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Coq-MathComp

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map_prod_XsubC I (rI : seq I) P F : (\prod_(i <- rI | P i) ('X - (F i)%:P))^f = \prod_(i <- rI | P i) ('X - (f (F i))%:P).
Proof. by rewrite rmorph_prod//; apply/eq_bigr => x /=; rewrite map_polyXsubC. Qed.
Lemma
map_prod_XsubC
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", "eq_bigr", "map_polyXsubC", "rmorph_prod", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_map_poly (ar : seq aR) P : \prod_(x <- map f ar | P x) ('X - x%:P) = (\prod_(x <- ar | P (f x)) ('X - x%:P))^f.
Proof. by rewrite big_map map_prod_XsubC. Qed.
Lemma
prod_map_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" ]
[ "big_map", "map", "map_prod_XsubC", "seq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_unity_root n z : n.-unity_root z -> n.-unity_root (f z).
Proof. move/(rmorph_root f); rewrite rootE rmorphB hornerD hornerN. by rewrite /= map_polyXn rmorph1 hornerC hornerXn subr_eq0 unity_rootE. Qed.
Lemma
rmorph_unity_root
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" ]
[ "hornerC", "hornerD", "hornerN", "hornerXn", "map_polyXn", "rmorph1", "rmorphB", "rmorph_root", "rootE", "subr_eq0", "unity_rootE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_is_linear : linear_for (f \; *%R) (horner_morph cfu).
Proof. exact: linearP. Qed.
Fact
horner_is_linear
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" ]
[ "cfu", "horner_morph", "linearP", "linear_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
in_alg_comm : commr_rmorph (in_alg A) a.
Proof. move=> r /=; by rewrite /GRing.comm comm_alg. Qed.
Lemma
in_alg_comm
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" ]
[ "comm", "comm_alg", "commr_rmorph", "in_alg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_alg
:= horner_morph in_alg_comm.
Definition
horner_alg
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" ]
[ "horner_morph", "in_alg_comm" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_algC c : horner_alg c%:P = c%:A.
Proof. exact: horner_morphC. Qed.
Lemma
horner_algC
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" ]
[ "horner_alg", "horner_morphC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_algX : horner_alg 'X = a.
Proof. exact: horner_morphX. Qed.
Lemma
horner_algX
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" ]
[ "horner_alg", "horner_morphX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_alg_initial : pf =1 horner_alg (pf 'X).
Proof. apply: poly_ind => [|p a IHp]; first by rewrite !rmorph0. rewrite !rmorphD !rmorphM /= -{}IHp horner_algC ?horner_algX. by rewrite -alg_polyC rmorph_alg. Qed.
Lemma
poly_alg_initial
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" ]
[ "alg_polyC", "apply", "horner_alg", "horner_algC", "horner_algX", "poly_ind", "rmorph0", "rmorphD", "rmorphM", "rmorph_alg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mapf_root (F : fieldType) (R : nzRingType) (f : {rmorphism F -> R}) (p : {poly F}) (x : F) : root (map_poly f p) (f x) = root p x.
Proof. by rewrite !rootE horner_map fmorph_eq0. Qed.
Lemma
mapf_root
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" ]
[ "fmorph_eq0", "horner_map", "map_poly", "poly", "root", "rootE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_morphX_comm : commr_rmorph (pf \o polyC) (pf 'X).
Proof. by move=> a; rewrite /GRing.comm /= -!rmorphM // commr_polyX. Qed.
Lemma
poly_morphX_comm
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" ]
[ "comm", "commr_polyX", "commr_rmorph", "polyC", "rmorphM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_initial : pf =1 horner_morph poly_morphX_comm.
Proof. apply: poly_ind => [|p a IHp]; first by rewrite !rmorph0. by rewrite !rmorphD !rmorphM /= -{}IHp horner_morphC ?horner_morphX. Qed.
Lemma
poly_initial
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", "horner_morph", "horner_morphC", "horner_morphX", "poly_ind", "poly_morphX_comm", "rmorph0", "rmorphD", "rmorphM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p ^:P"
:= (map_poly polyC p) : ring_scope.
Notation
p ^:P
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" ]
[ "map_poly", "polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_poly q p
:= p^:P.[q].
Definition
comp_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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p \Po q"
:= (comp_poly q p) : ring_scope.
Notation
p \Po q
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" ]
[ "comp_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_map_polyC p : size p^:P = size p.
