lemma coprimepP p q : reflect (forall d, d %| p -> d %| q -> d %= 1) (coprimep p q). Proof. apply: (iffP idP)=> [|h]. rewrite /coprimep; move/eqP=> hs d dvddp dvddq. have dvddg: d %| gcdp p q by rewrite dvdp_gcd dvddp dvddq. by apply: (dvdp_eqp1 dvddg); rewrite -size_poly_eq1; apply/eqP. case/andP: (dvdp_gcdlr p q)=> h1 h2. by rewrite /coprimep size_poly_eq1; apply: h. Qed. Lemma coprimepPn p q : p != 0 -> reflect (exists d, (d %| gcdp p q) && ~~ (d %= 1)) (~~ coprimep p q). Proof. move=> p0; apply: (iffP idP). by rewrite -gcdp_eqp1=> ng1; exists (gcdp p q); rewrite dvdpp /=. case=> d; case/andP=> dg; apply: contra; rewrite -gcdp_eqp1=> g1. by move: dg; rewrite (eqp_dvdr _ g1) dvdp1 size_poly_eq1. Qed. Lemma coprimep_dvdl q p r : r %| q -> coprimep p q -> coprimep p r. Proof. move=> rq cpq; apply/coprimepP=> d dp dr; move/coprimepP:cpq=> cpq'. by apply: cpq'; rewrite // (dvdp_trans dr). Qed. Lemma coprimep_dvdr p q r : r %| p -> coprimep p q -> coprimep r q. Proof. move=> rp; rewrite ![coprimep _ q]coprimep_sym. by move/coprimep_dvdl; apply. Qed. Lemma coprimep_modl p q : coprimep (p %% q) q = coprimep p q. Proof. symmetry; rewrite !coprimep_def. case: (ltnP (size p) (size q))=> hpq; first by rewrite modp_small. by rewrite gcdpE ltnNge hpq. Qed. Lemma coprimep_modr q p : coprimep q (p %% q) = coprimep q p. Proof. by rewrite ![coprimep q _]coprimep_sym coprimep_modl. Qed. Lemma rcoprimep_coprimep q p : rcoprimep q p = coprimep q p. Proof. by rewrite /coprimep /rcoprimep; rewrite (eqp_size (eqp_rgcd_gcd _ _)). Qed. Lemma eqp_coprimepr p q r : q %= r -> coprimep p q = coprimep p r. Proof. by rewrite -!gcdp_eqp1; move/(eqp_gcdr p) => h1; apply: (eqp_ltrans h1). Qed. Lemma eqp_coprimepl p q r : q %= r -> coprimep q p = coprimep r p. Proof. by rewrite !(coprimep_sym _ p); apply: eqp_coprimepr. Qed. (* This should be implemented with an extended remainder sequence *) Fixpoint egcdp_rec p q k {struct k} : {poly R} * {poly R} := if k is k'.+1 then if q == 0 then (1, 0) else let: (u, v) := egcdp_rec q (p %% q) k' in (lead_coef q ^+ scalp p q *: v, (u - v * (p %/ q))) else (1, 0). Definition egcdp p q := if size q <= size p then egcdp_rec p q (size q) else let e := egcdp_rec q p (size p) in (e.2, e.1). (* No provable egcd0p *) Lemma egcdp0 p : egcdp p 0 = (1, 0). Proof. by rewrite /egcdp size_poly0. Qed. Lemma egcdp_recP : forall k p q, q != 0 -> size q <= k -> size q <= size p -> let e := (egcdp_rec p q k) in [/\ size e.1 <= size q, size e.2 <= size p & gcdp p q %= e.1 * p + e.2 * q]. Proof. elim=> [|k ihk] p q /= qn0; first by rewrite leqn0 size_poly_eq0 (negPf qn0). move=> sqSn qsp; case: (eqVneq q 0)=> q0; first by rewrite q0 eqxx in qn0. rewrite (negPf qn0). have sp : size p > 0 by apply: leq_trans qsp; rewrite size_poly_gt0. case: (eqVneq (p %% q) 0) => [r0 | rn0] /=. rewrite r0 /egcdp_rec; case: k ihk sqSn => [|n] ihn sqSn /=. rewrite !scaler0 !mul0r subr0 add0r mul1r size_poly0 size_poly1. by rewrite dvdp_gcd_idr /dvdp ?r0. rewrite !eqxx mul0r scaler0 /= mul0r add0r subr0 mul1r size_poly0 size_poly1. by rewrite dvdp_gcd_idr /dvdp ?r0 //. have h1 : size (p %% q) <= k. by rewrite -ltnS; apply: leq_trans sqSn; rewrite ltn_modp. have h2 : size (p %% q) <= size q by rewrite ltnW // ltn_modp. have := (ihk q (p %% q) rn0 h1 h2). case: (egcdp_rec _ _)=> u v /= => [[ihn'1 ihn'2 ihn'3]]. rewrite gcdpE ltnNge qsp //= (eqp_ltrans (gcdpC _ _)); split; last first. - apply: (eqp_trans ihn'3). rewrite mulrBl addrCA -scalerAl scalerAr -mulrA -mulrBr. by rewrite divp_eq addrAC subrr add0r eqpxx. - apply: (leq_trans (size_add _ _)). case: (eqVneq v 0)=> [-> | vn0]. rewrite mul0r size_opp size_poly0 maxn0; apply: leq_trans ihn'1 _. exact: leq_modp. case: (eqVneq (p %/ q) 0)=> [-> | qqn0]. rewrite mulr0 size_opp size_poly0 maxn0; apply: leq_trans ihn'1 _. exact: leq_modp. rewrite geq_max (leq_trans ihn'1) ?leq_modp //= size_opp size_mul //. move: (ihn'2); rewrite -(leq_add2r (size (p %/ q))). have : size v + size (p %/ q) > 0 by rewrite addn_gt0 size_poly_gt0 vn0. have : size q + size (p %/ q) > 0 by rewrite addn_gt0 size_poly_gt0 qn0. do 2!move/prednK=> {1}<-; rewrite ltnS => h; apply: leq_trans h _. rewrite size_divp // addnBA; last by apply: leq_trans qsp; apply: leq_pred. rewrite addnC -addnBA ?leq_pred //; move: qn0; rewrite -size_poly_eq0 -lt0n. by move/prednK=> {1}<-; rewrite subSnn addn1. - by rewrite size_scale // lc_expn_scalp_neq0. Qed. Lemma egcdpP p q : p != 0 -> q != 0 -> forall (e := egcdp p q), [/\ size e.1 <= size q, size e.2 <= size p & gcdp p q %= e.1 * p + e.2 * q]. Proof. move=> pn0 qn0; rewrite /egcdp; case: (leqP (size q) (size p)) => /= hp. by apply: egcdp_recP. move/ltnW: hp => hp; case: (egcdp_recP pn0 (leqnn (size p)) hp) => h1 h2 h3. by split => //; rewrite (eqp_ltrans (gcdpC _ _)) addrC. Qed. Lemma egcdpE p q (e := egcdp p q) : gcdp p q %= e.1 * p + e.2 * q. Proof. rewrite {}/e; have [-> /= | qn0] := eqVneq q 0. by rewrite gcdp0 egcdp0 mul1r mulr0 addr0. have [p0 | pn0] := eqVneq p 0; last by case: (egcdpP pn0 qn0). rewrite p0 gcd0p mulr0 add0r /egcdp size_poly0 leqn0 size_poly_eq0 (negPf qn0). by rewrite /= mul1r. Qed. Lemma Bezoutp p q : exists u, u.1 * p + u.2 * q %= (gcdp p q). Proof. case: (eqVneq p 0) => [-> | pn0]. by rewrite gcd0p; exists (0, 1); rewrite mul0r mul1r add0r. case: (eqVneq q 0) => [-> | qn0]. by rewrite gcdp0; exists (1, 0); rewrite mul0r mul1r addr0. pose e := egcdp p q; exists e; rewrite eqp_sym. by case: (egcdpP pn0 qn0). Qed. Lemma Bezout_coprimepP : forall p q, reflect (exists u, u.1 * p + u.2 * q %= 1) (coprimep p q). Proof. move=> p q; rewrite -gcdp_eqp1; apply: (iffP idP)=> [g1|]. by case: (Bezoutp p q) => [[u v] Puv]; exists (u, v); apply: eqp_trans g1. case=> [[u v]]; rewrite eqp_sym=> Puv; rewrite /eqp (eqp_dvdr _ Puv). by rewrite dvdp_addr dvdp_mull ?dvdp_gcdl ?dvdp_gcdr //= dvd1p. Qed. Lemma coprimep_root p q x : coprimep p q -> root p x -> q.[x] != 0. Proof. case/Bezout_coprimepP=> [[u v] euv] px0. move/eqpP: euv => [[c1 c2]] /andP /= [c1n0 c2n0 e]. suffices: c1 * (v.[x] * q.