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 |
|---|---|---|---|---|---|---|
derivXnn : ('X^n)^`() = 'X^(n.-1) *+ n.
Proof.
case: n => [|n]; first exact: derivC.
apply/polyP=> i; rewrite coef_deriv coefMn !coefXn eqSS.
by case: eqP => [-> // | _]; rewrite !mul0rn.
Qed.
Fact deriv_is_semilinear : semilinear deriv.
Proof.
split=> [k p|p q]; apply/polyP => i.
by rewrite !(coef_deriv, coefZ) mulrnAr.
by rewrite !(coef_deriv, coefD) mulrnDl.
Qed.
HB.instance Definition _ := GRing.isSemilinear.Build R {poly R} {poly R} _ deriv
deriv_is_semilinear.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivXn
| |
deriv0: 0^`() = 0.
Proof. exact: linear0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
deriv0
| |
derivD: {morph deriv : p q / p + q}.
Proof. exact: linearD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivD
| |
derivMnn p : (p *+ n)^`() = p^`() *+ n.
Proof. exact: linearMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivMn
| |
derivZc p : (c *: p)^`() = c *: p^`().
Proof. exact: linearZ. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivZ
| |
deriv_mulCc p : (c%:P * p)^`() = c%:P * p^`().
Proof. by rewrite !mul_polyC derivZ. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
deriv_mulC
| |
derivMXaddCp c : (p * 'X + c%:P)^`() = p + p^`() * 'X.
Proof.
apply/polyP=> i; rewrite raddfD /= derivC addr0 coefD !(coefMX, coef_deriv).
by case: i; rewrite ?addr0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivMXaddC
| |
derivMp q : (p * q)^`() = p^`() * q + p * q^`().
Proof.
elim/poly_ind: p => [|p b IHp]; first by rewrite !(mul0r, add0r, derivC).
rewrite mulrDl -mulrA -commr_polyX mulrA -[_ * 'X]addr0 raddfD /= !derivMXaddC.
by rewrite deriv_mulC IHp !mulrDl -!mulrA !commr_polyX !addrA.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivM
| |
derivnn p := iter n deriv p.
Local Notation "a ^` ( n )" := (derivn n a) : ring_scope.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivn
| |
derivn0p : p^`(0) = p.
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivn0
| |
derivn1p : p^`(1) = p^`().
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivn1
| |
derivnSp n : p^`(n.+1) = p^`(n)^`().
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivnS
| |
derivSnp n : p^`(n.+1) = p^`()^`(n).
Proof. exact: iterSr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivSn
| |
coef_derivnn p i : p^`(n)`_i = p`_(n + i) *+ (n + i) ^_ n.
Proof.
elim: n i => [|n IHn] i; first by rewrite ffactn0 mulr1n.
by rewrite derivnS coef_deriv IHn -mulrnA ffactnSr addSnnS addKn.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
coef_derivn
| |
polyOver_derivn(ringS : semiringClosed R) :
{in polyOver ringS, forall p n, p^`(n) \is a polyOver ringS}.
Proof.
move=> p /polyOverP Kp /= n; apply/polyOverP=> i.
by rewrite coef_derivn rpredMn.
Qed.
Fact derivn_is_semilinear n : semilinear (derivn n).
Proof.
by elim: n => // n IHn; split=> [a p|p q]; rewrite derivnS IHn semilinearPZ.
Qed.
HB.instance Definition _ n :=
GRing.isSemilinear.Build R {poly R} {poly R} _ (derivn n)
(derivn_is_semilinear n).
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
polyOver_derivn
| |
derivnCc n : c%:P^`(n) = if n == 0 then c%:P else 0.
Proof. by case: n => // n; rewrite derivSn derivC linear0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivnC
| |
derivnDn : {morph derivn n : p q / p + q}.
Proof. exact: linearD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivnD
| |
derivnMnn m p : (p *+ m)^`(n) = p^`(n) *+ m.
Proof. exact: linearMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivnMn
| |
derivnZn : scalable (derivn n).
Proof. exact: linearZZ. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivnZ
| |
derivnXnm n : ('X^m)^`(n) = 'X^(m - n) *+ m ^_ n.
Proof.
apply/polyP=>i; rewrite coef_derivn coefMn !coefXn.
case: (ltnP m n) => [lt_m_n | le_m_n].
by rewrite eqn_leq leqNgt ltn_addr // mul0rn ffact_small.
by rewrite -{1 3}(subnKC le_m_n) eqn_add2l; case: eqP => [->|]; rewrite ?mul0rn.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivnXn
| |
derivnMXaddCn p c :
(p * 'X + c%:P)^`(n.+1) = p^`(n) *+ n.+1 + p^`(n.+1) * 'X.
