fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
gcdp_comp_polyr 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 //; [ exact: dvdp_gcdl | exact: dvdp_gcdr].
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | gcdp_comp_poly | |
coprimep_comp_polyr 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | coprimep_comp_poly | |
coprimep_addl_mulp q r : coprimep r (p * r + q) = coprimep r q.
Proof. by rewrite !coprimep_def (eqp_size (gcdp_addl_mul _ _ _)). Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | coprimep_addl_mul | |
irreducible_polyp :=
(size p > 1) * (forall q, size q != 1 -> q %| p -> q %= p) : Prop. | Definition | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | irreducible_poly | |
irredp_neq0p : irreducible_poly p -> p != 0.
Proof. by rewrite -size_poly_gt0 => [[/ltnW]]. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | irredp_neq0 | |
apply_irredpp (irr_p : irreducible_poly p) := irr_p.2. | Definition | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | apply_irredp | |
apply_irredp: irreducible_poly >-> Funclass. | Coercion | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | apply_irredp | |
modp_XsubCp c : p %% ('X - c%:P) = p.[c]%:P.
Proof.
have/factor_theorem [q /(canRL (subrK _)) Dp]: root (p - p.[c]%:P) c.
by rewrite /root !hornerE subrr.
rewrite modpE /= lead_coefXsubC unitr1 expr1n invr1 scale1r [in LHS]Dp.
rewrite RingMonic.rmodp_addl_mul_small // ?monicXsubC// size_XsubC size_polyC.
by case: (p.[c] == 0).
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modp_XsubC | |
coprimep_XsubCp c : coprimep p ('X - c%:P) = ~~ root p c.
Proof.
rewrite -coprimep_modl modp_XsubC /root -alg_polyC.
have [-> | /coprimepZl->] := eqVneq; last exact: coprime1p.
by rewrite scale0r /coprimep gcd0p size_XsubC.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | coprimep_XsubC | |
coprimep_XsubC2(a b : R) : b - a != 0 ->
coprimep ('X - a%:P) ('X - b%:P).
Proof. by move=> bBa_neq0; rewrite coprimep_XsubC rootE hornerXsubC. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | coprimep_XsubC2 | |
coprimepXp : coprimep p 'X = ~~ root p 0.
Proof. by rewrite -['X]subr0 coprimep_XsubC. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | coprimepX | |
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 -[in LHS](monicP monic_q) -(monicP monic_p) -!lead_coefZ eq_pq.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_monic | |
dvdp_mul_XsubCp 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdp_mul_XsubC | |
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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdp_prod_XsubC | |
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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | irredp_XsubC | |
irredp_XaddC(x : R) : irreducible_poly ('X + x%:P).
Proof. by rewrite -[x]opprK rmorphN; apply: irredp_XsubC. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | irredp_XaddC | |
irredp_XsubCPd 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. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | irredp_XsubCP | |
dvdp_exp_XsubCP(p : {poly R}) (c : R) (n : nat) :
reflect (exists2 k, (k <= n)%N & p %= ('X - c%:P) ^+ k)
(p %| ('X - c%:P) ^+ n).
Proof.
apply: (iffP idP) => [|[k lkn /eqp_dvdl->]]; last by rewrite dvdp_exp2l.
move=> /Pdiv.WeakIdomain.dvdpP[[/= a q] a_neq0].
have [m [r]] := multiplicity_XsubC p c; have [->|pN0]/= := eqVneq p 0.
rewrite mulr0 => _ _ /eqP; rewrite scale_poly_eq0 (negPf a_neq0)/=.
by rewrite expf_eq0/= andbC polyXsubC_eq0.
move=> rNc ->; rewrite mulrA => eq_qrm; exists m.
have: ('X - c%:P) ^+ m %| a *: ('X - c%:P) ^+ n by rewrite eq_qrm dvdp_mull.
by rewrite (eqp_dvdr _ (eqp_scale _ _))// dvdp_Pexp2l// size_XsubC.
suff /eqP : size r = 1%N.
by rewrite size_poly_eq1 => /eqp_mulr/eqp_trans->//; rewrite mul1r eqpxx.
have : r %| a *: ('X - c%:P) ^+ n by rewrite eq_qrm mulrAC dvdp_mull.
rewrite (eqp_dvdr _ (eqp_scale _ _))//.
move: rNc; rewrite -coprimep_XsubC => /(coprimep_expr n) /coprimepP.
by move=> /(_ _ (dvdpp _)); rewrite -size_poly_eq1 => /(_ _)/eqP.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdp_exp_XsubCP | |
divpEp : p %/ q = rdivp p q.