Proof. exact/(size_map_inj_poly polyC_inj). Qed.
Lemma
size_map_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" ]
[ "polyC_inj", "size", "size_map_inj_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyC_eq0 p : (p^:P == 0) = (p == 0).
Proof. by rewrite -!size_poly_eq0 size_map_polyC. Qed.
Lemma
map_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" ]
[ "size_map_polyC", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_polyC p x : root p^:P x%:P = root p x.
Proof. by rewrite rootE horner_map polyC_eq0. Qed.
Lemma
root_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" ]
[ "horner_map", "polyC_eq0", "root", "rootE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyE p q : p \Po q = \sum_(i < size p) p`_i *: q^+i.
Proof. by rewrite [p \Po q]horner_poly; apply: eq_bigr => i _; rewrite mul_polyC. Qed.
Lemma
comp_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", "eq_bigr", "horner_poly", "mul_polyC", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_comp_poly p q n : (p \Po q)`_n = \sum_(i < size p) p`_i * (q ^+ i)`_n.
Proof. by rewrite comp_polyE coef_sum; apply: eq_bigr => i; rewrite coefZ. Qed.
Lemma
coef_comp_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", "coefZ", "coef_sum", "comp_polyE", "eq_bigr", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_comp (ringS : semiringClosed R) : {in polyOver ringS &, forall p q, p \Po q \in polyOver ringS}.
Proof. move=> p q /polyOverP Sp Sq; rewrite comp_polyE rpred_sum // => i _. by rewrite polyOverZ ?rpredX. Qed.
Lemma
polyOver_comp
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" ]
[ "comp_polyE", "polyOver", "polyOverP", "polyOverZ", "rpredX", "rpred_sum" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyCr p c : p \Po c%:P = p.[c]%:P.
Proof. exact: horner_map. Qed.
Lemma
comp_polyCr
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" ]
[ "horner_map" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_poly0r p : p \Po 0 = (p`_0)%:P.
Proof. by rewrite comp_polyCr horner_coef0. Qed.
Lemma
comp_poly0r
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" ]
[ "comp_polyCr", "horner_coef0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyC c p : c%:P \Po p = c%:P.
Proof. by rewrite /(_ \Po p) map_polyC hornerC. Qed.
Lemma
comp_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" ]
[ "hornerC", "map_polyC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_poly_is_semilinear p : semilinear (comp_poly p).
Proof. split=> [a q|q r]; last by rewrite /comp_poly linearD /= hornerD. by rewrite /comp_poly linearZ /= hornerZ mul_polyC. Qed.
Fact
comp_poly_is_semilinear
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" ]
[ "comp_poly", "hornerD", "hornerZ", "last", "linearD", "linearZ", "mul_polyC", "semilinear", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_poly0 p : 0 \Po p = 0.
Proof. exact: raddf0. Qed.
Lemma
comp_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" ]
[ "raddf0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyD p q r : (p + q) \Po r = (p \Po r) + (q \Po r).
Proof. exact: raddfD. Qed.
Lemma
comp_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" ]
[ "raddfD" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyZ c p q : (c *: p) \Po q = c *: (p \Po q).
Proof. exact: linearZZ. Qed.
Lemma
comp_polyZ
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" ]
[ "linearZZ" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyXr p : p \Po 'X = p.
Proof. by rewrite -{2}/(idfun p) poly_initial. Qed.
Lemma
comp_polyXr
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_initial" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyX p : 'X \Po p = p.
Proof. by rewrite /(_ \Po p) map_polyX hornerX. Qed.
Lemma
comp_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" ]
[ "hornerX", "map_polyX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_poly_MXaddC c p q : (p * 'X + c%:P) \Po q = (p \Po q) * q + c%:P.
Proof. by rewrite /(_ \Po q) rmorphD rmorphM /= map_polyX map_polyC hornerMXaddC. Qed.
Lemma
comp_poly_MXaddC
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" ]
[ "hornerMXaddC", "map_polyC", "map_polyX", "rmorphD", "rmorphM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_comp_poly_leq p q : size (p \Po q) <= ((size p).-1 * (size q).-1).+1.
Proof. rewrite comp_polyE (leq_trans (size_sum _ _ _)) //; apply/bigmax_leqP => i _. rewrite (leq_trans (size_scale_leq _ _))//. rewrite (leq_trans (size_poly_exp_leq _ _))//. by rewrite ltnS mulnC leq_mul // -{2}(subnKC (valP i)) leq_addr. Qed.