Proof.
elim: n => [|n IHn]; first by rewrite derivn1 derivMXaddC.
rewrite derivnS IHn derivD derivM derivX mulr1 derivMn -!derivnS.
by rewrite addrA addrAC -mulrSr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivnMXaddC
| |
derivn_poly0p n : size p <= n -> p^`(n) = 0.
Proof.
move=> le_p_n; apply/polyP=> i; rewrite coef_derivn.
rewrite nth_default; first by rewrite mul0rn coef0.
exact/(leq_trans le_p_n)/leq_addr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
derivn_poly0
| |
lt_size_deriv(p : {poly R}) : p != 0 -> size p^`() < size p.
Proof. by move=> /polySpred->; apply: size_poly. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lt_size_deriv
| |
nderivnn p := \poly_(i < size p - n) (p`_(n + i) *+ 'C(n + i, n)).
Local Notation "a ^`N ( n )" := (nderivn n a) : ring_scope.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivn
| |
coef_nderivnn p i : p^`N(n)`_i = p`_(n + i) *+ 'C(n + i, n).
Proof.
rewrite coef_poly ltn_subRL; case: leqP => // le_p_ni.
by rewrite nth_default ?mul0rn.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
coef_nderivn
| |
nderivn_defn p : p^`(n) = p^`N(n) *+ n`!.
Proof.
by apply/polyP=> i; rewrite coefMn coef_nderivn coef_derivn -mulrnA bin_ffact.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivn_def
| |
polyOver_nderivn(ringS : semiringClosed R) :
{in polyOver ringS, forall p n, p^`N(n) \in polyOver ringS}.
Proof.
move=> p /polyOverP Sp /= n; apply/polyOverP=> i.
by rewrite coef_nderivn rpredMn.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
polyOver_nderivn
| |
nderivn0p : p^`N(0) = p.
Proof. by rewrite -[p^`N(0)](nderivn_def 0). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivn0
| |
nderivn1p : p^`N(1) = p^`().
Proof. by rewrite -[p^`N(1)](nderivn_def 1). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivn1
| |
nderivnCc n : (c%:P)^`N(n) = if n == 0 then c%:P else 0.
Proof.
apply/polyP=> i; rewrite coef_nderivn.
by case: n => [|n]; rewrite ?bin0 // coef0 coefC mul0rn.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivnC
| |
nderivnXnm n : ('X^m)^`N(n) = 'X^(m - n) *+ 'C(m, n).
Proof.
apply/polyP=> i; rewrite coef_nderivn coefMn !coefXn.
have [lt_m_n | le_n_m] := ltnP m n.
by rewrite eqn_leq leqNgt ltn_addr // mul0rn bin_small.
by rewrite -{1 3}(subnKC le_n_m) eqn_add2l; case: eqP => [->|]; rewrite ?mul0rn.
Qed.
Fact nderivn_is_semilinear n : semilinear (nderivn n).
Proof.
split=> [k p|p q]; apply/polyP => i.
by rewrite !(coef_nderivn, coefZ) mulrnAr.
by rewrite !(coef_nderivn, coefD) mulrnDl.
Qed.
HB.instance Definition _ n :=
GRing.isSemilinear.Build R {poly R} {poly R} _ (nderivn n)
(nderivn_is_semilinear n).
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivnXn
| |
nderivnDn : {morph nderivn n : p q / p + q}.
Proof. exact: linearD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivnD
| |
nderivnMnn m p : (p *+ m)^`N(n) = p^`N(n) *+ m.
Proof. exact: linearMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivnMn
| |
nderivnZn : scalable (nderivn n).
Proof. exact: linearZZ. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivnZ
| |
nderivnMXaddCn p c :
(p * 'X + c%:P)^`N(n.+1) = p^`N(n) + p^`N(n.+1) * 'X.
Proof.
apply/polyP=> i; rewrite coef_nderivn !coefD !coefMX coefC.
rewrite !addSn /= !coef_nderivn addr0 binS mulrnDr addrC; congr (_ + _).
by rewrite addSnnS; case: i; rewrite // addn0 bin_small.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivnMXaddC
| |
nderivn_poly0p n : size p <= n -> p^`N(n) = 0.