Proof. by rewrite divpE (eqP monq) unitr1 expr1n invr1 scale1r. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpE | |
modpEp : p %% q = rmodp p q.
Proof. by rewrite modpE (eqP monq) unitr1 expr1n invr1 scale1r. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpE | |
scalpEp : scalp p q = 0.
Proof. by rewrite scalpE (eqP monq) unitr1. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | scalpE | |
divp_eqp : p = (p %/ q) * q + (p %% q).
Proof. by rewrite -divp_eq (eqP monq) expr1n scale1r. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_eq | |
divppp : q %/ q = 1.
Proof. by rewrite divpp ?monic_neq0 // (eqP monq) expr1n. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpp | |
dvdp_eqp : (q %| p) = (p == (p %/ q) * q).
Proof. by rewrite dvdp_eq (eqP monq) expr1n scale1r. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdp_eq | |
dvdpPp : 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdpP | |
mulpKp : p * q %/ q = p.
Proof. by rewrite mulpK ?monic_neq0 // (eqP monq) expr1n scale1r. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | mulpK | |
mulKpp : q * p %/ q = p. Proof. by rewrite mulrC mulpK. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | mulKp | |
drop_poly_divpn p : drop_poly n p = p %/ 'X^n.
Proof. by rewrite RingMonic.drop_poly_rdivp divpE // monicXn. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | drop_poly_divp | |
take_poly_modpn p : take_poly n p = p %% 'X^n.
Proof. by rewrite RingMonic.take_poly_rmodp modpE // monicXn. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | take_poly_modp | |
divp_eqp : p = (p %/ d) * d + (p %% d).
Proof. by have := divp_eq p d; rewrite scalpE ulcd expr0 scale1r. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_eq | |
edivpPp 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 [LHS]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.
have [-> /esym /eqP|abs] := eqVneq (p %/ d) q.
by rewrite subrr mul0r subr_eq0 => /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 subr_eq0.
by move: abs; rewrite -subr_eq0; move/polySpred->; rewrite addSn /= leq_addl.
have hlt : size (r - p %% d) < size d.
apply: leq_ltn_trans (size_polyD _ _) _.
by rewrite gtn_max srd size_polyN ltn_modp -lead_coef_eq0.
by move=> e; have:= leq_trans hlt hleq; rewrite e ltnn.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | edivpP | |
divpPp q r : p = q * d + r -> size r < size d -> q = (p %/ d).
Proof. by move/edivpP=> h; case/h. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpP | |
modpPp q r : p = q * d + r -> size r < size d -> r = (p %% d).
Proof. by move/edivpP=> h; case/h. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpP | |
ulc_eqpPp q : lead_coef q \is a GRing.unit ->
reflect (exists2 c : R, c != 0 & p = c *: q) (p %= q).
Proof.
have [->|] := eqVneq (lead_coef q) 0; first by rewrite unitr0.
rewrite lead_coef_eq0 => nz_q ulcq; apply: (iffP idP).
have [->|nz_p] := eqVneq p 0; first by rewrite eqp_sym eqp0 (negPf 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | ulc_eqpP | |
dvdp_eqp : (d %| p) = (p == p %/ d * d).
Proof.
apply/eqP/eqP=> [modp0 | ->]; last exact: modp_mull.
by rewrite [p in LHS]divp_eq modp0 addr0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdp_eq | |
ucl_eqp_eqp 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | ucl_eqp_eq | |
modpZlc p : (c *: p) %% d = c *: (p %% d).
Proof.
have [-> | cn0] := eqVneq c 0; first by rewrite !scale0r mod0p.
have e : (c *: p) = (c *: (p %/ d)) * d + c *: (p %% d).
by rewrite -scalerAl -scalerDr -divp_eq.
suff s: size (c *: (p %% d)) < size d by case: (edivpP e s) => _ ->.
rewrite -mul_polyC; apply: leq_ltn_trans (size_polyMleq _ _) _.
rewrite size_polyC cn0 addSn add0n /= ltn_modp -lead_coef_eq0.
by apply: contraTneq ulcd => ->; rewrite unitr0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpZl | |
divpZlc p : (c *: p) %/ d = c *: (p %/ d).