Lemma
size_comp_poly_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" ]
[ "apply", "bigmax_leqP", "comp_polyE", "leq_addr", "leq_mul", "leq_trans", "ltnS", "mulnC", "size", "size_poly_exp_leq", "size_scale_leq", "size_sum", "subnKC", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_Xn_poly p n : 'X^n \Po p = p ^+ n.
Proof. by rewrite /(_ \Po p) map_polyXn hornerXn. Qed.
Lemma
comp_Xn_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" ]
[ "hornerXn", "map_polyXn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_comp_poly_Xn p n i : 0 < n -> (p \Po 'X^n)`_i = if n %| i then p`_(i %/ n) else 0.
Proof. move=> n_gt0; rewrite comp_polyE; under eq_bigr do rewrite -exprM mulnC. rewrite coef_sumMXn/=; case: dvdnP => [[j ->]|nD]; last first. by rewrite big1// => j /eqP ?; case: nD; exists j. under eq_bigl do rewrite eqn_mul2r gtn_eqF//. by rewrite big_ord1_eq if_nth ?leqVgt ?mulnK. Qed.
Lemma
coef_comp_poly_Xn
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_ord1_eq", "coef_sumMXn", "comp_polyE", "dvdnP", "eq_bigl", "eq_bigr", "eqn_mul2r", "exprM", "gtn_eqF", "if_nth", "last", "leqVgt", "mulnC", "mulnK", "n_gt0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_poly_Xn p n : 0 < n -> p \Po 'X^n = \poly_(i < size p * n) if n %| i then p`_(i %/ n) else 0.
Proof. move=> n_gt0; apply/polyP => i; rewrite coef_comp_poly_Xn // coef_poly. case: dvdnP => [[k ->]|]; last by rewrite if_same. by rewrite mulnK // ltn_mul2r n_gt0 if_nth ?leqVgt. Qed.
Lemma
comp_poly_Xn
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_comp_poly_Xn", "coef_poly", "dvdnP", "if_nth", "last", "leqVgt", "ltn_mul2r", "mulnK", "n_gt0", "polyP", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_comp_poly (aR rR : nzRingType) (f : {rmorphism aR -> rR}) p q : map_poly f (p \Po q) = map_poly f p \Po map_poly f q.
Proof. elim/poly_ind: p => [|p a IHp]; first by rewrite !raddf0. rewrite comp_poly_MXaddC !rmorphD !rmorphM /= !map_polyC map_polyX. by rewrite comp_poly_MXaddC -IHp. Qed.
Lemma
map_comp_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" ]
[ "comp_poly_MXaddC", "map_poly", "map_polyC", "map_polyX", "poly_ind", "raddf0", "rmorphD", "rmorphM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
n_gt0
:= prim_order_gt0 prim_z.
Let
n_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" ]
[ "prim_order_gt0", "prim_z" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_root_pcharF p : (p %| n)%N -> (p \in [pchar R]) = false.
Proof. move=> pn; apply: contraTF isT => pchar_p; have p_prime := pcharf_prime pchar_p. have /dvdnP[[|k] n_eq_kp] := pn; first by rewrite n_eq_kp in (n_gt0). have /eqP := prim_expr_order prim_z; rewrite n_eq_kp exprM. rewrite -pFrobenius_autE -(pFrobenius_aut1 pchar_p) -subr_eq0 -rmorphB/=. rewrite pFrobenius_autE expf...
Lemma
prim_root_pcharF
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", "divnn", "dvdn1", "dvdnP", "dvdn_divRL", "expf_eq0", "exprM", "ltngtP", "mulnC", "n_gt0", "pFrobenius_aut1", "pFrobenius_autE", "pchar", "pcharf_prime", "prim_expr_order", "prim_order_dvd", "prim_z", "prime_gt0", "prime_gt1", "rmorphB", "subr_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pchar_prim_root : [pchar R]^'.-nat n.
Proof. by apply/pnatP=> // p pp pn; rewrite inE/= prim_root_pcharF. Qed.
Lemma
pchar_prim_root
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", "nat", "pchar", "pnatP", "prim_root_pcharF" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_root_pi_eq0 m : \pi(n).-nat m -> m%:R != 0 :> R.
Proof. rewrite natf_neq0_pchar; apply: sub_in_pnat => p _. exact: pnatPpi pchar_prim_root. Qed.