[x]) != 0. by rewrite !mulf_eq0 !negb_or c1n0 /=; case/andP. move/(f_equal (fun t => horner t x)): e; rewrite /= !hornerZ hornerD. by rewrite !hornerM (eqP px0) mulr0 add0r hornerC mulr1; move->. Qed. Lemma Gauss_dvdpl p q d: coprimep d q -> (d %| p * q) = (d %| p). Proof. move/Bezout_coprimepP=>[[u v] Puv]; apply/idP/idP; last exact: dvdp_mulr. move: Puv; move/(eqp_mull p); rewrite mulr1 mulrDr eqp_sym=> peq dpq. rewrite (eqp_dvdr _ peq) dvdp_addr; first by rewrite mulrA mulrAC dvdp_mulr. by rewrite mulrA dvdp_mull ?dvdpp. Qed. Lemma Gauss_dvdpr p q d: coprimep d q -> (d %| q * p) = (d %| p). Proof. by rewrite mulrC; apply: Gauss_dvdpl. Qed. (* This could be simplified with the introduction of lcmp *) Lemma Gauss_dvdp m n p : coprimep m n -> (m * n %| p) = (m %| p) && (n %| p). Proof. case: (eqVneq m 0) => [-> | mn0]. by rewrite coprime0p => /eqp_dvdl->; rewrite !mul0r dvd0p dvd1p andbT. case: (eqVneq n 0) => [-> | nn0]. by rewrite coprimep0 => /eqp_dvdl->; rewrite !mulr0 dvd1p. move=> hc; apply/idP/idP. move/Gauss_dvdpl: hc => <- h; move/(dvdp_mull m): (h); rewrite dvdp_mul2l //. move->; move/(dvdp_mulr n): (h); rewrite dvdp_mul2r // andbT. exact: dvdp_mulr. case/andP => dmp dnp; move: (dnp); rewrite dvdp_eq. set c2 := _ ^+ _; set q2 := _ %/ _; move/eqP=> e2. have := (sym_eq (Gauss_dvdpl q2 hc)); rewrite -e2. have -> : m %| c2 *: p by rewrite -mul_polyC dvdp_mull. rewrite dvdp_eq; set c3 := _ ^+ _; set q3 := _ %/ _; move/eqP=> e3. apply: (@eq_dvdp (c3 * c2) q3). by rewrite mulf_neq0 // expf_neq0 // lead_coef_eq0. by rewrite mulrA -e3 -scalerAl -e2 scalerA. Qed. Lemma Gauss_gcdpr p m n : coprimep p m -> gcdp p (m * n) %= gcdp p n. Proof. move=> co_pm; apply/eqP; rewrite /eqp !dvdp_gcd !dvdp_gcdl /= andbC. rewrite dvdp_mull ?dvdp_gcdr // -(@Gauss_dvdpl _ m). by rewrite mulrC dvdp_gcdr. apply/coprimepP=> d; rewrite dvdp_gcd; case/andP=> hdp _ hdm. by move/coprimepP: co_pm; apply. Qed. Lemma Gauss_gcdpl p m n : coprimep p n -> gcdp p (m * n) %= gcdp p m. Proof. by move=> co_pn; rewrite mulrC Gauss_gcdpr. Qed. Lemma coprimep_mulr p q r : coprimep p (q * r) = (coprimep p q && coprimep p r). Proof. apply/coprimepP/andP=> [hp | [/coprimepP-hq hr]]. by split; apply/coprimepP=> d dp dq; rewrite hp //; [apply/dvdp_mulr | apply/dvdp_mull]. move=> d dp dqr; move/(_ _ dp) in hq. rewrite Gauss_dvdpl in dqr; first exact: hq. by move/coprimep_dvdr: hr; apply. Qed. Lemma coprimep_mull p q r: coprimep (q * r) p = (coprimep q p && coprimep r p). Proof. by rewrite ![coprimep _ p]coprimep_sym coprimep_mulr. Qed. Lemma modp_coprime k u n : k != 0 -> (k * u) %% n %= 1 -> coprimep k n. Proof. move=> kn0 hmod; apply/Bezout_coprimepP. exists (((lead_coef n)^+(scalp (k * u) n) *: u), (- (k * u %/ n))). rewrite -scalerAl mulrC (divp_eq (u * k) n) mulNr -addrAC subrr add0r. by rewrite mulrC. Qed. Lemma coprimep_pexpl k m n : 0 < k -> coprimep (m ^+ k) n = coprimep m n. Proof. case: k => // k _; elim: k => [|k IHk]; first by rewrite expr1. by rewrite exprS coprimep_mull -IHk andbb. Qed. Lemma coprimep_pexpr k m n : 0 < k -> coprimep m (n ^+ k) = coprimep m n. Proof. by move=> k_gt0; rewrite !(coprimep_sym m) coprimep_pexpl. Qed. Lemma coprimep_expl k m n : coprimep m n -> coprimep (m ^+ k) n. Proof. by case: k => [|k] co_pm; rewrite ?coprime1p // coprimep_pexpl. Qed. Lemma coprimep_expr k m n : coprimep m n -> coprimep m (n ^+ k). Proof. by rewrite !(coprimep_sym m); apply: coprimep_expl. Qed. Lemma gcdp_mul2l p q r : gcdp (p * q) (p * r) %= (p * gcdp q r). Proof. case: (eqVneq p 0)=> [->|hp]; first by rewrite !mul0r gcdp0 eqpxx. rewrite /eqp !dvdp_gcd !dvdp_mul2l // dvdp_gcdr dvdp_gcdl !andbT. move: (Bezoutp q r) => [[u v]] huv. rewrite eqp_sym in huv; rewrite (eqp_dvdr _ (eqp_mull _ huv)). rewrite mulrDr ![p * (_ * _)]mulrCA. by apply: dvdp_add; rewrite dvdp_mull// (dvdp_gcdr, dvdp_gcdl). Qed. Lemma gcdp_mul2r q r p : gcdp (q * p) (r * p) %= (gcdp q r * p). Proof. by rewrite ![_ * p]GRing.mulrC gcdp_mul2l. Qed. Lemma mulp_gcdr p q r : r * (gcdp p q) %= gcdp (r * p) (r * q). Proof. by rewrite eqp_sym gcdp_mul2l. Qed. Lemma mulp_gcdl p q r : (gcdp p q) * r %= gcdp (p * r) (q * r). Proof. by rewrite eqp_sym gcdp_mul2r. Qed. Lemma coprimep_div_gcd p q : (p != 0) || (q != 0) -> coprimep (p %/ (gcdp p q)) (q %/ gcdp p q). Proof. move=> hpq. have gpq0: gcdp p q != 0 by rewrite gcdp_eq0 negb_and. rewrite -gcdp_eqp1 -(@eqp_mul2r (gcdp p q)) // mul1r. have: gcdp p q %| p by rewrite dvdp_gcdl. have: gcdp p q %| q by rewrite dvdp_gcdr. rewrite !dvdp_eq eq_sym; move/eqP=> hq; rewrite eq_sym; move/eqP=> hp. rewrite (eqp_ltrans (mulp_gcdl _ _ _)) hq hp. have lcn0 k : (lead_coef (gcdp p q)) ^+ k != 0. by rewrite expf_neq0 ?lead_coef_eq0. by apply: eqp_gcd; rewrite ?eqp_scale. Qed. Lemma divp_eq0 p q : (p %/ q == 0) = [|| p == 0, q ==0 | size p < size q]. Proof. apply/eqP/idP=> [d0|]; last first. case/or3P; [by move/eqP->; rewrite div0p| by move/eqP->; rewrite divp0|]. by move/divp_small. case: (eqVneq p 0) => [->|pn0]; first by rewrite eqxx. case: (eqVneq q 0) => [-> | qn0]; first by rewrite eqxx orbT. move: (divp_eq p q); rewrite d0 mul0r add0r. move/(f_equal (fun x : {poly R} => size x)). by rewrite size_scale ?lc_expn_scalp_neq0 // => ->; rewrite ltn_modp qn0 !orbT. Qed. Lemma dvdp_div_eq0 p q : q %| p -> (p %/ q == 0) = (p == 0). Proof. move=> dvdp_qp; have [->|p_neq0] := altP (p =P 0); first by rewrite div0p eqxx. rewrite divp_eq0 ltnNge dvdp_leq // (negPf p_neq0) orbF /=. by apply: contraTF dvdp_qp=> /eqP ->; rewrite dvd0p. Qed. Lemma Bezout_coprimepPn p q : p != 0 -> q != 0 -> reflect (exists2 uv : {poly R} * {poly R}, (0 < size uv.1 < size q) && (0 < size uv.2 < size p) & uv.1 * p = uv.2 * q) (~~ (coprimep p q)). move=> pn0 qn0; apply: (iffP idP); last first. case=> [[u v] /= /andP [/andP [ps1 s1] /andP [ps2 s2]] e]. have: ~~(size (q * p) <= size (u * p)). rewrite -ltnNge !size_mul // -?size_poly_gt0 // (polySpred pn0) !addnS. by rewrite ltn_add2r. apply: contra => ?; apply: dvdp_leq; rewrite ?mulf_neq0 // -?size_poly_gt0 //. by rewrite mulrC Gauss_dvdp // dvdp_mull // e dvdp_mull. rewrite coprimep_def neq_ltn. case/orP; first by rewrite ltnS leqn0 size_poly_eq0 gcdp_eq0 -[p == 0]negbK pn0. case sg: (size (gcdp p q)) => [|n] //; case: n sg=> [|n] // sg _. move: (dvdp_gcdl p q); rewrite dvdp_eq; set c1 := _ ^+ _; move/eqP=> hu1. move: (dvdp_gcdr p q); rewrite dvdp_eq; set c2 := _ ^+ _; move/eqP=> hv1. exists (c1 *: (q %/ gcdp p q), c2 *: (p %/ gcdp p q)); last first. by rewrite -!{1}scalerAl !scalerAr hu1 hv1 mulrCA. rewrite !{1}size_scale ?lc_expn_scalp_neq0 //= !size_poly_gt0 !divp_eq0. rewrite gcdp_eq0 !(negPf pn0) !(negPf qn0) /= -!leqNgt leq_gcdpl //. rewrite leq_gcdpr //= !ltn_divpl -?size_poly_eq0 ?sg //. rewrite !size_mul // -?size_poly_eq0 ?sg // ![(_ + n.