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Coq-MathComp

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coef_add_poly p q i : (add_poly p q)`_i = p`_i + q`_i.
Proof. rewrite unlock coef_poly; case: leqP => //. by rewrite geq_max => /andP[le_p_i le_q_i]; rewrite !nth_default ?add0r. Qed.
Fact
coef_add_poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "add0r", "add_poly", "coef_poly", "geq_max", "leqP", "nth_default" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_polyA : associative add_poly.
Proof. by move=> p q r; apply/polyP=> i; rewrite !coef_add_poly addrA. Qed.
Fact
add_polyA
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "add_poly", "addrA", "apply", "coef_add_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_polyC : commutative add_poly.
Proof. by move=> p q; apply/polyP=> i; rewrite !coef_add_poly addrC. Qed.
Fact
add_polyC
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "add_poly", "addrC", "apply", "coef_add_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_poly0 : left_id 0%:P add_poly.
Proof. by move=> p; apply/polyP=> i; rewrite coef_add_poly coefC if_same add0r. Qed.
Fact
add_poly0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "add0r", "add_poly", "apply", "coefC", "coef_add_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyC0 : 0%:P = 0 :> {poly R}.
Proof. by []. Qed.
Lemma
polyC0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "poly" ]
Properties of the zero polynomial
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyseq0 : (0 : {poly R}) = [::] :> seq R.
Proof. by rewrite polyseqC eqxx. Qed.
Lemma
polyseq0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "eqxx", "poly", "polyseqC", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly0 : size (0 : {poly R}) = 0%N.
Proof. by rewrite polyseq0. Qed.
Lemma
size_poly0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "poly", "polyseq0", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef0 i : (0 : {poly R})`_i = 0.
Proof. by rewrite coefC if_same. Qed.
Lemma
coef0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coefC", "poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef0 : lead_coef 0 = 0 :> R.
Proof. exact: lead_coefC. Qed.
Lemma
lead_coef0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "lead_coef", "lead_coefC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly_eq0 p : (size p == 0) = (p == 0).
Proof. by rewrite size_eq0 -polyseq0. Qed.
Lemma
size_poly_eq0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "polyseq0", "size", "size_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly_leq0 p : (size p <= 0) = (p == 0).
Proof. by rewrite leqn0 size_poly_eq0. Qed.
Lemma
size_poly_leq0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "leqn0", "size", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly_leq0P p : reflect (p = 0) (size p <= 0).
Proof. by apply: (iffP idP); rewrite size_poly_leq0; move/eqP. Qed.
Lemma
size_poly_leq0P
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "size", "size_poly_leq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly_gt0 p : (0 < size p) = (p != 0).
Proof. by rewrite lt0n size_poly_eq0. Qed.
Lemma
size_poly_gt0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "lt0n", "size", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt_size_poly_neq0 p n : size p > n -> p != 0.
Proof. by move=> /(leq_ltn_trans _) h; rewrite -size_poly_eq0 lt0n_neq0 ?h. Qed.
Lemma
gt_size_poly_neq0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "leq_ltn_trans", "lt0n_neq0", "size", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_poly p : nilp p = (p == 0).
Proof. exact: size_poly_eq0. Qed.
Lemma
nil_poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "nilp", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly0Vpos p : {p = 0} + {size p > 0}.
Proof. by rewrite lt0n size_poly_eq0; case: eqVneq; [left | right]. Qed.
Lemma
poly0Vpos
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "eqVneq", "lt0n", "size", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polySpred p : p != 0 -> size p = (size p).-1.+1.
Proof. by rewrite -size_poly_eq0 -lt0n => /prednK. Qed.
Lemma
polySpred
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "lt0n", "prednK", "size", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_eq0 p : (lead_coef p == 0) = (p == 0).
Proof. rewrite -nil_poly /lead_coef nth_last. by case: p => [[|x s] /= /negbTE // _]; rewrite eqxx. Qed.
Lemma
lead_coef_eq0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "eqxx", "lead_coef", "nil_poly", "nth_last" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyC_eq0 c : (c%:P == 0) = (c == 0).