Lemma
prim_root_pi_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" ]
[ "apply", "nat", "natf_neq0_pchar", "pchar_prim_root", "pi", "pnatPpi", "sub_in_pnat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_root_dvd_eq0 m : (m %| n)%N -> m%:R != 0 :> R.
Proof. case: m => [|m mn]; first by rewrite dvd0n gtn_eqF. by rewrite prim_root_pi_eq0 ?(sub_in_pnat (in1W (pi_of_dvd mn _))) ?pnat_pi. Qed.
Lemma
prim_root_dvd_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" ]
[ "dvd0n", "gtn_eqF", "pi_of_dvd", "pnat_pi", "prim_root_pi_eq0", "sub_in_pnat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_root_natf_neq0 : n%:R != 0 :> R.
Proof. by rewrite prim_root_dvd_eq0. Qed.
Lemma
prim_root_natf_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" ]
[ "prim_root_dvd_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_root_charF
:= prim_root_pcharF (only parsing).
Notation
prim_root_charF
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" ]
[ "prim_root_pcharF" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_prim_root
:= pchar_prim_root (only parsing).
Notation
char_prim_root
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" ]
[ "pchar_prim_root" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_poly_is_linear p : linear (comp_poly p).
Proof. exact: linearP. Qed.
Fact
comp_poly_is_linear
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" ]
[ "comp_poly", "linear", "linearP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyB p q r : (p - q) \Po r = (p \Po r) - (q \Po r).
Proof. exact: raddfB. Qed.
Lemma
comp_polyB
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" ]
[ "raddfB" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_polyXaddC_K p z : (p \Po ('X + z%:P)) \Po ('X - z%:P) = p.
Proof. have addzK: ('X + z%:P) \Po ('X - z%:P) = 'X. by rewrite raddfD /= comp_polyC comp_polyX subrK. elim/poly_ind: p => [|p c IHp]; first by rewrite !comp_poly0. rewrite comp_poly_MXaddC linearD /= comp_polyC {1}/comp_poly rmorphM /=. by rewrite hornerM_comm /comm_poly -!/(_ \Po _) ?IHp ?addzK ?commr_polyX. Qed.
Lemma
comp_polyXaddC_K
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" ]
[ "comm_poly", "commr_polyX", "comp_poly", "comp_poly0", "comp_polyC", "comp_polyX", "comp_poly_MXaddC", "hornerM_comm", "linearD", "poly_ind", "raddfD", "rmorphM", "subrK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
even_poly p : {poly R}
:= \poly_(i < uphalf (size p)) p`_i.*2.
Definition
even_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" ]
[ "poly", "size" ]
Even part of a polynomial
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_even_poly p : size (even_poly p) <= uphalf (size p).
Proof. exact: size_poly. Qed.
Lemma
size_even_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" ]
[ "even_poly", "size", "size_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_even_poly p i : (even_poly p)`_i = p`_i.*2.
Proof. by rewrite coef_poly gtn_uphalf_double if_nth ?leqVgt. Qed.
Lemma
coef_even_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" ]
[ "coef_poly", "even_poly", "gtn_uphalf_double", "if_nth", "leqVgt" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
even_polyE s p : size p <= s.*2 -> even_poly p = \poly_(i < s) p`_i.*2.
Proof. move=> pLs2; apply/polyP => i; rewrite coef_even_poly !coef_poly if_nth //. by case: ltnP => //= ?; rewrite (leq_trans pLs2) ?leq_double. Qed.
Lemma
even_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", "coef_even_poly", "coef_poly", "even_poly", "if_nth", "leq_double", "leq_trans", "ltnP", "polyP", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_even_poly_eq p : odd (size p) -> size (even_poly p) = uphalf (size p).
Proof. move=> p_even; rewrite size_poly_eq// double_pred odd_uphalfK//=. by rewrite lead_coef_eq0 -size_poly_eq0; case: size p_even. Qed.
Lemma
size_even_poly_eq
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" ]
[ "double_pred", "even_poly", "lead_coef_eq0", "odd", "odd_uphalfK", "size", "size_poly_eq", "size_poly_eq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
even_polyD p q : even_poly (p + q) = even_poly p + even_poly q.
Proof. by apply/polyP => i; rewrite !(coef_even_poly, coefD). Qed.