Proof.
move=> le_p_n; apply/polyP=> i; rewrite coef_nderivn.
rewrite nth_default; first by rewrite mul0rn coef0.
exact/(leq_trans le_p_n)/leq_addr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderivn_poly0
| |
nderiv_taylorp x h :
GRing.comm x h -> p.[x + h] = \sum_(i < size p) p^`N(i).[x] * h ^+ i.
Proof.
move/commrX=> cxh; elim/poly_ind: p => [|p c IHp].
by rewrite size_poly0 big_ord0 horner0.
rewrite hornerMXaddC size_MXaddC.
have [-> | nz_p] := eqVneq p 0.
rewrite horner0 !simp; have [-> | _] := c =P 0; first by rewrite big_ord0.
by rewrite size_poly0 big_ord_recl big_ord0 nderivn0 hornerC !simp.
rewrite big_ord_recl nderivn0 !simp hornerMXaddC addrAC; congr (_ + _).
rewrite mulrDr {}IHp !big_distrl polySpred //= big_ord_recl /= mulr1 -addrA.
rewrite nderivn0 /bump /(addn 1) /=; congr (_ + _).
rewrite !big_ord_recr /= nderivnMXaddC -mulrA -exprSr -polySpred // !addrA.
congr (_ + _); last by rewrite (nderivn_poly0 (leqnn _)) !simp.
rewrite addrC -big_split /=; apply: eq_bigr => i _.
rewrite nderivnMXaddC hornerD (hornerM_comm _ (comm_polyX _)) hornerX.
by rewrite mulrDl -!mulrA -exprSr cxh.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderiv_taylor
| |
nderiv_taylor_widen p x h :
GRing.comm x h -> size p <= n ->
p.[x + h] = \sum_(i < n) p^`N(i).[x] * h ^+ i.
Proof.
move/nderiv_taylor=> -> le_p_n.
rewrite (big_ord_widen n (fun i => p^`N(i).[x] * h ^+ i)) // big_mkcond.
apply: eq_bigr => i _; case: leqP => // /nderivn_poly0->.
by rewrite horner0 simp.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
nderiv_taylor_wide
| |
monic_pred:= fun p => lead_coef p == 1.
Arguments monic_pred _ /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monic_pred
| |
monic:= [qualify p | monic_pred p].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monic
| |
monicEp : (p \is monic) = (lead_coef p == 1). Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monicE
| |
monicPp : reflect (lead_coef p = 1) (p \is monic).
Proof. exact: eqP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monicP
| |
monic1: 1 \is monic. Proof. exact/eqP/lead_coef1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monic1
| |
monicX: 'X \is monic. Proof. exact/eqP/lead_coefX. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monicX
| |
monicXnn : 'X^n \is monic. Proof. exact/eqP/lead_coefXn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monicXn
| |
monic_neq0p : p \is monic -> p != 0.
Proof. by rewrite -lead_coef_eq0 => /eqP->; apply: oner_neq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monic_neq0
| |
lead_coef_monicMp q : p \is monic -> lead_coef (p * q) = lead_coef q.
Proof.
have [-> | nz_q] := eqVneq q 0; first by rewrite mulr0.
by move/monicP=> mon_p; rewrite lead_coef_proper_mul mon_p mul1r ?lead_coef_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lead_coef_monicM
| |
lead_coef_Mmonicp q : q \is monic -> lead_coef (p * q) = lead_coef p.
Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite mul0r.
by move/monicP=> mon_q; rewrite lead_coef_proper_mul mon_q mulr1 ?lead_coef_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lead_coef_Mmonic
| |
size_monicMp q :
p \is monic -> q != 0 -> size (p * q) = (size p + size q).-1.
Proof.
move/monicP=> mon_p nz_q.
by rewrite size_proper_mul // mon_p mul1r lead_coef_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_monicM
| |
size_Mmonicp q :
p != 0 -> q \is monic -> size (p * q) = (size p + size q).-1.
Proof.
move=> nz_p /monicP mon_q.
by rewrite size_proper_mul // mon_q mulr1 lead_coef_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_Mmonic
| |
monicMlp q : p \is monic -> (p * q \is monic) = (q \is monic).
Proof. by move=> mon_p; rewrite !monicE lead_coef_monicM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monicMl
| |
monicMrp q : q \is monic -> (p * q \is monic) = (p \is monic).
Proof. by move=> mon_q; rewrite !monicE lead_coef_Mmonic. Qed.
Fact monic_mulr_closed : mulr_closed monic.
Proof. by split=> [|p q mon_p]; rewrite (monic1, monicMl). Qed.