Proof.
have [-> | cn0] := eqVneq c 0; first by rewrite !scale0r div0p.
have e : (c *: p) = (c *: (p %/ d)) * d + c *: (p %% d).
by rewrite -scalerAl -scalerDr -divp_eq.
suff s: size (c *: (p %% d)) < size d by case: (edivpP e s) => ->.
rewrite -mul_polyC; apply: leq_ltn_trans (size_polyMleq _ _) _.
rewrite size_polyC cn0 addSn add0n /= ltn_modp -lead_coef_eq0.
by apply: contraTneq ulcd => ->; rewrite unitr0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpZl | |
eqp_modplp q : p %= q -> (p %% d) %= (q %% d).
Proof.
case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e].
by apply/eqpP; exists (c1, c2); rewrite ?c1n0 //= -!modpZl e.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_modpl | |
eqp_divlp q : p %= q -> (p %/ d) %= (q %/ d).
Proof.
case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e].
by apply/eqpP; exists (c1, c2); rewrite ?c1n0 // -!divpZl e.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_divl | |
modpNp : (- p) %% d = - (p %% d).
Proof. by rewrite -mulN1r -[RHS]mulN1r -polyCN !mul_polyC modpZl. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpN | |
divpNp : (- p) %/ d = - (p %/ d).
Proof. by rewrite -mulN1r -[RHS]mulN1r -polyCN !mul_polyC divpZl. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpN | |
modpDp q : (p + q) %% d = p %% d + q %% d.
Proof.
have/edivpP [] // : (p + q) = (p %/ d + q %/ d) * d + (p %% d + q %% d).
by rewrite mulrDl addrACA -!divp_eq.
apply: leq_ltn_trans (size_polyD _ _) _.
rewrite gtn_max !ltn_modp andbb -lead_coef_eq0.
by apply: contraTneq ulcd => ->; rewrite unitr0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpD | |
divpDp q : (p + q) %/ d = p %/ d + q %/ d.
Proof.
have/edivpP [] // : (p + q) = (p %/ d + q %/ d) * d + (p %% d + q %% d).
by rewrite mulrDl addrACA -!divp_eq.
apply: leq_ltn_trans (size_polyD _ _) _.
rewrite gtn_max !ltn_modp andbb -lead_coef_eq0.
by apply: contraTneq ulcd => ->; rewrite unitr0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpD | |
mulpKq : (q * d) %/ d = q.
Proof.
case/esym/edivpP: (addr0 (q * d)); rewrite // size_poly0 size_poly_gt0.
by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | mulpK | |
mulKpq : (d * q) %/ d = q. Proof. by rewrite mulrC; apply: mulpK. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | mulKp | |
divp_addl_mul_smallq r : size r < size d -> (q * d + r) %/ d = q.
Proof. by move=> srd; rewrite divpD (divp_small srd) addr0 mulpK. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_addl_mul_small | |
modp_addl_mul_smallq r : size r < size d -> (q * d + r) %% d = r.
Proof. by move=> srd; rewrite modpD modp_mull add0r modp_small. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modp_addl_mul_small | |
divp_addl_mulq r : (q * d + r) %/ d = q + r %/ d.
Proof. by rewrite divpD mulpK. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_addl_mul | |
divpp: d %/ d = 1. Proof. by rewrite -[d in d %/ _]mul1r mulpK. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpp | |
leq_divMpm : size (m %/ d * d) <= size m.
Proof.
case: (eqVneq d 0) ulcd => [->|dn0 _]; first by rewrite lead_coef0 unitr0.
have [->|q0] := eqVneq (m %/ d) 0; first by rewrite mul0r size_poly0 leq0n.
rewrite {2}(divp_eq m) size_polyDl // size_mul // (polySpred q0) addSn /=.
by rewrite ltn_addl // ltn_modp.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | leq_divMp | |
dvdpPp : 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdpP | |
divpKp : d %| p -> p %/ d * d = p.
Proof. by rewrite dvdp_eq; move/eqP. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpK | |
divpKCp : d %| p -> d * (p %/ d) = p.
Proof. by move=> ?; rewrite mulrC divpK. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpKC | |
dvdp_eq_divp q : d %| p -> (q == p %/ d) = (q * d == p).