Lemma
even_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" ]
[ "apply", "coefD", "coef_even_poly", "even_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
even_polyZ k p : even_poly (k *: p) = k *: even_poly p.
Proof. by apply/polyP => i; rewrite !(coefZ, coef_even_poly). Qed.
Lemma
even_polyZ
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", "coefZ", "coef_even_poly", "even_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
even_polyC (c : R) : even_poly c%:P = c%:P.
Proof. by apply/polyP => i; rewrite coef_even_poly !coefC; case: i. Qed.
Lemma
even_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" ]
[ "apply", "coefC", "coef_even_poly", "even_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_poly p : {poly R}
:= \poly_(i < (size p)./2) p`_i.*2.+1.
Definition
odd_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" ]
[ "poly", "size" ]
Odd part of a polynomial
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_odd_poly p : size (odd_poly p) <= (size p)./2.
Proof. exact: size_poly. Qed.
Lemma
size_odd_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" ]
[ "odd_poly", "size", "size_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_odd_poly p i : (odd_poly p)`_i = p`_i.*2.+1.
Proof. by rewrite coef_poly gtn_half_double if_nth ?leqVgt. Qed.
Lemma
coef_odd_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" ]
[ "coef_poly", "gtn_half_double", "if_nth", "leqVgt", "odd_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_polyE s p : size p <= s.*2.+1 -> odd_poly p = \poly_(i < s) p`_i.*2.+1.
Proof. move=> pLs2; apply/polyP => i; rewrite coef_odd_poly !coef_poly if_nth //. by case: ltnP => //= ?; rewrite (leq_trans pLs2) ?ltnS ?leq_double. Qed.
Lemma
odd_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", "coef_odd_poly", "coef_poly", "if_nth", "leq_double", "leq_trans", "ltnP", "ltnS", "odd_poly", "polyP", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_polyC (c : R) : odd_poly c%:P = 0.
Proof. by apply/polyP => i; rewrite coef_odd_poly !coefC; case: i. Qed.
Lemma
odd_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" ]
[ "apply", "coefC", "coef_odd_poly", "odd_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_polyD p q : odd_poly (p + q) = odd_poly p + odd_poly q.
Proof. by apply/polyP => i; rewrite !(coef_odd_poly, coefD). Qed.
Lemma
odd_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" ]
[ "apply", "coefD", "coef_odd_poly", "odd_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_polyZ k p : odd_poly (k *: p) = k *: odd_poly p.
Proof. by apply/polyP => i; rewrite !(coefZ, coef_odd_poly). Qed.
Lemma
odd_polyZ
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", "coefZ", "coef_odd_poly", "odd_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_odd_poly_eq p : ~~ odd (size p) -> size (odd_poly p) = (size p)./2.
Proof. have [->|p_neq0] := eqVneq p 0; first by rewrite odd_polyC size_poly0. move=> p_odd; rewrite size_poly_eq// -subn1 doubleB subn2 even_halfK//. rewrite prednK ?lead_coef_eq0// ltn_predRL. by move: p_neq0 p_odd; rewrite -size_poly_eq0; case: (size p) => [|[]]. Qed.
Lemma
size_odd_poly_eq
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" ]
[ "doubleB", "eqVneq", "even_halfK", "lead_coef_eq0", "ltn_predRL", "odd", "odd_poly", "odd_polyC", "prednK", "size", "size_poly0", "size_poly_eq", "size_poly_eq0", "subn1", "subn2" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_polyMX p : odd_poly (p * 'X) = even_poly p.
Proof. have [->|pN0] := eqVneq p 0; first by rewrite mul0r even_polyC odd_polyC. by apply/polyP => i; rewrite !coef_poly size_mulX // coefMX. Qed.
Lemma
odd_polyMX
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", "coefMX", "coef_poly", "eqVneq", "even_poly", "even_polyC", "mul0r", "odd_poly", "odd_polyC", "polyP", "size_mulX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
even_polyMX p : even_poly (p * 'X) = odd_poly p * 'X.
Proof. have [->|pN0] := eqVneq p 0; first by rewrite mul0r even_polyC odd_polyC mul0r. by apply/polyP => -[|i]; rewrite !(coefMX, coef_poly, if_same, size_mulX). Qed.
Lemma
even_polyMX
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", "coefMX", "coef_poly", "eqVneq", "even_poly", "even_polyC", "mul0r", "odd_poly", "odd_polyC", "polyP", "size_mulX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_even_poly p : \sum_(i < size p | ~~ odd i) p`_i *: 'X^i = even_poly p \Po 'X^2.