Proof. by rewrite -nil_poly polyseqC; case: (c == 0). Qed.
Lemma
polyC_eq0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "nil_poly", "polyseqC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly1P p : reflect (exists2 c, c != 0 & p = c%:P) (size p == 1).
Proof. apply: (iffP eqP) => [pC | [c nz_c ->]]; last by rewrite size_polyC nz_c. have def_p: p = (p`_0)%:P by rewrite -size1_polyC ?pC. by exists p`_0; rewrite // -polyC_eq0 -def_p -size_poly_eq0 pC. Qed.
Lemma
size_poly1P
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "def_p", "last", "polyC_eq0", "size", "size1_polyC", "size_polyC", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_polyC_leq1 c : (size c%:P <= 1)%N.
Proof. by rewrite size_polyC; case: (c == 0). Qed.
Lemma
size_polyC_leq1
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "size", "size_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_sizeP p i : reflect (forall j, i <= j -> p`_j = 0) (size p <= i).
Proof. apply: (iffP idP) => [hp j hij| hp]. by apply: nth_default; apply: leq_trans hij. case: (eqVneq p) (lead_coef_eq0 p) => [->|p0]; first by rewrite size_poly0. rewrite leqNgt; apply/contraFN => hs. by apply/eqP/hp; rewrite -ltnS (ltn_predK hs). Qed.
Lemma
leq_sizeP
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "eqVneq", "lead_coef_eq0", "leqNgt", "leq_trans", "ltnS", "ltn_predK", "nth_default", "p0", "size", "size_poly0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefD p q i : (p + q)`_i = p`_i + q`_i.
Proof. exact: coef_add_poly. Qed.
Lemma
coefD
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coef_add_poly" ]
Size, leading coef, morphism properties of coef
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefMn p n i : (p *+ n)`_i = p`_i *+ n.
Proof. exact: (raddfMn (coefp i)). Qed.
Lemma
coefMn
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coefp", "raddfMn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_sum I (r : seq I) (P : pred I) (F : I -> {poly R}) k : (\sum_(i <- r | P i) F i)`_k = \sum_(i <- r | P i) (F i)`_k.
Proof. exact: (raddf_sum (coefp k)). Qed.
Lemma
coef_sum
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coefp", "poly", "raddf_sum", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyCD : {morph polyC : a b / a + b}.
Proof. by move=> a b; apply/polyP=> [[|i]]; rewrite coefD !coefC ?addr0. Qed.
Lemma
polyCD
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "addr0", "apply", "coefC", "coefD", "polyC", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyCMn n : {morph polyC : c / c *+ n}.
Proof. exact: raddfMn. Qed.
Lemma
polyCMn
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "polyC", "raddfMn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_polyD p q : size (p + q) <= maxn (size p) (size q).
Proof. by rewrite -[+%R]/add_poly unlock; exact: size_poly. Qed.
Lemma
size_polyD
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "add_poly", "maxn", "size", "size_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_polyDl p q : size p > size q -> size (p + q) = size p.
Proof. move=> ltqp; rewrite -[+%R]/add_poly unlock size_poly_eq (maxn_idPl (ltnW _))//. by rewrite addrC nth_default ?simp ?nth_last //; case: p ltqp => [[]]. Qed.
Lemma
size_polyDl
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "add_poly", "addrC", "ltnW", "maxn_idPl", "nth_default", "nth_last", "simp", "size", "size_poly_eq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_sum I (r : seq I) (P : pred I) (F : I -> {poly R}) : size (\sum_(i <- r | P i) F i) <= \max_(i <- r | P i) size (F i).
Proof. elim/big_rec2: _ => [|i p q _ IHp]; first by rewrite size_poly0. by rewrite -(maxn_idPr IHp) maxnA leq_max size_polyD. Qed.
Lemma
size_sum
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "big_rec2", "leq_max", "maxnA", "maxn_idPr", "poly", "seq", "size", "size_poly0", "size_polyD" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefDl p q : size p > size q -> lead_coef (p + q) = lead_coef p.