Proof.
move/divpK=> {2}<-; apply/eqP/eqP; first by move->.
apply/mulIf; rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->.
by rewrite unitr0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdp_eq_div | |
dvdp_eq_mulp 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdp_eq_mul | |
divp_mulAp 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_mulA | |
divp_mulACm n : d %| m -> m %/ d * n = m * n %/ d.
Proof. by move=> hdm; rewrite mulrC (mulrC m); apply: divp_mulA. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_mulAC | |
divp_mulCAp q : d %| p -> d %| q -> p * (q %/ d) = q * (p %/ d).
Proof. by move=> hdp hdq; rewrite mulrC divp_mulAC // divp_mulA. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_mulCA | |
modp_mulp q : (p * (q %% d)) %% d = (p * q) %% d.
Proof. by rewrite [q in RHS]divp_eq mulrDr modpD mulrA modp_mull add0r. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modp_mul | |
leq_trunc_divp:= leq_divMp. | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | leq_trunc_divp | |
expp_subm 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | expp_sub | |
divp_pmul2lp q : lead_coef q \in GRing.unit -> d * p %/ (d * q) = p %/ q.
Proof.
move=> uq; rewrite {1}(divp_eq uq p) mulrDr mulrCA divp_addl_mul //; last first.
by rewrite lead_coefM unitrM_comm ?ulcd //; red; rewrite mulrC.
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.
have [-> | rn0] := eqVneq (p %% q) 0.
by rewrite mulr0 size_poly0 size_poly_gt0.
by rewrite !size_mul // (polySpred dn0) !addSn /= ltn_add2l ltn_modp.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_pmul2l | |
divp_pmul2rp q : lead_coef p \in GRing.unit -> q * d %/ (p * d) = q %/ p.
Proof. by move=> uq; rewrite -!(mulrC d) divp_pmul2l. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_pmul2r | |
divp_divlr 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).
have [-> | qn0] := eqVneq ((q %/ p) %% r) 0.
rewrite mul0r add0r size_mul // (polySpred rn0) addnS /=.
by apply: leq_trans (leq_addr _ _); rewrite ltn_modp.
rewrite size_polyDl 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_divl | |
divpACp q : lead_coef p \in GRing.unit -> q %/ d %/ p = q %/ p %/ d.
Proof. by move=> ulcp; rewrite !divp_divl // mulrC. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpAC | |
modpZrc p : c \in GRing.unit -> p %% (c *: d) = (p %% d).
Proof.
case: (eqVneq d 0) => [-> | dn0 cn0]; 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpZr | |
divpZrc p : c \in GRing.unit -> p %/ (c *: d) = c^-1 *: (p %/ d).
Proof.
case: (eqVneq d 0) => [-> | dn0 cn0]; first 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. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpZr | |
divp_eqp q : p = (p %/ q) * q + (p %% q).
Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite modp0 mulr0 add0r.
by apply: IdomainUnit.divp_eq; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_eq | |
divp_modpPp 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_modpP | |
divpPp q d r : p = q * d + r -> size r < size d ->
q = (p %/ d).
Proof. by move/divp_modpP=> h; case/h. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpP | |
modpPp q d r : p = q * d + r -> size r < size d -> r = (p %% d).
Proof. by move/divp_modpP=> h; case/h. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpP | |
eqpfPp q : p %= q -> p = (lead_coef p / lead_coef q) *: q.
Proof.
have [->|nz_q] := eqVneq q 0; first by rewrite eqp0 scaler0 => /eqP ->.
by apply/IdomainUnit.ucl_eqp_eq; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqpfP | |
dvdp_eqq p : (q %| p) = (p == p %/ q * q).
Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite dvd0p mulr0 eq_sym.
by apply: IdomainUnit.dvdp_eq; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | dvdp_eq | |
eqpf_eqp 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] := eqVneq q 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqpf_eq | |
modpZlc p q : (c *: p) %% q = c *: (p %% q).
Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite !modp0.
by apply: IdomainUnit.modpZl; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpZl | |
mulpKp q : q != 0 -> p * q %/ q = p.
Proof. by move=> qn0; rewrite IdomainUnit.mulpK // unitfE lead_coef_eq0. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | mulpK | |
mulKpp q : q != 0 -> q * p %/ q = p.
Proof. by rewrite mulrC; apply: mulpK. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | mulKp | |
divpZlc p q : (c *: p) %/ q = c *: (p %/ q).
Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite !divp0 scaler0.
by apply: IdomainUnit.divpZl; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpZl | |
modpZrc p d : c != 0 -> p %% (c *: d) = (p %% d).
Proof.
case: (eqVneq d 0) => [-> | dn0 cn0]; 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpZr | |
divpZrc p d : c != 0 -> p %/ (c *: d) = c^-1 *: (p %/ d).
Proof.
case: (eqVneq d 0) => [-> | dn0 cn0]; first 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpZr | |
eqp_modpld 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 // -!modpZl e.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_modpl | |
eqp_divld 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 // -!divpZl e.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_divl | |
eqp_modprd 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 modpZr ?eqpxx // mulf_eq0 negb_or invr_eq0 c1n0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_modpr | |
eqp_modp1 p2 q1 q2 : p1 %= p2 -> q1 %= q2 -> p1 %% q1 %= p2 %% q2.
Proof. move=> e1 e2; exact: eqp_trans (eqp_modpl _ e1) (eqp_modpr _ e2). Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_mod | |
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 divpZr ?eqp_scale // ?invr_eq0 mulf_eq0 negb_or invr_eq0 c1n0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_divr | |
eqp_divp1 p2 q1 q2 : p1 %= p2 -> q1 %= q2 -> p1 %/ q1 %= p2 %/ q2.
Proof. move=> e1 e2; exact: eqp_trans (eqp_divl _ e1) (eqp_divr _ e2). Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_div | |
eqp_gdcorp 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 => _ //.
exact/ihn/eqp_div/eqp_gcdl.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_gdcor | |
eqp_gdcolp 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 /=.
case: (eqVneq q 0) eqr => [-> | nq0 eqr] /=.
by rewrite eqp_sym eqp0 => ->; rewrite eqpxx.
by case: (eqVneq r 0) eqr nq0 => [->|]; rewrite ?eqpxx // eqp0 => ->.
rewrite (eqp_coprimepr _ eqr) (eqp_coprimepl _ esp); case: ifP=> _ //.
exact/ihn/eqp_div/eqp_gcd.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_gdcol | |
eqp_rgdco_gdcoq 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.
case: (eqVneq t 0) tq => [-> | nt0 etq].
by rewrite eqp_sym eqp0 => ->; rewrite eqpxx.
by case: (eqVneq q 0) etq nt0 => [->|]; rewrite ?eqpxx // 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 | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | eqp_rgdco_gdco | |
modpDd p q : (p + q) %% d = p %% d + q %% d.
Proof.
have [-> | dn0] := eqVneq d 0; first by rewrite !modp0.
by apply: IdomainUnit.modpD; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpD | |
modpNp q : (- p) %% q = - (p %% q).
Proof. by apply/eqP; rewrite -addr_eq0 -modpD addNr mod0p. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modpN | |
modNpp q : (- p) %% q = - (p %% q). Proof. exact: modpN. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modNp | |
divpDd p q : (p + q) %/ d = p %/ d + q %/ d.
Proof.
have [-> | dn0] := eqVneq d 0; first by rewrite !divp0 addr0.
by apply: IdomainUnit.divpD; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpD | |
divpNp q : (- p) %/ q = - (p %/ q).
Proof. by apply/eqP; rewrite -addr_eq0 -divpD addNr div0p. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpN | |
divp_addl_mul_smalld q r : size r < size d -> (q * d + r) %/ d = q.
Proof.
move=> srd; rewrite divpD (divp_small srd) addr0 mulpK // -size_poly_gt0.
exact: leq_trans srd.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_addl_mul_small | |
modp_addl_mul_smalld q r : size r < size d -> (q * d + r) %% d = r.
Proof. by move=> srd; rewrite modpD modp_mull add0r modp_small. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | modp_addl_mul_small | |
divp_addl_muld q r : d != 0 -> (q * d + r) %/ d = q + r %/ d.
Proof. by move=> dn0; rewrite divpD mulpK. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divp_addl_mul | |
divppd : d != 0 -> d %/ d = 1.
Proof.
by move=> dn0; apply: IdomainUnit.divpp; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | divpp | |
leq_divMpd m : size (m %/ d * d) <= size m.
Proof.
have [-> | dn0] := eqVneq d 0; first by rewrite mulr0 size_poly0.
by apply: IdomainUnit.leq_divMp; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop ssralg poly"
] | algebra/polydiv.v | leq_divMp |
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