Proof. apply/polyP => i; rewrite coef_comp_poly_Xn// coef_sumMXn coef_even_poly. rewrite (big_ord1_cond_eq _ _ (negb \o _))/= -dvdn2 andbC -muln2. by case: dvdnP => //= -[k ->]; rewrite mulnK// if_nth ?leqVgt. Qed.
Lemma
sum_even_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", "big_ord1_cond_eq", "coef_comp_poly_Xn", "coef_even_poly", "coef_sumMXn", "dvdn2", "dvdnP", "even_poly", "if_nth", "leqVgt", "muln2", "mulnK", "odd", "polyP", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_odd_poly p : \sum_(i < size p | odd i) p`_i *: 'X^i = (odd_poly p \Po 'X^2) * 'X.
Proof. apply/polyP => i; rewrite coefMX coef_comp_poly_Xn// coef_sumMXn coef_odd_poly/=. case: i => [|i]//=; first by rewrite big_andbC big1// => -[[|j]//]. rewrite big_ord1_cond_eq/= -dvdn2 andbC -muln2. by case: dvdnP => //= -[k ->]; rewrite mulnK// if_nth ?leqVgt. Qed.
Lemma
sum_odd_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", "big1", "big_andbC", "big_ord1_cond_eq", "coefMX", "coef_comp_poly_Xn", "coef_odd_poly", "coef_sumMXn", "dvdn2", "dvdnP", "if_nth", "leqVgt", "muln2", "mulnK", "odd", "odd_poly", "polyP", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_even_odd p : even_poly p \Po 'X^2 + (odd_poly p \Po 'X^2) * 'X = p.
Proof. rewrite -sum_even_poly -sum_odd_poly addrC -(bigID _ xpredT). by rewrite -[RHS]coefK poly_def. Qed.
Lemma
poly_even_odd
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", "bigID", "coefK", "even_poly", "odd_poly", "poly_def", "sum_even_poly", "sum_odd_poly" ]
Decomposition in odd and even part
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_poly m p
:= \poly_(i < m) p`_i.
Definition
take_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" ]
[]
take and drop for polynomials
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_take_poly m p : size (take_poly m p) <= m.
Proof. exact: size_poly. Qed.
Lemma
size_take_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" ]
[ "size", "size_poly", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_take_poly m p i : (take_poly m p)`_i = if i < m then p`_i else 0.
Proof. exact: coef_poly. Qed.
Lemma
coef_take_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" ]
[ "coef_poly", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_poly_id m p : size p <= m -> take_poly m p = p.
Proof. move=> /leq_trans gep; apply/polyP => i; rewrite coef_poly if_nth//=. by case: ltnP => // /gep->. Qed.
Lemma
take_poly_id
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_poly", "if_nth", "leq_trans", "ltnP", "polyP", "size", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_polyD m p q : take_poly m (p + q) = take_poly m p + take_poly m q.
Proof. by apply/polyP => i; rewrite !(coefD, coef_poly); case: leqP; rewrite ?add0r. Qed.
Lemma
take_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" ]
[ "add0r", "apply", "coefD", "coef_poly", "leqP", "polyP", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_polyZ k m p : take_poly m (k *: p) = k *: take_poly m p.
Proof. apply/polyP => i; rewrite !(coefZ, coef_take_poly); case: leqP => //. by rewrite mulr0. Qed.
Lemma
take_polyZ
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", "coefZ", "coef_take_poly", "leqP", "mulr0", "polyP", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_poly_sum m I r P (p : I -> {poly R}) : take_poly m (\sum_(i <- r | P i) p i) = \sum_(i <- r| P i) take_poly m (p i).
Proof. exact: linear_sum. Qed.
Lemma
take_poly_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" ]
[ "linear_sum", "poly", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_poly0l p : take_poly 0 p = 0.
Proof. exact/size_poly_leq0P/size_take_poly. Qed.
Lemma
take_poly0l
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_poly_leq0P", "size_take_poly", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_poly0r m : take_poly m 0 = 0.
Proof. exact: linear0. Qed.
Lemma
take_poly0r
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" ]
[ "linear0", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_polyMXn m n p : take_poly m (p * 'X^n) = take_poly (m - n) p * 'X^n.