Proof. move=> ltqp; rewrite /lead_coef coefD size_polyDl //. by rewrite addrC nth_default ?simp // -ltnS (ltn_predK ltqp). Qed.
Lemma
lead_coefDl
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "addrC", "coefD", "lead_coef", "ltnS", "ltn_predK", "nth_default", "simp", "size", "size_polyDl" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefDr p q : size q > size p -> lead_coef (p + q) = lead_coef q.
Proof. by move/lead_coefDl<-; rewrite addrC. Qed.
Lemma
lead_coefDr
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "addrC", "lead_coef", "lead_coefDl", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_poly_def p q
:= \poly_(i < (size p + size q).-1) (\sum_(j < i.+1) p`_j * q`_(i - j)).
Definition
mul_poly_def
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "size" ]
Polynomial semiring structure.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_poly_key : unit.
Proof. by []. Qed.
Fact
mul_poly_key
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_poly
:= locked_with mul_poly_key mul_poly_def.
Definition
mul_poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mul_poly_def", "mul_poly_key" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_poly_unlockable
:= [unlockable fun mul_poly].
Canonical
mul_poly_unlockable
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mul_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_mul_poly p q i : (mul_poly p q)`_i = \sum_(j < i.+1) p`_j * q`_(i - j).
Proof. rewrite unlock coef_poly ltn_predRL; case: leqP => // le_pq_i1. rewrite big1 // => j _; have [lq_p_j|lt_j_p] := leqP (size p) j. by rewrite nth_default ?mul0r. rewrite [q`__]nth_default ?mulr0 // leq_subRL -ltnS //. by rewrite (leq_trans _ le_pq_i1) // ltn_add2r. Qed.
Fact
coef_mul_poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "big1", "coef_poly", "leqP", "leq_subRL", "leq_trans", "ltnS", "ltn_add2r", "ltn_predRL", "mul0r", "mul_poly", "mulr0", "nth_default", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_mul_poly_rev p q i : (mul_poly p q)`_i = \sum_(j < i.+1) p`_(i - j) * q`_j.
Proof. rewrite coef_mul_poly (reindex_inj rev_ord_inj) /=. by apply: eq_bigr => j _; rewrite (sub_ordK j). Qed.
Fact
coef_mul_poly_rev
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "coef_mul_poly", "eq_bigr", "mul_poly", "reindex_inj", "rev_ord_inj", "sub_ordK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_polyA : associative mul_poly.
Proof. move=> p q r; apply/polyP=> i; rewrite coef_mul_poly coef_mul_poly_rev. pose coef3 j k := p`_j * (q`_(i - j - k) * r`_k). transitivity (\sum_(j < i.+1) \sum_(k < i.+1 | k <= i - j) coef3 j k). apply: eq_bigr => /= j _; rewrite coef_mul_poly_rev big_distrr /=. by rewrite (big_ord_narrow_leq (leq_subr _ _)). r...
Fact
mul_polyA
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "addnC", "apply", "big_distrl", "big_distrr", "big_ord_narrow_leq", "coef_mul_poly", "coef_mul_poly_rev", "eq_bigl", "eq_bigr", "exchange_big_dep", "leq_ord", "leq_subr", "ltnS", "mul_poly", "mulrA", "polyP", "subSn", "subnDA", "subn_gt0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_1poly : left_id 1%:P mul_poly.
Proof. move=> p; apply/polyP => i; rewrite coef_mul_poly big_ord_recl subn0. by rewrite big1 => [j _|]; rewrite coefC !simp. Qed.
Fact
mul_1poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "big1", "big_ord_recl", "coefC", "coef_mul_poly", "mul_poly", "polyP", "simp", "subn0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_poly1 : right_id 1%:P mul_poly.
Proof. move=> p; apply/polyP => i; rewrite coef_mul_poly_rev big_ord_recl subn0. by rewrite big1 => [j _|]; rewrite coefC !simp. Qed.