+2)%N]addnS /=. by rewrite -{1}(addn0 (size p)) -{1}(addn0 (size q)) !ltn_add2l. Qed. Lemma dvdp_pexp2r m n k : k > 0 -> (m ^+ k %| n ^+ k) = (m %| n). Proof. move=> k_gt0; apply/idP/idP; last exact: dvdp_exp2r. case: (eqVneq n 0) => [-> | nn0] //; case: (eqVneq m 0) => [-> | mn0]. move/prednK: k_gt0=> {1}<-; rewrite exprS mul0r //= !dvd0p expf_eq0. by case/andP=> _ ->. set d := gcdp m n; have := (dvdp_gcdr m n); rewrite -/d dvdp_eq. set c1 := _ ^+ _; set n' := _ %/ _; move/eqP=> def_n. have := (dvdp_gcdl m n); rewrite -/d dvdp_eq. set c2 := _ ^+ _; set m' := _ %/ _; move/eqP=> def_m. have dn0 : d != 0 by rewrite gcdp_eq0 negb_and nn0 orbT. have c1n0 : c1 != 0 by rewrite !expf_neq0 // lead_coef_eq0. have c2n0 : c2 != 0 by rewrite !expf_neq0 // lead_coef_eq0. rewrite -(@dvdp_scaler (c1 ^+ k)) ?expf_neq0 ?lead_coef_eq0 //. have c2k_n0 : c2 ^+ k != 0 by rewrite !expf_neq0 // lead_coef_eq0. rewrite -(@dvdp_scalel (c2 ^+k)) // -!exprZn def_m def_n !exprMn. rewrite dvdp_mul2r ?expf_neq0 //. have: coprimep (m' ^+ k) (n' ^+ k). rewrite coprimep_pexpl // coprimep_pexpr //; apply: coprimep_div_gcd. by rewrite nn0 orbT. move/coprimepP=> hc hd. have /size_poly1P [c cn0 em'] : size m' == 1%N. case: (eqVneq m' 0) => [m'0 |m'_n0]. move/eqP: def_m; rewrite m'0 mul0r scale_poly_eq0. by rewrite (negPf mn0) (negPf c2n0). have := (hc _ (dvdpp _) hd); rewrite -size_poly_eq1. rewrite polySpred; last by rewrite expf_eq0 negb_and m'_n0 orbT. rewrite size_exp eqSS muln_eq0; move: k_gt0; rewrite lt0n; move/negPf->. by rewrite orbF -{2}(@prednK (size m')) ?lt0n // size_poly_eq0. rewrite -(@dvdp_scalel c2) // def_m em' mul_polyC dvdp_scalel //. by rewrite -(@dvdp_scaler c1) // def_n dvdp_mull. Qed. Lemma root_gcd p q x : root (gcdp p q) x = root p x && root q x. Proof. rewrite /= !root_factor_theorem; apply/idP/andP=> [dg| [dp dq]]. by split; apply: dvdp_trans dg _; rewrite ?(dvdp_gcdl, dvdp_gcdr). have:= (Bezoutp p q)=> [[[u v]]]; rewrite eqp_sym=> e. by rewrite (eqp_dvdr _ e) dvdp_addl dvdp_mull. Qed. Lemma root_biggcd : forall x (ps : seq {poly R}), root (\big[gcdp/0]_(p <- ps) p) x = all (fun p => root p x) ps. Proof. move=> x; elim; first by rewrite big_nil root0. by move=> p ps ihp; rewrite big_cons /= root_gcd ihp. Qed. (* "gdcop Q P" is the Greatest Divisor of P which is coprime to Q *) (* if P null, we pose that gdcop returns 1 if Q null, 0 otherwise*) Fixpoint gdcop_rec q p k := if k is m.+1 then if coprimep p q then p else gdcop_rec q (divp p (gcdp p q)) m else (q == 0)%:R. Definition gdcop q p := gdcop_rec q p (size p). Variant gdcop_spec q p : {poly R} -> Type := GdcopSpec r of (dvdp r p) & ((coprimep r q) || (p == 0)) & (forall d, dvdp d p -> coprimep d q -> dvdp d r) : gdcop_spec q p r. Lemma gdcop0 q : gdcop q 0 = (q == 0)%:R. Proof. by rewrite /gdcop size_poly0. Qed. Lemma gdcop_recP : forall q p k, size p <= k -> gdcop_spec q p (gdcop_rec q p k). Proof. move=> q p k; elim: k p => [p | k ihk p] /=. rewrite leqn0 size_poly_eq0; move/eqP->. case q0: (_ == _); split; rewrite ?coprime1p // ?eqxx ?orbT //. by move=> d _; rewrite (eqP q0) coprimep0 dvdp1 size_poly_eq1. move=> hs; case cop : (coprimep _ _); first by split; rewrite ?dvdpp ?cop. case (eqVneq p 0) => [-> | p0]. by rewrite div0p; apply: ihk; rewrite size_poly0 leq0n. case: (eqVneq q 0) => [-> | q0]. rewrite gcdp0 divpp ?p0 //= => {hs ihk}; case: k=> /=. rewrite eqxx; split; rewrite ?dvd1p ?coprimep0 ?eqpxx //=. by move=> d _; rewrite coprimep0 dvdp1 size_poly_eq1. move=> n; rewrite coprimep0 polyC_eqp1 //; rewrite lc_expn_scalp_neq0. split; first by rewrite (@eqp_dvdl 1) ?dvd1p // polyC_eqp1 lc_expn_scalp_neq0. by rewrite coprimep0 polyC_eqp1 // ?lc_expn_scalp_neq0. by move=> d _; rewrite coprimep0; move/eqp_dvdl->; rewrite dvd1p. move: (dvdp_gcdl p q); rewrite dvdp_eq; move/eqP=> e. have sgp : size (gcdp p q) <= size p. by apply: dvdp_leq; rewrite ?gcdp_eq0 ?p0 ?q0 // dvdp_gcdl. have : p %/ gcdp p q != 0; last move/negPf=>p'n0. move: (dvdp_mulIl (p %/ gcdp p q) (gcdp p q)); move/dvdpN0; apply; rewrite -e. by rewrite scale_poly_eq0 negb_or lc_expn_scalp_neq0. have gn0 : gcdp p q != 0. move: (dvdp_mulIr (p %/ gcdp p q) (gcdp p q)); move/dvdpN0; apply; rewrite -e. by rewrite scale_poly_eq0 negb_or lc_expn_scalp_neq0. have sp' : size (p %/ (gcdp p q)) <= k. rewrite size_divp ?sgp // leq_subLR (leq_trans hs)//. rewrite -subn_gt0 addnK -subn1 ltn_subRL addn0 ltnNge leq_eqVlt. by rewrite [_ == _]cop ltnS leqn0 size_poly_eq0 (negPf gn0). case (ihk _ sp')=> r' dr'p'; first rewrite p'n0 orbF=> cr'q maxr'. constructor=> //=; rewrite ?(negPf p0) ?orbF //. exact/(dvdp_trans dr'p')/divp_dvd/dvdp_gcdl. move=> d dp cdq; apply: maxr'; last by rewrite cdq. case dpq: (d %| gcdp p q). move: (dpq); rewrite dvdp_gcd dp /= => dq; apply: dvdUp; move: cdq. apply: contraLR=> nd1; apply/coprimepPn; last first. by exists d; rewrite dvdp_gcd dvdpp dq nd1. move/negP: p0; move/negP; apply: contra=> d0; move: dp; rewrite (eqP d0). by rewrite dvd0p. move: (dp); apply: contraLR=> ndp'. rewrite (@eqp_dvdr ((lead_coef (gcdp p q) ^+ scalp p (gcdp p q))*:p)). by rewrite e; rewrite Gauss_dvdpl //; apply: (coprimep_dvdl (dvdp_gcdr _ _)). by rewrite eqp_sym eqp_scale // lc_expn_scalp_neq0. Qed. Lemma gdcopP q p : gdcop_spec q p (gdcop q p). Proof. by rewrite /gdcop; apply: gdcop_recP. Qed. Lemma coprimep_gdco p q : (q != 0)%B -> coprimep (gdcop p q) p. Proof. by move=> q_neq0; case: gdcopP=> d; rewrite (negPf q_neq0) orbF. Qed. Lemma size2_dvdp_gdco p q d : p != 0 -> size d = 2%N -> (d %| (gdcop q p)) = (d %| p) && ~~(d %| q). Proof. case: (eqVneq d 0) => [-> | dn0]; first by rewrite size_poly0. move=> p0 sd; apply/idP/idP. case: gdcopP=> r rp crq maxr dr; move/negPf: (p0)=> p0f. rewrite (dvdp_trans dr) //=. move: crq; apply: contraL=> dq; rewrite p0f orbF; apply/coprimepPn. by move: p0; apply: contra=> r0; move: rp; rewrite (eqP r0) dvd0p. by exists d; rewrite dvdp_gcd dr dq -size_poly_eq1 sd. case/andP=> dp dq; case: gdcopP=> r rp crq maxr; apply: maxr=> //. apply/coprimepP=> x xd xq. move: (dvdp_leq dn0 xd); rewrite leq_eqVlt sd; case/orP; last first. rewrite ltnS leq_eqVlt; case/orP; first by rewrite -size_poly_eq1. rewrite ltnS leqn0 size_poly_eq0; move/eqP=> x0; move: xd; rewrite x0 dvd0p. by rewrite (negPf dn0). by rewrite -sd dvdp_size_eqp //; move/(eqp_dvdl q); rewrite xq (negPf dq). Qed. Lemma dvdp_gdco p q : (gdcop p q) %| q. Proof. by case: gdcopP. Qed. Lemma root_gdco p q x : p != 0 -> root (gdcop q p) x = root p x && ~~(root q x). Proof. move=> p0 /=; rewrite !root_factor_theorem. apply: size2_dvdp_gdco; rewrite ?p0 //. by rewrite size_addl size_polyX // size_opp size_polyC ltnS; case: (x != 0). Qed. Lemma dvdp_comp_poly r p q : (p %| q) -> (p \Po r) %| (q \Po r). Proof. case: (eqVneq p 0) => [-> | pn0]. by rewrite comp_poly0 !dvd0p; move/eqP->; rewrite comp_poly0. rewrite dvdp_eq; set c := _ ^+ _; set s := _ %/ _; move/eqP=> Hq. apply: (@eq_dvdp c (s \Po r)); first by rewrite expf_neq0 // lead_coef_eq0. by rewrite -comp_polyZ Hq comp_polyM. Qed. Lemma gcdp_comp_poly r p q : gcdp p q \Po r %= gcdp (p \Po r) (q \Po r). Proof. apply/andP; split. by rewrite dvdp_gcd !dvdp_comp_poly ?dvdp_gcdl ?dvdp_gcdr. case: (Bezoutp p q) => [[u v]] /andP []. move/(dvdp_comp_poly r) => Huv _. rewrite (dvdp_trans _ Huv) // comp_polyD !comp_polyM. by rewrite dvdp_add // dvdp_mull // (dvdp_gcdl,dvdp_gcdr). Qed. Lemma coprimep_comp_poly r p q : coprimep p q -> coprimep (p \Po r) (q \Po r). Proof. rewrite -!gcdp_eqp1 -!size_poly_eq1 -!dvdp1; move/(dvdp_comp_poly r). rewrite comp_polyC => Hgcd. by apply: dvdp_trans Hgcd; case/andP: (gcdp_comp_poly r p q). Qed. Lemma coprimep_addl_mul p q r : coprimep r (p * r + q) = coprimep r q. Proof. by rewrite !coprimep_def (eqp_size (gcdp_addl_mul _ _ _)). Qed. Definition irreducible_poly p := (size p > 1) * (forall q, size q != 1%N -> q %| p -> q %= p) : Prop. Lemma irredp_neq0 p : irreducible_poly p -> p != 0. Proof. by rewrite -size_poly_eq0 -lt0n => [[/ltnW]]. Qed. Definition apply_irredp p (irr_p : irreducible_poly p) := irr_p.2. Coercion apply_irredp : irreducible_poly >-> Funclass. Lemma modp_XsubC p c : p %% ('X - c%:P) = p.