Proof. have [->|/eqP p_neq0] := p =P 0; first by rewrite !(mul0r, take_poly0r). apply/polyP => i; rewrite !(coef_take_poly, coefMXn). by have [iLn|nLi] := leqP n i; rewrite ?if_same// ltn_sub2rE. Qed.
Lemma
take_polyMXn
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", "coefMXn", "coef_take_poly", "leqP", "ltn_sub2rE", "mul0r", "polyP", "take_poly", "take_poly0r" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_polyMXn_0 n p : take_poly n (p * 'X^n) = 0.
Proof. by rewrite take_polyMXn subnn take_poly0l mul0r. Qed.
Lemma
take_polyMXn_0
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", "subnn", "take_poly", "take_poly0l", "take_polyMXn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_polyDMXn n p q : size p <= n -> take_poly n (p + q * 'X^n) = p.
Proof. by move=> ?; rewrite take_polyD take_poly_id// take_polyMXn_0 addr0. Qed.
Lemma
take_polyDMXn
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", "size", "take_poly", "take_polyD", "take_polyMXn_0", "take_poly_id" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_poly m p
:= \poly_(i < size p - m) p`_(i + m).
Definition
drop_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" ]
[ "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_drop_poly m p i : (drop_poly m p)`_i = p`_(i + m).
Proof. by rewrite coef_poly ltn_subRL addnC if_nth ?leqVgt. Qed.
Lemma
coef_drop_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" ]
[ "addnC", "coef_poly", "drop_poly", "if_nth", "leqVgt", "ltn_subRL" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_poly_eq0 m p : size p <= m -> drop_poly m p = 0.
Proof. move=> sLm; apply/polyP => i; rewrite coef_poly coef0 ltn_subRL addnC. by rewrite if_nth ?leqVgt// nth_default// (leq_trans _ (leq_addl _ _)). Qed.
Lemma
drop_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" ]
[ "addnC", "apply", "coef0", "coef_poly", "drop_poly", "if_nth", "leqVgt", "leq_addl", "leq_trans", "ltn_subRL", "nth_default", "polyP", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_drop_poly n p : size (drop_poly n p) = (size p - n)%N.
Proof. have [pLn|nLp] := leqP (size p) n. by rewrite (eqP pLn) drop_poly_eq0 ?size_poly0. have p_neq0 : p != 0 by rewrite -size_poly_gt0 (leq_trans _ nLp). by rewrite size_poly_eq// predn_sub subnK ?lead_coef_eq0// -ltnS -polySpred. Qed.
Lemma
size_drop_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" ]
[ "drop_poly", "drop_poly_eq0", "lead_coef_eq0", "leqP", "leq_trans", "ltnS", "polySpred", "predn_sub", "size", "size_poly0", "size_poly_eq", "size_poly_gt0", "subnK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_drop_poly n p : \sum_(n <= i < size p) p`_i *: 'X^i = drop_poly n p * 'X^n.
Proof. rewrite (big_addn 0) big_mkord /drop_poly poly_def mulr_suml. by apply: eq_bigr => i _; rewrite exprD scalerAl. Qed.
Lemma
sum_drop_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", "big_addn", "big_mkord", "drop_poly", "eq_bigr", "exprD", "mulr_suml", "poly_def", "scalerAl", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_polyD m p q : drop_poly m (p + q) = drop_poly m p + drop_poly m q.
Proof. by apply/polyP => i; rewrite coefD !coef_drop_poly coefD. Qed.
Lemma
drop_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" ]
[ "apply", "coefD", "coef_drop_poly", "drop_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_polyZ k m p : drop_poly m (k *: p) = k *: drop_poly m p.
Proof. by apply/polyP => i; rewrite coefZ !coef_drop_poly coefZ. Qed.
Lemma
drop_polyZ
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", "coefZ", "coef_drop_poly", "drop_poly", "polyP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_poly_sum m I r P (p : I -> {poly R}) : drop_poly m (\sum_(i <- r | P i) p i) = \sum_(i <- r | P i) drop_poly m (p i).
Proof. exact: linear_sum. Qed.
Lemma
drop_poly_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" ]
[ "drop_poly", "linear_sum", "poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_poly0l p : drop_poly 0 p = p.
Proof. by apply/polyP => i; rewrite coef_poly subn0 addn0 if_nth ?leqVgt. Qed.