HB.instance Definition _ := GRing.isMulClosed.Build {poly R} monic_pred
monic_mulr_closed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monicMr
| |
monic_expp n : p \is monic -> p ^+ n \is monic.
Proof. exact: rpredX. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monic_exp
| |
monic_prodI rI (P : pred I) (F : I -> {poly R}):
(forall i, P i -> F i \is monic) -> \prod_(i <- rI | P i) F i \is monic.
Proof. exact: rpred_prod. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monic_prod
| |
monicXaddCc : 'X + c%:P \is monic.
Proof. exact/eqP/lead_coefXaddC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monicXaddC
| |
monicXnaddCn c : 0 < n -> 'X^n + c%:P \is monic.
Proof. by move=> n_gt0; rewrite monicE lead_coefXnaddC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monicXnaddC
| |
lreg_lead0p : GRing.lreg (lead_coef p) -> p != 0.
Proof. by move/lreg_neq0; rewrite lead_coef_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lreg_lead0
| |
rreg_lead0p : GRing.rreg (lead_coef p) -> p != 0.
Proof. by move/rreg_neq0; rewrite lead_coef_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rreg_lead0
| |
lreg_sizec p : GRing.lreg c -> size (c *: p) = size p.
Proof.
move=> reg_c; have [-> | nz_p] := eqVneq p 0; first by rewrite scaler0.
rewrite -mul_polyC size_proper_mul; first by rewrite size_polyC lreg_neq0.
by rewrite lead_coefC mulrI_eq0 ?lead_coef_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lreg_size
| |
lreg_polyZ_eq0c p : GRing.lreg c -> (c *: p == 0) = (p == 0).
Proof. by rewrite -!size_poly_eq0 => /lreg_size->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lreg_polyZ_eq0
| |
lead_coef_lregc p : GRing.lreg c -> lead_coef (c *: p) = c * lead_coef p.
Proof. by move=> reg_c; rewrite !lead_coefE coefZ lreg_size. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lead_coef_lreg
| |
rreg_sizec p : GRing.rreg c -> size (p * c%:P) = size p.
Proof.
move=> reg_c; have [-> | nz_p] := eqVneq p 0; first by rewrite mul0r.
rewrite size_proper_mul; first by rewrite size_polyC rreg_neq0 ?addn1.
by rewrite lead_coefC mulIr_eq0 ?lead_coef_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rreg_size
| |
rreg_polyMC_eq0c p : GRing.rreg c -> (p * c%:P == 0) = (p == 0).
Proof. by rewrite -!size_poly_eq0 => /rreg_size->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rreg_polyMC_eq0
| |
rreg_div0q r d :
GRing.rreg (lead_coef d) -> size r < size d ->
(q * d + r == 0) = (q == 0) && (r == 0).
Proof.
move=> /mulIr_eq0 reg_d lt_r_d; rewrite addrC.
have [-> | nz_q] := eqVneq q 0; first by rewrite mul0r addr0.
have qd0: lead_coef q * lead_coef d != 0 by rewrite reg_d lead_coef_eq0.
apply/negbTE; rewrite -size_poly_eq0 addrC size_polyDl.
by rewrite size_poly_eq0 -lead_coef_eq0 lead_coef_proper_mul.
apply: leq_trans lt_r_d _; rewrite size_proper_mul //.
move: nz_q; rewrite -size_poly_eq0.
by case: (size q) => [//|] ? _; rewrite addSn /= leq_addl.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rreg_div0
| |
monic_comregp :
p \is monic -> GRing.comm p (lead_coef p)%:P /\ GRing.rreg (lead_coef p).
Proof. by move/monicP->; split; [apply: commr1 | apply: rreg1]. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
monic_comreg
| |
rootp : pred R := fun x => p.[x] == 0.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
root
| |
mem_rootp x : x \in root p = (p.[x] == 0).
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
mem_root
| |
rootEp x : (root p x = (p.[x] == 0)) * ((x \in root p) = (p.[x] == 0)).
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rootE
| |
rootPp x : reflect (p.[x] = 0) (root p x).
Proof. exact: eqP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rootP
| |
rootPtp x : reflect (p.[x] == 0) (root p x).
Proof. exact: idP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rootPt
| |
rootPfp x : reflect ((p.[x] == 0) = false) (~~ root p x).
Proof. exact: negPf. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rootPf
| |
rootCa x : root a%:P x = (a == 0).
Proof. by rewrite rootE hornerC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rootC
| |
root0x : root 0 x.
Proof. by rewrite rootC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
root0
| |
root1x : ~~ root 1 x.