[c]%:P. Proof. have: root (p - p.[c]%:P) c by rewrite /root !hornerE subrr. case/factor_theorem=> q /(canRL (subrK _)) Dp; rewrite modpE /= lead_coefXsubC. rewrite GRing.unitr1 expr1n invr1 scale1r {1}Dp. rewrite RingMonic.rmodp_addl_mul_small // ?monicXsubC // size_XsubC size_polyC. by case: (p.[c] == 0). Qed. Lemma coprimep_XsubC p c : coprimep p ('X - c%:P) = ~~ root p c. Proof. rewrite -coprimep_modl modp_XsubC /root -alg_polyC. have [-> | /coprimep_scalel->] := altP eqP; last exact: coprime1p. by rewrite scale0r /coprimep gcd0p size_XsubC. Qed. Lemma coprimepX p : coprimep p 'X = ~~ root p 0. Proof. by rewrite -['X]subr0 coprimep_XsubC. Qed. Lemma eqp_monic : {in monic &, forall p q, (p %= q) = (p == q)}. Proof. move=> p q monic_p monic_q; apply/idP/eqP=> [|-> //]. case/eqpP=> [[a b] /= /andP[a_neq0 _] eq_pq]. apply: (@mulfI _ a%:P); first by rewrite polyC_eq0. rewrite !mul_polyC eq_pq; congr (_ *: q); apply: (mulIf (oner_neq0 _)). by rewrite -{1}(monicP monic_q) -(monicP monic_p) -!lead_coefZ eq_pq. Qed. Lemma dvdp_mul_XsubC p q c : (p %| ('X - c%:P) * q) = ((if root p c then p %/ ('X - c%:P) else p) %| q). Proof. case: ifPn => [| not_pc0]; last by rewrite Gauss_dvdpr ?coprimep_XsubC. rewrite root_factor_theorem -eqp_div_XsubC mulrC => /eqP{1}->. by rewrite dvdp_mul2l ?polyXsubC_eq0. Qed. Lemma dvdp_prod_XsubC (I : Type) (r : seq I) (F : I -> R) p : p %| \prod_(i <- r) ('X - (F i)%:P) -> {m | p %= \prod_(i <- mask m r) ('X - (F i)%:P)}. Proof. elim: r => [|i r IHr] in p *. by rewrite big_nil dvdp1; exists nil; rewrite // big_nil -size_poly_eq1. rewrite big_cons dvdp_mul_XsubC root_factor_theorem -eqp_div_XsubC. case: eqP => [{2}-> | _] /IHr[m Dp]; last by exists (false :: m). by exists (true :: m); rewrite /= mulrC big_cons eqp_mul2l ?polyXsubC_eq0. Qed. Lemma irredp_XsubC (x : R) : irreducible_poly ('X - x%:P). Proof. split=> [|d size_d d_dv_Xx]; first by rewrite size_XsubC. have: ~ d %= 1 by apply/negP; rewrite -size_poly_eq1. have [|m /=] := @dvdp_prod_XsubC _ [:: x] id d; first by rewrite big_seq1. by case: m => [|[] [|_ _] /=]; rewrite (big_nil, big_seq1). Qed. Lemma irredp_XsubCP d p : irreducible_poly p -> d %| p -> {d %= 1} + {d %= p}. Proof. move=> irred_p dvd_dp; have [] := boolP (_ %= 1); first by left. by rewrite -size_poly_eq1=> /irred_p /(_ dvd_dp); right. Qed. End IDomainPseudoDivision. Hint Resolve eqpxx divp0 divp1 mod0p modp0 modp1 dvdp_mull dvdp_mulr dvdpp. Hint Resolve dvdp0. End CommonIdomain. Module Idomain. Include IdomainDefs. Export IdomainDefs. Include WeakIdomain. Include CommonIdomain. End Idomain. Module IdomainMonic. Import Ring ComRing UnitRing IdomainDefs Idomain. Section MonicDivisor. Variable R : idomainType. Variable q : {poly R}. Hypothesis monq : q \is monic. Implicit Type p d r : {poly R}. Lemma divpE p : p %/ q = rdivp p q. Proof. by rewrite divpE (eqP monq) unitr1 expr1n invr1 scale1r. Qed. Lemma modpE p : p %% q = rmodp p q. Proof. by rewrite modpE (eqP monq) unitr1 expr1n invr1 scale1r. Qed. Lemma scalpE p : scalp p q = 0%N. Proof. by rewrite scalpE (eqP monq) unitr1. Qed. Lemma divp_eq p : p = (p %/ q) * q + (p %% q). Proof. by rewrite -divp_eq (eqP monq) expr1n scale1r. Qed. Lemma divpp p : q %/ q = 1. Proof. by rewrite divpp ?monic_neq0 // (eqP monq) expr1n. Qed. Lemma dvdp_eq p : (q %| p) = (p == (p %/ q) * q). Proof. by rewrite dvdp_eq (eqP monq) expr1n scale1r. Qed. Lemma dvdpP p : reflect (exists qq, p = qq * q) (q %| p). Proof. apply: (iffP idP); first by rewrite dvdp_eq; move/eqP=> e; exists (p %/ q). by case=> qq ->; rewrite dvdp_mull // dvdpp. Qed. Lemma mulpK p : p * q %/ q = p. Proof. by rewrite mulpK ?monic_neq0 // (eqP monq) expr1n scale1r. Qed. Lemma mulKp p : q * p %/ q = p. Proof. by rewrite mulrC; apply: mulpK. Qed. End MonicDivisor. End IdomainMonic. Module IdomainUnit. Import Ring ComRing UnitRing IdomainDefs Idomain. Section UnitDivisor. Variable R : idomainType. Variable d : {poly R}. Hypothesis ulcd : lead_coef d \in GRing.unit. Implicit Type p q r : {poly R}. Lemma divp_eq p : p = (p %/ d) * d + (p %% d). Proof. by have := (divp_eq p d); rewrite scalpE ulcd expr0 scale1r. Qed. Lemma edivpP p q r : p = q * d + r -> size r < size d -> q = (p %/ d) /\ r = p %% d. Proof. move=> ep srd; have := (divp_eq p); rewrite {1}ep. move/eqP; rewrite -subr_eq -addrA addrC eq_sym -subr_eq -mulrBl; move/eqP. have lcdn0 : lead_coef d != 0 by apply: contraTneq ulcd => ->; rewrite unitr0. case abs: (p %/ d - q == 0). move: abs; rewrite subr_eq0; move/eqP->; rewrite subrr mul0r; move/eqP. by rewrite eq_sym subr_eq0; move/eqP->. have hleq : size d <= size ((p %/ d - q) * d). rewrite size_proper_mul; last first. by rewrite mulf_eq0 (negPf lcdn0) orbF lead_coef_eq0 abs. move: abs; rewrite -size_poly_eq0; move/negbT; rewrite -lt0n; move/prednK<-. by rewrite addSn /= leq_addl. have hlt : size (r - p %% d) < size d. apply: leq_ltn_trans (size_add _ _) _; rewrite size_opp. by rewrite gtn_max srd ltn_modp /= -lead_coef_eq0. by move=> e; have:= (leq_trans hlt hleq); rewrite e ltnn. Qed. Lemma divpP p q r : p = q * d + r -> size r < size d -> q = (p %/ d). Proof. by move/edivpP=> h; case/h. Qed. Lemma modpP p q r : p = q * d + r -> size r < size d -> r = (p %% d). Proof. by move/edivpP=> h; case/h. Qed. Lemma ulc_eqpP p q : lead_coef q \is a GRing.unit -> reflect (exists2 c : R, c != 0 & p = c *: q) (p %= q). Proof. case: (altP (lead_coef q =P 0)) => [->|]; first by rewrite unitr0. rewrite lead_coef_eq0 => nz_q ulcq; apply: (iffP idP). case: (altP (p =P 0)) => [->|nz_p]. by rewrite eqp_sym eqp0 (negbTE nz_q). move/eqp_eq=> eq; exists (lead_coef p / lead_coef q). by rewrite mulf_neq0 // ?invr_eq0 lead_coef_eq0. by apply/(scaler_injl ulcq); rewrite scalerA mulrCA divrr // mulr1. by case=> c nz_c ->; apply/eqpP; exists (1, c); rewrite ?scale1r ?oner_eq0. Qed. Lemma dvdp_eq p : (d %| p) = (p == p %/ d * d). Proof. apply/eqP/eqP=> [modp0 | ->]; last exact: modp_mull. by rewrite {1}(divp_eq p) modp0 addr0. Qed. Lemma ucl_eqp_eq p q : lead_coef q \is a