Lemma
drop_poly0l
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" ]
[ "addn0", "apply", "coef_poly", "drop_poly", "if_nth", "leqVgt", "polyP", "subn0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_poly0r m : drop_poly m 0 = 0.
Proof. exact: linear0. Qed.
Lemma
drop_poly0r
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" ]
[ "drop_poly", "linear0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_polyMXn m n p : drop_poly m (p * 'X^n) = drop_poly (m - n) p * 'X^(n - m).
Proof. have [->|p_neq0] := eqVneq p 0; first by rewrite mul0r !drop_poly0r mul0r. apply/polyP => i; rewrite !(coefMXn, coef_drop_poly) ltn_subRL [(m + i)%N]addnC. have [i_small|i_big]// := ltnP; congr nth. by have [mn|/ltnW mn] := leqP m n; rewrite (eqP mn) (addn0, subn0) (subnBA, addnBA). Qed.
Lemma
drop_polyMXn
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" ]
[ "addn0", "addnBA", "addnC", "apply", "coefMXn", "coef_drop_poly", "drop_poly", "drop_poly0r", "eqVneq", "leqP", "ltnP", "ltnW", "ltn_subRL", "mul0r", "nth", "polyP", "subn0", "subnBA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_polyMXn_id n p : drop_poly n (p * 'X^ n) = p.
Proof. by rewrite drop_polyMXn subnn drop_poly0l expr0 mulr1. Qed.
Lemma
drop_polyMXn_id
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" ]
[ "drop_poly", "drop_poly0l", "drop_polyMXn", "expr0", "mulr1", "subnn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_polyDMXn n p q : size p <= n -> drop_poly n (p + q * 'X^n) = q.
Proof. by move=> ?; rewrite drop_polyD drop_poly_eq0// drop_polyMXn_id add0r. Qed.
Lemma
drop_polyDMXn
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", "drop_poly", "drop_polyD", "drop_polyMXn_id", "drop_poly_eq0", "size" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_take_drop n p : take_poly n p + drop_poly n p * 'X^n = p.
Proof. apply/polyP => i; rewrite coefD coefMXn coef_take_poly coef_drop_poly. by case: ltnP => ni; rewrite ?addr0 ?add0r//= subnK. Qed.
Lemma
poly_take_drop
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", "addr0", "apply", "coefD", "coefMXn", "coef_drop_poly", "coef_take_poly", "drop_poly", "ltnP", "polyP", "subnK", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqp_take_drop n p q : take_poly n p = take_poly n q -> drop_poly n p = drop_poly n q -> p = q.
Proof. by move=> tpq dpq; rewrite -[p](poly_take_drop n) -[q](poly_take_drop n) tpq dpq. Qed.
Lemma
eqp_take_drop
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" ]
[ "drop_poly", "poly_take_drop", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
even_poly_is_linear : linear (@even_poly R).
Proof. exact: linearP. Qed.
Fact
even_poly_is_linear
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" ]
[ "even_poly", "linear", "linearP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_poly_is_linear : linear (@odd_poly R).
Proof. exact: linearP. Qed.
Fact
odd_poly_is_linear
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" ]
[ "linear", "linearP", "odd_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
take_poly_is_linear m : linear (@take_poly R m).
Proof. exact: linearP. Qed.
Fact
take_poly_is_linear
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" ]
[ "linear", "linearP", "take_poly" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
drop_poly_is_linear m : linear (@drop_poly R m).
Proof. exact: linearP. Qed.
Fact
drop_poly_is_linear
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" ]
[ "drop_poly", "linear", "linearP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefE
:= (coef0, coef1, coefC, coefX, coefXn, coef_sumMXn, coefZ, coefMC, coefCM, coefXnM, coefMXn, coefXM, coefMX, coefMNn, coefMn, coefN, coefB, coefD, coef_even_poly, coef_odd_poly, coef_take_poly, coef_drop_poly, coef_cons, coef_Poly, coef_poly, coef_deriv, coef_nderivn, coef_derivn, coef_map, coef_sum, ...
Definition
coefE
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" ]
[ "coef0", "coef1", "coefB", "coefC", "coefCM", "coefD", "coefMC", "coefMNn", "coefMX", "coefMXn", "coefMn", "coefN", "coefX", "coefXM", "coefXn", "coefXnM", "coefZ", "coef_Poly", "coef_comp_poly", "coef_comp_poly_Xn", "coef_cons", "coef_deriv", "coef_derivn", "coef_drop_...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d