Proof. by rewrite rootC oner_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
root1
| |
rootXx : root 'X x = (x == 0).
Proof. by rewrite rootE hornerX. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
rootX
| |
root_size_gt1a p : p != 0 -> root p a -> 1 < size p.
Proof.
rewrite ltnNge => nz_p; apply: contraL => /size1_polyC Dp.
by rewrite Dp rootC -polyC_eq0 -Dp.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
root_size_gt1
| |
size_add:= size_polyD (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="renamed to `size_polyDl`")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_add
| |
size_addl:= size_polyDl (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="renamed to `size_polyMleq`")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_addl
| |
size_mul_leq:= size_polyMleq (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="renamed to `size_poly_prod_leq`")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_mul_leq
| |
size_prod_leq:= size_poly_prod_leq (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="renamed to `size_poly_exp_leq`")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_prod_leq
| |
size_exp_leq:= size_poly_exp_leq (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pchar_poly instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_exp_leq
| |
char_poly:= pchar_poly (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
char_poly
| |
opp_poly_defp := \poly_(i < size p) - p`_i.
Fact opp_poly_key : unit. Proof. by []. Qed.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
opp_poly_def
| |
opp_poly:= locked_with opp_poly_key opp_poly_def.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
opp_poly
| |
opp_poly_unlockable:= [unlockable fun opp_poly].
Fact coef_opp_poly p i : (opp_poly p)`_i = - p`_i.
Proof.
rewrite unlock coef_poly /=.
by case: leqP => // le_p_i; rewrite nth_default ?oppr0.
Qed.
Fact add_polyN : left_inverse 0%:P opp_poly (@add_poly _).
Proof.
by move=> p; apply/polyP => i; rewrite coefD coef_opp_poly coef0 addNr.
Qed.
HB.instance Definition _ := GRing.Nmodule_isZmodule.Build (polynomial R)
add_polyN.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
opp_poly_unlockable
| |
coefNp i : (- p)`_i = - p`_i.
Proof. exact: coef_opp_poly. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
coefN
| |
coefBp q i : (p - q)`_i = p`_i - q`_i.
Proof. by rewrite coefD coefN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
coefB
| |
coefMNnp n i : (p *- n)`_i = p`_i *- n.
Proof. by rewrite coefN coefMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
coefMNn
| |
polyCN: {morph (@polyC R) : c / - c}.
Proof. exact: raddfN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
polyCN
| |
polyCB: {morph (@polyC R) : a b / a - b}.
Proof. exact: raddfB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
polyCB
| |
size_polyNp : size (- p) = size p.
Proof.
by apply/eqP; rewrite eqn_leq -{3}(opprK p) -[-%R]/opp_poly unlock !size_poly.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_polyN
| |
lead_coefNp : lead_coef (- p) = - lead_coef p.
Proof. by rewrite /lead_coef size_polyN coefN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lead_coefN
| |
size_Msignp n : size ((-1) ^+ n * p) = size p.
Proof.
by rewrite -signr_odd; case: (odd n); rewrite ?mul1r // mulN1r size_polyN.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_Msign
| |
polyseqXsubCa : 'X - a%:P = [:: - a; 1] :> seq R.
Proof. by rewrite -polyCN polyseqXaddC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
polyseqXsubC
| |
size_XsubCa : size ('X - a%:P) = 2.
Proof. by rewrite polyseqXsubC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_XsubC
| |
lead_coefXsubCa : lead_coef ('X - a%:P) = 1.
Proof. by rewrite lead_coefE polyseqXsubC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lead_coefXsubC
| |
polyXsubC_eq0a : ('X - a%:P == 0) = false.
Proof. by rewrite -nil_poly polyseqXsubC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
polyXsubC_eq0
| |
lead_coefXnsubCn c : 0 < n -> lead_coef ('X^n - c%:P) = 1.
Proof. by move=> n_gt0; rewrite -polyCN lead_coefXnaddC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
lead_coefXnsubC
| |
size_XnsubCn c : 0 < n -> size ('X^n - c%:P) = n.+1.
Proof. by move=> *; rewrite -polyCN size_XnaddC. Qed.
#[deprecated(since="mathcomp 2.3.0",note="Use size_XnsubC instead.")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_XnsubC
| |
size_Xn_sub_1n : n > 0 -> size ('X^n - 1 : {poly R}) = n.+1.
Proof. exact/size_XnsubC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finset tuple div ssralg",
"From mathcomp Require Import countalg binomial"
] |
algebra/poly.v
|
size_Xn_sub_1
|
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