GRing.unit -> p %= q -> p = (lead_coef p / lead_coef q) *: q. Proof. move=> ulcq /eqp_eq; move/(congr1 ( *:%R (lead_coef q)^-1 )). by rewrite !scalerA mulrC divrr // scale1r mulrC. Qed. Lemma modp_scalel c p : (c *: p) %% d = c *: (p %% d). Proof. case: (altP (c =P 0)) => [-> | cn0]; first by rewrite !scale0r mod0p. have e : (c *: p) = (c *: (p %/ d)) * d + c *: (p %% d). by rewrite -scalerAl -scalerDr -divp_eq. have s: size (c *: (p %% d)) < size d. rewrite -mul_polyC; apply: leq_ltn_trans (size_mul_leq _ _) _. rewrite size_polyC cn0 addSn add0n /= ltn_modp. by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0. by case: (edivpP e s) => _ ->. Qed. Lemma divp_scalel c p : (c *: p) %/ d = c *: (p %/ d). Proof. case: (altP (c =P 0)) => [-> | cn0]; first by rewrite !scale0r div0p. have e : (c *: p) = (c *: (p %/ d)) * d + c *: (p %% d). by rewrite -scalerAl -scalerDr -divp_eq. have s: size (c *: (p %% d)) < size d. rewrite -mul_polyC; apply: leq_ltn_trans (size_mul_leq _ _) _. rewrite size_polyC cn0 addSn add0n /= ltn_modp. by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0. by case: (edivpP e s) => ->. Qed. Lemma eqp_modpl p q : p %= q -> (p %% d) %= (q %% d). Proof. case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e]. by apply/eqpP; exists (c1, c2); rewrite ?c1n0 //= -!modp_scalel e. Qed. Lemma eqp_divl p q : p %= q -> (p %/ d) %= (q %/ d). Proof. case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e]. by apply/eqpP; exists (c1, c2); rewrite ?c1n0 // -!divp_scalel e. Qed. Lemma modp_opp p : (- p) %% d = - (p %% d). Proof. by rewrite -mulN1r -[- (_ %% _)]mulN1r -polyC_opp !mul_polyC modp_scalel. Qed. Lemma divp_opp p : (- p) %/ d = - (p %/ d). Proof. by rewrite -mulN1r -[- (_ %/ _)]mulN1r -polyC_opp !mul_polyC divp_scalel. Qed. Lemma modp_add p q : (p + q) %% d = p %% d + q %% d. Proof. have hs : size (p %% d + q %% d) < size d. apply: leq_ltn_trans (size_add _ _) _. rewrite gtn_max !ltn_modp andbb -lead_coef_eq0. by apply: contraTneq ulcd => ->; rewrite unitr0. have he : (p + q) = (p %/ d + q %/ d) * d + (p %% d + q %% d). rewrite {1}(divp_eq p) {1}(divp_eq q) addrAC addrA -mulrDl. by rewrite [_ %% _ + _]addrC addrA. by case: (edivpP he hs). Qed. Lemma divp_add p q : (p + q) %/ d = p %/ d + q %/ d. Proof. have hs : size (p %% d + q %% d) < size d. apply: leq_ltn_trans (size_add _ _) _. rewrite gtn_max !ltn_modp andbb -lead_coef_eq0. by apply: contraTneq ulcd => ->; rewrite unitr0. have he : (p + q) = (p %/ d + q %/ d) * d + (p %% d + q %% d). rewrite {1}(divp_eq p) {1}(divp_eq q) addrAC addrA -mulrDl. by rewrite [_ %% _ + _]addrC addrA. by case: (edivpP he hs). Qed. Lemma mulpK q : (q * d) %/ d = q. Proof. case/edivpP: (sym_eq (addr0 (q * d))); rewrite // size_poly0 size_poly_gt0. by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0. Qed. Lemma mulKp q : (d * q) %/ d = q. Proof. by rewrite mulrC; apply: mulpK. Qed. Lemma divp_addl_mul_small q r : size r < size d -> (q * d + r) %/ d = q. Proof. by move=> srd; rewrite divp_add (divp_small srd) addr0 mulpK. Qed. Lemma modp_addl_mul_small q r : size r < size d -> (q * d + r) %% d = r. Proof. by move=> srd; rewrite modp_add modp_mull add0r modp_small. Qed. Lemma divp_addl_mul q r : (q * d + r) %/ d = q + r %/ d. Proof. by rewrite divp_add mulpK. Qed. Lemma divpp : d %/ d = 1. Proof. by rewrite -{1}(mul1r d) mulpK. Qed. Lemma leq_trunc_divp m : size (m %/ d * d) <= size m. Proof. have dn0 : d != 0. by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0. case q0 : (m %/ d == 0); first by rewrite (eqP q0) mul0r size_poly0 leq0n. rewrite {2}(divp_eq m) size_addl // size_mul ?q0 //; move/negbT: q0. rewrite -size_poly_gt0; move/prednK<-; rewrite addSn /=. by move: dn0; rewrite -(ltn_modp m); move/ltn_addl->. Qed. Lemma dvdpP p : reflect (exists q, p = q * d) (d %| p). Proof. apply: (iffP idP) => [| [k ->]]; last by apply/eqP; rewrite modp_mull. by rewrite dvdp_eq; move/eqP->; exists (p %/ d). Qed. Lemma divpK p : d %| p -> p %/ d * d = p. Proof. by rewrite dvdp_eq; move/eqP. Qed. Lemma divpKC p : d %| p -> d * (p %/ d) = p. Proof. by move=> ?; rewrite mulrC divpK. Qed. Lemma dvdp_eq_div p q : d %| p -> (q == p %/ d) = (q * d == p). Proof. move/divpK=> {2}<-; apply/eqP/eqP; first by move->. suff dn0 : d != 0 by move/(mulIf dn0). by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0. Qed. Lemma dvdp_eq_mul p q : d %| p -> (p == q * d) = (p %/ d == q). Proof. by move=> dv_d_p; rewrite eq_sym -dvdp_eq_div // eq_sym. Qed. Lemma divp_mulA p q : d %| q -> p * (q %/ d) = p * q %/ d. Proof. move=> hdm; apply/eqP; rewrite eq_sym -dvdp_eq_mul. by rewrite -mulrA divpK. by move/divpK: hdm<-; rewrite mulrA dvdp_mull // dvdpp. Qed. Lemma divp_mulAC m n : d %| m -> m %/ d * n = m * n %/ d. Proof. by move=> hdm; rewrite mulrC (mulrC m); apply: divp_mulA. Qed. Lemma divp_mulCA p q : d %| p -> d %| q -> p * (q %/ d) = q * (p %/ d). Proof. by move=> hdp hdq; rewrite mulrC divp_mulAC // divp_mulA. Qed. Lemma modp_mul p q : (p * (q %% d)) %% d = (p * q) %% d. Proof. have -> : q %% d = q - q %/ d * d by rewrite {2}(divp_eq q) -addrA addrC subrK. rewrite mulrDr modp_add // -mulNr mulrA -{2}[_ %% _]addr0; congr (_ + _). by apply/eqP; apply: dvdp_mull; apply: dvdpp. Qed. End UnitDivisor. Section MoreUnitDivisor. Variable R : idomainType. Variable d : {poly R}. Hypothesis ulcd : lead_coef d \in GRing.unit. Implicit Types p q : {poly R}. Lemma expp_sub m n : n <= m -> (d ^+ (m - n))%N = d ^+ m %/ d ^+ n. Proof. by move/subnK=> {2}<-; rewrite exprD mulpK // lead_coef_exp unitrX. Qed. Lemma divp_pmul2l p q : lead_coef q \in GRing.unit -> d * p %/ (d * q) = p %/ q. Proof. move=> uq. have udq: lead_coef (d * q) \in GRing.unit. by rewrite lead_coefM unitrM_comm ?ulcd //; red; rewrite mulrC. rewrite {1}(divp_eq uq p) mulrDr mulrCA divp_addl_mul //. have dn0 : d != 0. by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0. have qn0 : q != 0. by rewrite -lead_coef_eq0; apply: contraTneq uq => ->; rewrite unitr0. have dqn0 : d * q != 0 by rewrite mulf_eq0 negb_or dn0. suff : size (d * (p %% q)) < size (d * q). by rewrite ltnNge -divpN0 // negbK => /eqP ->; rewrite addr0. case: (altP ( (p %% q) =P 0)) => [-> | rn0]. by rewrite mulr0 size_poly0 size_poly_gt0. rewrite !size_mul //; move: dn0; rewrite -size_poly_gt0. by move/prednK<-; rewrite !addSn /= ltn_add2l ltn_modp. Qed. Lemma divp_pmul2r p q : lead_coef p \in GRing.unit -> q * d %/ (p * d) = q %/ p. Proof. by move=> uq; rewrite -!