Fact
mul_poly1
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "big1", "big_ord_recl", "coefC", "coef_mul_poly_rev", "mul_poly", "polyP", "simp", "subn0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_polyDl : left_distributive mul_poly +%R.
Proof. move=> p q r; apply/polyP=> i; rewrite coefD !coef_mul_poly -big_split. by apply: eq_bigr => j _; rewrite coefD mulrDl. Qed.
Fact
mul_polyDl
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "big_split", "coefD", "coef_mul_poly", "eq_bigr", "mul_poly", "mulrDl", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_polyDr : right_distributive mul_poly +%R.
Proof. move=> p q r; apply/polyP=> i; rewrite coefD !coef_mul_poly -big_split. by apply: eq_bigr => j _; rewrite coefD mulrDr. Qed.
Fact
mul_polyDr
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "big_split", "coefD", "coef_mul_poly", "eq_bigr", "mul_poly", "mulrDr", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_0poly : left_zero 0%:P mul_poly.
Proof. move=> p; apply/polyP => i; rewrite coef_mul_poly big_ord_recl subn0. by rewrite big1 => [j _|]; rewrite coefC !simp // coefC; case: ifP. Qed.
Fact
mul_0poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "big1", "big_ord_recl", "coefC", "coef_mul_poly", "mul_poly", "polyP", "simp", "subn0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_poly0 : right_zero 0%:P mul_poly.
Proof. move=> p; apply/polyP => i; rewrite coef_mul_poly_rev big_ord_recl subn0. by rewrite big1 => [j _|]; rewrite coefC !simp // coefC; case: ifP. Qed.
Fact
mul_poly0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "big1", "big_ord_recl", "coefC", "coef_mul_poly_rev", "mul_poly", "polyP", "simp", "subn0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly1_neq0 : 1%:P != 0 :> {poly R}.
Proof. by rewrite polyC_eq0 oner_neq0. Qed.
Fact
poly1_neq0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "oner_neq0", "poly", "polyC_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyC1 : 1%:P = 1 :> {poly R}.
Proof. by []. Qed.
Lemma
polyC1
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyseq1 : (1 : {poly R}) = [:: 1] :> seq R.
Proof. by rewrite polyseqC oner_neq0. Qed.
Lemma
polyseq1
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "oner_neq0", "poly", "polyseqC", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly1 : size (1 : {poly R}) = 1.
Proof. by rewrite polyseq1. Qed.
Lemma
size_poly1
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "poly", "polyseq1", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef1 i : (1 : {poly R})`_i = (i == 0)%:R.
Proof. by case: i => [|i]; rewrite polyseq1 /= ?nth_nil. Qed.
Lemma
coef1
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "nth_nil", "poly", "polyseq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef1 : lead_coef 1 = 1 :> R.
Proof. exact: lead_coefC. Qed.
Lemma
lead_coef1
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "lead_coef", "lead_coefC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefM p q i : (p * q)`_i = \sum_(j < i.+1) p`_j * q`_(i - j).
Proof. exact: coef_mul_poly. Qed.
Lemma
coefM
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coef_mul_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefMr p q i : (p * q)`_i = \sum_(j < i.+1) p`_(i - j) * q`_j.
Proof. exact: coef_mul_poly_rev. Qed.
Lemma
coefMr
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coef_mul_poly_rev" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef0M p q : (p * q)`_0 = p`_0 * q`_0.
Proof. by rewrite coefM big_ord1. Qed.
Lemma
coef0M
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "big_ord1", "coefM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefp0_is_monoid_morphism : monoid_morphism (coefp 0).
Proof. by split; [exact: polyCK | exact: coef0M]. Qed.
Fact
coefp0_is_monoid_morphism
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coef0M", "coefp", "monoid_morphism", "polyCK", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefp0_multiplicative
:= (fun g => (g.2, g.1)) coefp0_is_monoid_morphism.
Definition
coefp0_multiplicative
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coefp0_is_monoid_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef0_prod I rI (F : I -> {poly R}) P : (\prod_(i <- rI| P i) F i)`_0 = \prod_(i <- rI | P i) (F i)`_0.