(mulrC d) divp_pmul2l. Qed. Lemma divp_divl r p q : lead_coef r \in GRing.unit -> lead_coef p \in GRing.unit -> q %/ p %/ r = q %/ (p * r). Proof. move=> ulcr ulcp. have e : q = (q %/ p %/ r) * (p * r) + ((q %/ p) %% r * p + q %% p). by rewrite addrA (mulrC p) mulrA -mulrDl; rewrite -divp_eq //; apply: divp_eq. have pn0 : p != 0. by rewrite -lead_coef_eq0; apply: contraTneq ulcp => ->; rewrite unitr0. have rn0 : r != 0. by rewrite -lead_coef_eq0; apply: contraTneq ulcr => ->; rewrite unitr0. have s : size ((q %/ p) %% r * p + q %% p) < size (p * r). case: (altP ((q %/ p) %% r =P 0)) => [-> | qn0]. rewrite mul0r add0r size_mul // (polySpred rn0) addnS /=. by apply: leq_trans (leq_addr _ _); rewrite ltn_modp. rewrite size_addl mulrC. by rewrite !size_mul // (polySpred pn0) !addSn /= ltn_add2l ltn_modp. rewrite size_mul // (polySpred qn0) addnS /=. by apply: leq_trans (leq_addr _ _); rewrite ltn_modp. case: (edivpP _ e s) => //; rewrite lead_coefM unitrM_comm ?ulcp //. by red; rewrite mulrC. Qed. Lemma divpAC p q : lead_coef p \in GRing.unit -> q %/ d %/ p = q %/ p %/ d. Proof. by move=> ulcp; rewrite !divp_divl // mulrC. Qed. Lemma modp_scaler c p : c \in GRing.unit -> p %% (c *: d) = (p %% d). Proof. move=> cn0; case: (eqVneq d 0) => [-> | dn0]; first by rewrite scaler0 !modp0. have e : p = (c^-1 *: (p %/ d)) * (c *: d) + (p %% d). by rewrite scalerCA scalerA mulVr // scale1r -(divp_eq ulcd). suff s : size (p %% d) < size (c *: d). by rewrite (modpP _ e s) // -mul_polyC lead_coefM lead_coefC unitrM cn0. by rewrite size_scale ?ltn_modp //; apply: contraTneq cn0 => ->; rewrite unitr0. Qed. Lemma divp_scaler c p : c \in GRing.unit -> p %/ (c *: d) = c^-1 *: (p %/ d). Proof. move=> cn0; case: (eqVneq d 0) => [-> | dn0]. by rewrite scaler0 !divp0 scaler0. have e : p = (c^-1 *: (p %/ d)) * (c *: d) + (p %% d). by rewrite scalerCA scalerA mulVr // scale1r -(divp_eq ulcd). suff s : size (p %% d) < size (c *: d). by rewrite (divpP _ e s) // -mul_polyC lead_coefM lead_coefC unitrM cn0. by rewrite size_scale ?ltn_modp //; apply: contraTneq cn0 => ->; rewrite unitr0. Qed. End MoreUnitDivisor. End IdomainUnit. Module Field. Import Ring ComRing UnitRing. Include IdomainDefs. Export IdomainDefs. Include CommonIdomain. Section FieldDivision. Variable F : fieldType. Implicit Type p q r d : {poly F}. Lemma divp_eq p q : p = (p %/ q) * q + (p %% q). Proof. case: (eqVneq q 0) => [-> | qn0]; first by rewrite modp0 mulr0 add0r. by apply: IdomainUnit.divp_eq; rewrite unitfE lead_coef_eq0. Qed. Lemma divp_modpP p q d r : p = q * d + r -> size r < size d -> q = (p %/ d) /\ r = p %% d. Proof. move=> he hs; apply: IdomainUnit.edivpP => //; rewrite unitfE lead_coef_eq0. by rewrite -size_poly_gt0; apply: leq_trans hs. Qed. Lemma divpP p q d r : p = q * d + r -> size r < size d -> q = (p %/ d). Proof. by move/divp_modpP=> h; case/h. Qed. Lemma modpP p q d r : p = q * d + r -> size r < size d -> r = (p %% d). Proof. by move/divp_modpP=> h; case/h. Qed. Lemma eqpfP p q : p %= q -> p = (lead_coef p / lead_coef q) *: q. Proof. have [->|nz_q] := altP (q =P 0). by rewrite eqp0 => /eqP ->; rewrite scaler0. move/IdomainUnit.ucl_eqp_eq; apply; rewrite unitfE. by move: nz_q; rewrite -lead_coef_eq0 => nz_qT. Qed. Lemma dvdp_eq q p : (q %| p) = (p == p %/ q * q). Proof. case: (eqVneq q 0) => [-> | qn0]; first by rewrite dvd0p mulr0 eq_sym. by apply: IdomainUnit.dvdp_eq; rewrite unitfE lead_coef_eq0. Qed. Lemma eqpf_eq p q : reflect (exists2 c, c != 0 & p = c *: q) (p %= q). Proof. apply: (iffP idP); last first. case=> c nz_c ->; apply/eqpP. by exists (1, c); rewrite ?scale1r ?oner_eq0. have [->|nz_q] := altP (q =P 0). by rewrite eqp0=> /eqP ->; exists 1; rewrite ?scale1r ?oner_eq0. case/IdomainUnit.ulc_eqpP; first by rewrite unitfE lead_coef_eq0. by move=> c nz_c ->; exists c. Qed. Lemma modp_scalel c p q : (c *: p) %% q = c *: (p %% q). Proof. case: (eqVneq q 0) => [-> | qn0]; first by rewrite !modp0. by apply: IdomainUnit.modp_scalel; rewrite unitfE lead_coef_eq0. Qed. Lemma mulpK p q : q != 0 -> p * q %/ q = p. Proof. by move=> qn0; rewrite IdomainUnit.mulpK // unitfE lead_coef_eq0. Qed. Lemma mulKp p q : q != 0 -> q * p %/ q = p. Proof. by rewrite mulrC; apply: mulpK. Qed. Lemma divp_scalel c p q : (c *: p) %/ q = c *: (p %/ q). Proof. case: (eqVneq q 0) => [-> | qn0]; first by rewrite !divp0 scaler0. by apply: IdomainUnit.divp_scalel; rewrite unitfE lead_coef_eq0. Qed. Lemma modp_scaler c p d : c != 0 -> p %% (c *: d) = (p %% d). Proof. move=> cn0; case: (eqVneq d 0) => [-> | dn0]; first by rewrite scaler0 !modp0. have e : p = (c^-1 *: (p %/ d)) * (c *: d) + (p %% d). by rewrite scalerCA scalerA mulVf // scale1r -divp_eq. suff s : size (p %% d) < size (c *: d) by rewrite (modpP e s). by rewrite size_scale ?ltn_modp. Qed. Lemma divp_scaler c p d : c != 0 -> p %/ (c *: d) = c^-1 *: (p %/ d). Proof. move=> cn0; case: (eqVneq d 0) => [-> | dn0]. by rewrite scaler0 !divp0 scaler0. have e : p = (c^-1 *: (p %/ d)) * (c *: d) + (p %% d). by rewrite scalerCA scalerA mulVf // scale1r -divp_eq. suff s : size (p %% d) < size (c *: d) by rewrite (divpP e s). by rewrite size_scale ?ltn_modp. Qed. Lemma eqp_modpl d p q : p %= q -> (p %% d) %= (q %% d). Proof. case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e]. by apply/eqpP; exists (c1, c2); rewrite ?c1n0 // -!modp_scalel e. Qed. Lemma eqp_divl d p q : p %= q -> (p %/ d) %= (q %/ d). Proof. case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e]. by apply/eqpP; exists (c1, c2); rewrite ?c1n0 // -!divp_scalel e. Qed. Lemma eqp_modpr d p q : p %= q -> (d %% p) %= (d %% q). Proof. case/eqpP=> [[c1 c2]] /andP [c1n0 c2n0 e]. have -> : p = (c1^-1 * c2) *: q by rewrite -scalerA -e scalerA mulVf // scale1r. by rewrite modp_scaler ?eqpxx // mulf_eq0 negb_or invr_eq0 c1n0. Qed. Lemma eqp_mod p1 p2 q1 q2 : p1 %= p2 -> q1 %= q2 -> p1 %% q1 %= p2 %% q2. Proof. move=> e1 e2; apply: eqp_trans (eqp_modpr _ e2). by apply: eqp_trans (eqp_modpl _ e1); apply: eqpxx. Qed. Lemma eqp_divr (d m n : {poly F}) : m %= n -> (d %/ m) %= (d %/ n). Proof. case/eqpP=> [[c1 c2]] /andP [c1n0 c2n0 e]. have -> : m = (c1^-1 * c2) *: n by rewrite -scalerA -e scalerA mulVf // scale1r. by rewrite divp_scaler ?eqp_scale // ?invr_eq0 mulf_eq0 negb_or invr_eq0 c1n0. Qed. Lemma eqp_div p1 p2 q1 q2 : p1 %= p2 -> q1 %= q2 -> p1 %/ q1 %= p2 %/ q2. Proof. move=> e1 e2; apply: eqp_trans (eqp_divr _ e2). by apply: eqp_trans (eqp_divl _ e1); apply: eqpxx. Qed. Lemma eqp_gdcor p q r : q %= r -> gdcop p q %= gdcop p r. Proof. move=> eqr; rewrite /gdcop (eqp_size eqr). move: (size r)=> n; elim: n p q r eqr => [|n ihn] p q r; first by rewrite eqpxx. move=> eqr /=; rewrite (eqp_coprimepl p eqr); case: ifP => _ //; apply: ihn. by apply: eqp_div => //; apply: eqp_gcdl. Qed. Lemma eqp_gdcol p q r : q %= r -> gdcop q p %= gdcop r p. Proof. move=> eqr; rewrite /gdcop; move: (size p)=> n. elim: n p q r eqr {1 3}p (eqpxx p) => [|n ihn] p q r eqr s esp /=. move: eqr; case: (eqVneq q 0)=> [-> | nq0 eqr] /=. by rewrite eqp_sym eqp0; move->; rewrite eqxx eqpxx. suff rn0 : r != 0 by rewrite (negPf nq0) (negPf rn0) eqpxx. by apply: contraTneq eqr => ->; rewrite eqp0. rewrite (eqp_coprimepr _ eqr) (eqp_coprimepl _ esp); case: ifP=> _ //. by apply: ihn => //; apply: eqp_div => //; apply: eqp_gcd. Qed. Lemma eqp_rgdco_gdco q p : rgdcop q p %= gdcop q p. Proof. rewrite /rgdcop /gdcop; move: (size p)=> n. elim: n p q {1 3}p {1 3}q (eqpxx p) (eqpxx q) => [|n ihn] p q s t /= sp tq. move: tq; case: (eqVneq t 0)=> [-> | nt0 etq]. by rewrite eqp_sym eqp0; move->; rewrite eqxx eqpxx. suff qn0 : q != 0 by rewrite (negPf nt0) (negPf qn0) eqpxx. by apply: contraTneq etq => ->; rewrite eqp0. rewrite rcoprimep_coprimep (eqp_coprimepl t sp) (eqp_coprimepr p tq). case: ifP=> // _; apply: ihn => //; apply: eqp_trans (eqp_rdiv_div _ _) _. by apply: eqp_div => //; apply: eqp_trans (eqp_rgcd_gcd _ _) _; apply: eqp_gcd. Qed. Lemma modp_opp p q : (- p) %% q = - (p %% q). Proof. case: (eqVneq q 0) => [-> | qn0]; first by rewrite !modp0. by apply: IdomainUnit.modp_opp; rewrite unitfE lead_coef_eq0. Qed. Lemma divp_opp p q : (- p) %/ q = - (p %/ q). Proof. case: (eqVneq q 0) => [-> | qn0]; first by rewrite !divp0 oppr0. by apply: IdomainUnit.divp_opp; rewrite unitfE lead_coef_eq0. Qed. Lemma modp_add d p q : (p + q) %% d = p %% d + q %% d. Proof. case: (eqVneq d 0) => [-> | dn0]; first by rewrite !modp0. by apply: IdomainUnit.modp_add; rewrite unitfE lead_coef_eq0. Qed. Lemma modNp p q : (- p) %% q = - (p %% q). Proof. by apply/eqP; rewrite -addr_eq0 -modp_add addNr mod0p. Qed. Lemma divp_add d p q : (p + q) %/ d = p %/ d + q %/ d. Proof. case: (eqVneq d 0) => [-> | dn0]; first by rewrite !divp0 addr0. by apply: IdomainUnit.divp_add; rewrite unitfE lead_coef_eq0. Qed. Lemma divp_addl_mul_small d q r : size r < size d -> (q * d + r) %/ d = q. Proof. move=> srd; rewrite divp_add (divp_small srd) addr0 mulpK //. by rewrite -size_poly_gt0; apply: leq_trans srd. Qed. Lemma modp_addl_mul_small d q r : size r < size d -> (q * d + r) %% d = r. Proof. by move=> srd; rewrite modp_add modp_mull add0r modp_small. Qed. Lemma divp_addl_mul d q r : d != 0 -> (q * d + r) %/ d = q + r %/ d. Proof. by move=> dn0; rewrite divp_add mulpK. Qed. Lemma divpp d : d != 0 -> d %/ d = 1. Proof. by move=> dn0; apply: IdomainUnit.divpp; rewrite unitfE lead_coef_eq0. Qed. Lemma leq_trunc_divp d m : size (m %/ d * d) <= size m. Proof. case: (eqVneq d 0) => [-> | dn0]; first by rewrite mulr0 size_poly0. by apply: IdomainUnit.leq_trunc_divp; rewrite unitfE lead_coef_eq0. Qed. Lemma divpK d p : d %| p -> p %/ d * d = p. Proof. case: (eqVneq d 0) => [-> | dn0]; first by move/dvd0pP->; rewrite mulr0. by apply: IdomainUnit.divpK; rewrite unitfE lead_coef_eq0. Qed. Lemma divpKC d p : d %| p -> d * (p %/ d) = p. Proof. by move=> ?; rewrite mulrC divpK. Qed. Lemma dvdp_eq_div d p q : d != 0 -> d %| p -> (q == p %/ d) = (q * d == p). Proof. by move=> dn0; apply: IdomainUnit.dvdp_eq_div; rewrite unitfE lead_coef_eq0. Qed. Lemma dvdp_eq_mul d p q : d != 0 -> d %| p -> (p == q * d) = (p %/ d == q). Proof. by move=> dn0 dv_d_p; rewrite eq_sym -dvdp_eq_div // eq_sym. Qed. Lemma divp_mulA d p q : d %| q -> p * (q %/ d) = p * q %/ d. Proof. case: (eqVneq d 0) => [-> | dn0]; first by move/dvd0pP->; rewrite !divp0 mulr0. by apply: IdomainUnit.divp_mulA; rewrite unitfE lead_coef_eq0. Qed. Lemma divp_mulAC d m n : d %| m -> m %/ d * n = m * n %/ d. Proof. by move=> hdm; rewrite mulrC (mulrC m); apply: divp_mulA. Qed. Lemma divp_mulCA d p q : d %| p -> d %| q -> p * (q %/ d) = q * (p %/ d). Proof. by move=> hdp hdq; rewrite mulrC divp_mulAC // divp_mulA. Qed. Lemma expp_sub d m n : d != 0 -> m >= n -> (d ^+ (m - n))%N = d ^+ m %/ d ^+ n. Proof. by move=> dn0 /subnK=> {2}<-; rewrite exprD mulpK // expf_neq0. Qed. Lemma divp_pmul2l d q p : d != 0 -> q != 0 -> d * p %/ (d * q) = p %/ q. Proof. by move=> dn0 qn0; apply: IdomainUnit.divp_pmul2l; rewrite unitfE lead_coef_eq0. Qed. Lemma divp_pmul2r d p q : d != 0 -> p != 0 -> q * d %/ (p * d) = q %/ p. Proof. by move=> dn0 qn0; rewrite -!(mulrC d) divp_pmul2l. Qed. Lemma divp_divl r p q : q %/ p %/ r = q %/ (p * r). Proof. case: (eqVneq r 0) => [-> | rn0]; first by rewrite mulr0 !divp0. case: (eqVneq p 0) => [-> | pn0]; first by rewrite mul0r !divp0 div0p. by apply: IdomainUnit.divp_divl; rewrite unitfE lead_coef_eq0. Qed. Lemma divpAC d p q : q %/ d %/ p = q %/ p %/ d. Proof. by rewrite !divp_divl // mulrC. Qed. Lemma edivp_def p q : edivp p q = (0%N, p %/ q, p %% q). Proof. rewrite Idomain.edivp_def; congr (_, _, _); rewrite /scalp 2!unlock /=. case (eqVneq q 0) => [-> | qn0]; first by rewrite eqxx lead_coef0 unitr0. rewrite (negPf qn0) /= unitfE lead_coef_eq0 qn0 /=. by case: (redivp_rec _ _ _ _) => [[]]. Qed. Lemma divpE p q : p %/ q = (lead_coef q)^-(rscalp p q) *: (rdivp p q). Proof. case: (eqVneq q 0) => [-> | qn0]; first by rewrite rdivp0 divp0 scaler0. by rewrite Idomain.divpE unitfE lead_coef_eq0 qn0. Qed. Lemma modpE p q : p %% q = (lead_coef q)^-(rscalp p q) *: (rmodp p q). Proof. case: (eqVneq q 0) => [-> | qn0]. by rewrite rmodp0 modp0 /rscalp unlock eqxx lead_coef0 expr0 invr1 scale1r. by rewrite Idomain.modpE unitfE lead_coef_eq0 qn0. Qed. Lemma scalpE p q : scalp p q = 0%N. Proof. case: (eqVneq q 0) => [-> | qn0]; first by rewrite scalp0. by rewrite Idomain.scalpE unitfE lead_coef_eq0 qn0. Qed. (* Just to have it without importing the weak theory *) Lemma dvdpE p q : p %| q = rdvdp p q. Proof. exact: Idomain.dvdpE. Qed. Variant edivp_spec m d : nat * {poly F} * {poly F} -> Type := EdivpSpec n q r of m = q * d + r & (d != 0) ==> (size r < size d) : edivp_spec m d (n, q, r). Lemma edivpP m d : edivp_spec m d (edivp m d). Proof. rewrite edivp_def; constructor; first exact: divp_eq. by apply/implyP=> dn0; rewrite ltn_modp. Qed. Lemma edivp_eq d q r : size r < size d -> edivp (q * d + r) d = (0%N, q, r). Proof. move=> srd; apply: Idomain.edivp_eq; rewrite // unitfE lead_coef_eq0. by rewrite -size_poly_gt0; apply: leq_trans srd. Qed. Lemma modp_mul p q m : (p * (q %% m)) %% m = (p * q) %% m. Proof. have ->: q %% m = q - q %/ m * m by rewrite {2}(divp_eq q m) -addrA addrC subrK. rewrite mulrDr modp_add // -mulNr mulrA -{2}[_ %% _]addr0; congr (_ + _). by apply/eqP; apply: dvdp_mull; apply: dvdpp. Qed. Lemma dvdpP p q : reflect (exists qq, p = qq * q) (q %| p). Proof. case: (eqVneq q 0)=> [-> | qn0]; last first. by apply: IdomainUnit.dvdpP; rewrite unitfE lead_coef_eq0. rewrite dvd0p. by apply: (iffP idP) => [/eqP->| [? ->]]; [exists 1|]; rewrite mulr0. Qed. Lemma Bezout_eq1_coprimepP : forall p q, reflect (exists u, u.1 * p + u.2 * q = 1) (coprimep p q). Proof. move=> p q; apply: (iffP idP)=> [hpq|]; last first. by case=> [[u v]] /= e; apply/Bezout_coprimepP; exists (u, v); rewrite e eqpxx. case/Bezout_coprimepP: hpq => [[u v]] /=. case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0] e. exists (c2^-1 *: (c1 *: u), c2^-1 *: (c1 *: v)); rewrite /= -!scalerAl. by rewrite -!scalerDr e scalerA mulVf // scale1r. Qed. Lemma dvdp_gdcor p q : q != 0 -> p %| (gdcop q p) * (q ^+ size p). Proof. move=> q_neq0; rewrite /gdcop. elim: (size p) {-2 5}p (leqnn (size p))=> {p} [|n ihn] p. rewrite size_poly_leq0; move/eqP->. by rewrite size_poly0 /= dvd0p expr0 mulr1 (negPf q_neq0). move=> hsp /=; have [->|p_neq0] := altP (p =P 0). rewrite size_poly0 /= dvd0p expr0 mulr1 div0p /=. case: ifP=> // _; have := (ihn 0). by rewrite size_poly0 expr0 mulr1 dvd0p=> /(_ isT). have [|ncop_pq] := boolP (coprimep _ _); first by rewrite dvdp_mulr ?dvdpp. have g_gt1: (1 < size (gcdp p q))%N. have [|//|/eqP] := ltngtP; last by rewrite -coprimep_def (negPf ncop_pq). by rewrite ltnS leqn0 size_poly_eq0 gcdp_eq0 (negPf p_neq0). have sd : (size (p %/ gcdp p q) < size p)%N. rewrite size_divp -?size_poly_eq0 -(subnKC g_gt1) // add2n /=. by rewrite -[size _]prednK ?size_poly_gt0 // ltnS subSS leq_subr. rewrite -{1}[p](divpK (dvdp_gcdl _ q)) -(subnKC sd) addSnnS exprD mulrA. rewrite dvdp_mul ?ihn //; first by rewrite -ltnS (leq_trans sd). by rewrite exprS dvdp_mulr // dvdp_gcdr. Qed. Lemma reducible_cubic_root p q : size p <= 4 -> 1 < size q < size p -> q %| p -> {r | root p r}. Proof. move=> p_le4 /andP[]; rewrite leq_eqVlt eq_sym. have [/poly2_root[x qx0] _ _ | _ /= q_gt2 p_gt_q] := size q =P 2. by exists x; rewrite -!dvdp_XsubCl in qx0 *; apply: (dvdp_trans qx0). case/dvdpP/sig_eqW=> r def_p; rewrite def_p. suffices /poly2_root[x rx0]: size r = 2 by exists x; rewrite rootM rx0. have /norP[nz_r nz_q]: ~~ [|| r == 0 | q == 0]. by rewrite -mulf_eq0 -def_p -size_poly_gt0 (leq_ltn_trans _ p_gt_q). rewrite def_p size_mul // -subn1 leq_subLR ltn_subRL in p_gt_q p_le4. by apply/eqP; rewrite -(eqn_add2r (size q)) eqn_leq (leq_trans p_le4). Qed. Lemma cubic_irreducible p : 1 < size p <= 4 -> (forall x, ~~ root p x) -> irreducible_poly p. Proof. move=> /andP[p_gt1 p_le4] root'p; split=> // q sz_q_neq1 q_dv_p. have nz_p: p != 0 by rewrite -size_poly_gt0 ltnW. have nz_q: q != 0 by apply: contraTneq q_dv_p => ->; rewrite dvd0p. have q_gt1: size q > 1 by rewrite ltn_neqAle eq_sym sz_q_neq1 size_poly_gt0. rewrite -dvdp_size_eqp // eqn_leq dvdp_leq //= leqNgt; apply/negP=> p_gt_q. by have [|x /idPn//] := reducible_cubic_root p_le4 _ q_dv_p; rewrite q_gt1. Qed. Section FieldRingMap. Variable rR : ringType. Variable f : {rmorphism F -> rR}. Local Notation "p ^f" := (map_poly f p) : ring_scope. Implicit Type a b : {poly F}. Lemma redivp_map a b : redivp a^f b^f = (rscalp a b, (rdivp a b)^f, (rmodp a b)^f). Proof. rewrite /rdivp /rscalp /rmodp !unlock map_poly_eq0 size_map_poly. case: eqP; rewrite /= -(rmorph0 (map_poly_rmorphism f)) //; move/eqP=> q_nz. move: (size a) => m; elim: m 0%N 0 a => [|m IHm] qq r a /=. rewrite -!mul_polyC !size_map_poly !lead_coef_map // -(map_polyXn f). by rewrite -!(map_polyC f) -!rmorphM -rmorphB -rmorphD; case: (_ < _). rewrite -!mul_polyC !size_map_poly !lead_coef_map // -(map_polyXn f). by rewrite -!(map_polyC f) -!rmorphM -rmorphB -rmorphD /= IHm; case: (_ < _). Qed. End FieldRingMap. Section FieldMap. Variable rR : idomainType. Variable f : {rmorphism F -> rR}. Local Notation "p ^f" := (map_poly f p) : ring_scope. Implicit Type a b : {poly F}. Lemma edivp_map a b : edivp a^f b^f = (0%N, (a %/ b)^f, (a %% b)^f). Proof. case: (eqVneq b 0) => [-> | bn0]. rewrite (rmorph0 (map_poly_rmorphism f)) WeakIdomain.edivp_def !modp0 !divp0. by rewrite (rmorph0 (map_poly_rmorphism f)) scalp0. rewrite unlock redivp_map lead_coef_map rmorph_unit; last first. by rewrite unitfE lead_coef_eq0. rewrite modpE divpE !map_polyZ !rmorphV ?rmorphX // unitfE. by rewrite expf_neq0 // lead_coef_eq0. Qed. Lemma scalp_map p q : scalp p^f q^f = scalp p q. Proof. by rewrite /scalp edivp_map edivp_def. Qed. Lemma map_divp p q : (p %/ q)^f = p^f %/ q^f. Proof. by rewrite /divp edivp_map edivp_def. Qed. Lemma map_modp p q : (p %% q)^f = p^f %% q^f. Proof. by rewrite /modp edivp_map edivp_def. Qed. Lemma egcdp_map p q : egcdp (map_poly f p) (map_poly f q) = (map_poly f (egcdp p q).1, map_poly f (egcdp p q).2). Proof. wlog le_qp: p q / size q <= size p. move=> IH; have [/IH// | lt_qp] := leqP (size q) (size p). have /IH := ltnW lt_qp; rewrite /egcdp !size_map_poly ltnW // leqNgt lt_qp /=. by case: (egcdp_rec _ _ _) => u v [-> ->]. rewrite /egcdp !size_map_poly {}le_qp; move: (size q) => n. elim: n => /= [|n IHn] in p q *; first by rewrite rmorph1 rmorph0. rewrite map_poly_eq0; have [_ | nz_q] := ifPn; first by rewrite rmorph1 rmorph0. rewrite -map_modp (IHn q (p %% q)); case: (egcdp_rec _ _ n) => u v /=. by rewrite map_polyZ lead_coef_map -rmorphX scalp_map rmorphB rmorphM -map_divp. Qed. Lemma dvdp_map p q : (p^f %| q^f) = (p %| q). Proof. by rewrite /dvdp -map_modp map_poly_eq0. Qed. Lemma eqp_map p q : (p^f %= q^f) = (p %= q). Proof. by rewrite /eqp !dvdp_map. Qed. Lemma gcdp_map p q : (gcdp p q)^f = gcdp p^f q^f. Proof. wlog lt_p_q: p q / size p < size q. move=> IHpq; case: (ltnP (size p) (size q)) => [|le_q_p]; first exact: IHpq. rewrite gcdpE (gcdpE p^f) !size_map_poly ltnNge le_q_p /= -map_modp. case: (eqVneq q 0) => [-> | q_nz]; first by rewrite rmorph0 !gcdp0. by rewrite IHpq ?ltn_modp. elim: {q}_.+1 p {-2}q (ltnSn (size q)) lt_p_q => // m IHm p q le_q_m lt_p_q. rewrite gcdpE (gcdpE p^f) !size_map_poly lt_p_q -map_modp. case: (eqVneq p 0) => [-> | q_nz]; first by rewrite rmorph0 !gcdp0. by rewrite IHm ?(leq_trans lt_p_q) ?ltn_modp. Qed. Lemma coprimep_map p q : coprimep p^f q^f = coprimep p q. Proof. by rewrite -!gcdp_eqp1 -eqp_map rmorph1 gcdp_map. Qed. Lemma gdcop_rec_map p q n : (gdcop_rec p q n)^f = (gdcop_rec p^f q^f n). Proof. elim: n p q => [|n IH] => /= p q. by rewrite map_poly_eq0; case: eqP; rewrite ?rmorph1 ?rmorph0. rewrite /coprimep -gcdp_map size_map_poly. by case: eqP => Hq0 //; rewrite -map_divp -IH. Qed. Lemma gdcop_map p q : (gdcop p q)^f = (gdcop p^f q^f). Proof. by rewrite /gdcop gdcop_rec_map !size_map_poly. Qed. End FieldMap. End FieldDivision. End Field. Module ClosedField. Import Field. Section closed. Variable F : closedFieldType. Lemma root_coprimep (p q : {poly F}): (forall x, root p x -> q.[x] != 0) -> coprimep p q. Proof. move=> Ncmn; rewrite -gcdp_eqp1 -size_poly_eq1; apply/closed_rootP. by case=> r; rewrite root_gcd !rootE=> /andP [/Ncmn/negbTE->]. Qed. Lemma coprimepP (p q : {poly F}): reflect (forall x, root p x -> q.[x] != 0) (coprimep p q). Proof. by apply: (iffP idP)=> [/coprimep_root|/root_coprimep]. Qed. End closed. End ClosedField. End Pdiv. Export Pdiv.Field.