Proof. exact: (rmorph_prod (coefp 0)). Qed.
Lemma
coef0_prod
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coefp", "poly", "rmorph_prod" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_polyMleq p q : size (p * q) <= (size p + size q).-1.
Proof. by rewrite -[*%R]/mul_poly unlock size_poly. Qed.
Lemma
size_polyMleq
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mul_poly", "size", "size_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_lead_coef p q : lead_coef p * lead_coef q = (p * q)`_(size p + size q).-2.
Proof. pose dp := (size p).-1; pose dq := (size q).-1. have [-> | nz_p] := eqVneq p 0; first by rewrite lead_coef0 !mul0r coef0. have [-> | nz_q] := eqVneq q 0; first by rewrite lead_coef0 !mulr0 coef0. have ->: (size p + size q).-2 = (dp + dq)%N. by do 2!rewrite polySpred // addSn addnC. have lt_p_pq: dp < (dp + dq)...
Lemma
mul_lead_coef
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "addKn", "addSn", "addnBA", "addnC", "addnS", "big1", "bigD1", "coef0", "coefM", "eqVneq", "last", "lead_coef", "lead_coef0", "leq_addr", "ltnS", "mul0r", "mulr0", "neq_ltn", "nth_default", "nz_p", "polySpred", "simp", "size", "subSS", "val_eqE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_proper_mul p q : lead_coef p * lead_coef q != 0 -> size (p * q) = (size p + size q).-1.
Proof. apply: contraNeq; rewrite mul_lead_coef eqn_leq size_polyMleq -ltnNge => lt_pq. by rewrite nth_default // -subn1 -(leq_add2l 1) -leq_subLR leq_sub2r. Qed.
Lemma
size_proper_mul
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "contraNeq", "eqn_leq", "lead_coef", "leq_add2l", "leq_sub2r", "leq_subLR", "ltnNge", "mul_lead_coef", "nth_default", "size", "size_polyMleq", "subn1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_proper_mul p q : let c := lead_coef p * lead_coef q in c != 0 -> lead_coef (p * q) = c.
Proof. by move=> /= nz_c; rewrite mul_lead_coef -size_proper_mul. Qed.
Lemma
lead_coef_proper_mul
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "lead_coef", "mul_lead_coef", "size_proper_mul" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly_prod_leq (I : finType) (P : pred I) (F : I -> {poly R}) : size (\prod_(i | P i) F i) <= (\sum_(i | P i) size (F i)).+1 - #|P|.
Proof. rewrite -sum1_card. elim/big_rec3: _ => [|i n m p _ IHp]; first by rewrite size_poly1. have [-> | nz_p] := eqVneq p 0; first by rewrite mulr0 size_poly0. rewrite (leq_trans (size_polyMleq _ _)) // subnS -!subn1 leq_sub2r //. rewrite -addnS -addnBA ?leq_add2l // ltnW // -subn_gt0 (leq_trans _ IHp) //. by rewrite ...
Lemma
size_poly_prod_leq
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "addnBA", "addnS", "big_rec3", "eqVneq", "leq_add2l", "leq_sub2r", "leq_trans", "ltnW", "mulr0", "nz_p", "poly", "polySpred", "size", "size_poly0", "size_poly1", "size_polyMleq", "subn1", "subnS", "subn_gt0", "sum1_card" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefCM c p i : (c%:P * p)`_i = c * p`_i.
Proof. by rewrite coefM big_ord_recl subn0 big1 => [j _|]; rewrite coefC !simp. Qed.
Lemma
coefCM
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "big1", "big_ord_recl", "coefC", "coefM", "simp", "subn0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefMC c p i : (p * c%:P)`_i = p`_i * c.
Proof. by rewrite coefMr big_ord_recl subn0 big1 => [j _|]; rewrite coefC !simp. Qed.
Lemma
coefMC
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "big1", "big_ord_recl", "coefC", "coefMr", "simp", "subn0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyCM : {morph polyC : a b / a * b}.
Proof. by move=> a b; apply/polyP=> [[|i]]; rewrite coefCM !coefC ?simp. Qed.
Lemma
polyCM
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "coefC", "coefCM", "polyC", "polyP", "simp" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_poly_exp_leq p n : size (p ^+ n) <= ((size p).-1 * n).+1.
Proof. elim: n => [|n IHn]; first by rewrite size_poly1. have [-> | nzp] := poly0Vpos p; first by rewrite exprS mul0r size_poly0. rewrite exprS (leq_trans (size_polyMleq _ _)) //. by rewrite -{1}(prednK nzp) mulnS -addnS leq_add2l. Qed.
Lemma
size_poly_exp_leq
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "addnS", "exprS", "leq_add2l", "leq_trans", "mul0r", "mulnS", "poly0Vpos", "prednK", "size", "size_poly0", "size_poly1", "size_polyMleq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyC_is_monoid_morphism : monoid_morphism polyC.
Proof. by split; last apply: polyCM. Qed.
Fact
polyC_is_monoid_morphism
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "last", "monoid_morphism", "polyC", "polyCM", "split" ]
Polynomial ring structure.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyC_multiplicative
:= (fun g => (g.2, g.1)) polyC_is_monoid_morphism.
Definition
polyC_multiplicative
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "polyC_is_monoid_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyC_exp n : {morph polyC : c / c ^+ n}.
Proof. exact: rmorphXn. Qed.
Lemma
polyC_exp
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "polyC", "rmorphXn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyC_natr n : n%:R%:P = n%:R :> {poly R}.
Proof. exact: rmorph_nat. Qed.
Lemma
polyC_natr
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "poly", "rmorph_nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pchar_poly : [pchar {poly R}] =i [pchar R].
Proof. move=> p; rewrite !inE; congr (_ && _). apply/eqP/eqP=> [/(congr1 val) /=|]; last by rewrite -polyC_natr => ->. by rewrite polyseq0 -polyC_natr polyseqC; case: eqP. Qed.
Lemma
pchar_poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "inE", "last", "pchar", "poly", "polyC_natr", "polyseq0", "polyseqC", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_poly_def a (p : {poly R})
:= \poly_(i < size p) (a * p`_i).
Definition
scale_poly_def
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "poly", "size" ]
Algebra structure of polynomials.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_poly_key : unit.
Proof. by []. Qed.
Fact
scale_poly_key
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_poly
:= locked_with scale_poly_key scale_poly_def.
Definition
scale_poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "scale_poly_def", "scale_poly_key" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_poly_unlockable
:= [unlockable fun scale_poly].
Canonical
scale_poly_unlockable
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "scale_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_polyE a p : scale_poly a p = a%:P * p.
Proof. apply/polyP=> n; rewrite unlock coef_poly coefCM. by case: leqP => // le_p_n; rewrite nth_default ?mulr0. Qed.
Fact
scale_polyE
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "coefCM", "coef_poly", "leqP", "mulr0", "nth_default", "polyP", "scale_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_polyA a b p : scale_poly a (scale_poly b p) = scale_poly (a * b) p.
Proof. by rewrite !scale_polyE mulrA polyCM. Qed.
Fact
scale_polyA
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mulrA", "polyCM", "scale_poly", "scale_polyE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_0poly p : scale_poly 0 p = 0.
Proof. by rewrite scale_polyE mul0r. Qed.
Fact
scale_0poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mul0r", "scale_poly", "scale_polyE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_1poly : left_id 1 scale_poly.
Proof. by move=> p; rewrite scale_polyE mul1r. Qed.
Fact
scale_1poly
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mul1r", "scale_poly", "scale_polyE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_polyDr a : {morph scale_poly a : p q / p + q}.
Proof. by move=> p q; rewrite !scale_polyE mulrDr. Qed.
Fact
scale_polyDr
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mulrDr", "scale_poly", "scale_polyE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_polyDl p : {morph scale_poly^~ p : a b / a + b}.
Proof. by move=> a b /=; rewrite !scale_polyE raddfD mulrDl. Qed.
Fact
scale_polyDl
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mulrDl", "raddfD", "scale_poly", "scale_polyE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_polyAl a p q : scale_poly a (p * q) = scale_poly a p * q.
Proof. by rewrite !scale_polyE mulrA. Qed.
Fact
scale_polyAl
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mulrA", "scale_poly", "scale_polyE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_polyC a p : a%:P * p = a *: p.
Proof. by rewrite -scale_polyE. Qed.
Lemma
mul_polyC
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "scale_polyE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_polyC a b : a *: b%:P = (a * b)%:P.
Proof. by rewrite -mul_polyC polyCM. Qed.
Lemma
scale_polyC
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mul_polyC", "polyCM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
alg_polyC a : a%:A = a%:P :> {poly R}.
Proof. by rewrite -mul_polyC mulr1. Qed.
Lemma
alg_polyC
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "mul_polyC", "mulr1", "poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefZ a p i : (a *: p)`_i = a * p`_i.
Proof. rewrite -[*:%R]/scale_poly unlock coef_poly. by case: leqP => // le_p_n; rewrite nth_default ?mulr0. Qed.
Lemma
coefZ
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coef_poly", "leqP", "mulr0", "nth_default", "scale_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_scale_leq a p : size (a *: p) <= size p.
Proof. by rewrite -[*:%R]/scale_poly unlock size_poly. Qed.
Lemma
size_scale_leq
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "scale_poly", "size", "size_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyX_def
:= Poly [:: 0; 1].
Definition
polyX_def
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "Poly" ]
The indeterminate, at last!
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyX_key : unit.
Proof. by []. Qed.
Fact
polyX_key
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyX : {poly R}
:= locked_with polyX_key polyX_def.
Definition
polyX
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "poly", "polyX_def", "polyX_key" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyX_unlockable
:= [unlockable of polyX].
Canonical
polyX_unlockable
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "polyX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'X"
:= polyX.
Notation
'X
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "polyX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyseqX : 'X = [:: 0; 1] :> seq R.
Proof. by rewrite unlock !polyseq_cons nil_poly eqxx /= polyseq1. Qed.
Lemma
polyseqX
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "eqxx", "nil_poly", "polyseq1", "polyseq_cons", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_polyX : size 'X = 2.
Proof. by rewrite polyseqX. Qed.
Lemma
size_polyX
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "polyseqX", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyX_eq0 : ('X == 0) = false.
Proof. by rewrite -size_poly_eq0 size_polyX. Qed.
Lemma
polyX_eq0
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "size_polyX", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefX i : 'X`_i = (i == 1)%:R.
Proof. by case: i => [|[|i]]; rewrite polyseqX //= nth_nil. Qed.
Lemma
coefX
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "nth_nil", "polyseqX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefX : lead_coef 'X = 1.
Proof. by rewrite /lead_coef polyseqX. Qed.
Lemma
lead_coefX
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "lead_coef", "polyseqX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_polyX p : GRing.comm p 'X.
Proof. apply/polyP=> i; rewrite coefMr coefM. by apply: eq_bigr => j _; rewrite coefX commr_nat. Qed.
Lemma
commr_polyX
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "apply", "coefM", "coefMr", "coefX", "comm", "commr_nat", "eq_bigr", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefMX p i : (p * 'X)`_i = (if (i == 0)%N then 0 else p`_i.-1).
Proof. rewrite coefMr big_ord_recl coefX ?simp. case: i => [|i]; rewrite ?big_ord0 //= big_ord_recl polyseqX subn1 /=. by rewrite big1 ?simp // => j _; rewrite nth_nil !simp. Qed.
Lemma
coefMX
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "big1", "big_ord0", "big_ord_recl", "coefMr", "coefX", "nth_nil", "polyseqX", "simp", "subn1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefXM p i : ('X * p)`_i = (if (i == 0)%N then 0 else p`_i.-1).
Proof. by rewrite -commr_polyX coefMX. Qed.
Lemma
coefXM
algebra
algebra/poly.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]
[ "coefMX", "commr_polyX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d