phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
cons_poly_def p a : cons_poly a p = p * 'X + a%:P.	Proof. apply/polyP=> i; rewrite coef_cons coefD coefMX coefC. by case: ifP; rewrite !simp. Qed.	Lemma	cons_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" ]	[ "apply", "coefC", "coefD", "coefMX", "coef_cons", "cons_poly", "polyP", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_ind (K : {poly R} -> Type) : K 0 -> (forall p c, K p -> K (p * 'X + c%:P)) -> (forall p, K p).	Proof. move=> K0 Kcons p; rewrite -[p]polyseqK. by elim: {p}(p : seq R) => //= p c IHp; rewrite cons_poly_def; apply: Kcons. Qed.	Lemma	poly_ind	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", "cons_poly_def", "poly", "polyseqK", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyseqXaddC a : 'X + a%:P = [:: a; 1] :> seq R.	Proof. by rewrite -['X]mul1r -cons_poly_def polyseq_cons polyseq1. Qed.	Lemma	polyseqXaddC	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" ]	[ "cons_poly_def", "mul1r", "polyseq1", "polyseq_cons", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_XaddC b : size ('X + b%:P) = 2.	Proof. by rewrite polyseqXaddC. Qed.	Lemma	size_XaddC	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" ]	[ "polyseqXaddC", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefXaddC a : lead_coef ('X + a%:P) = 1.	Proof. by rewrite lead_coefE polyseqXaddC. Qed.	Lemma	lead_coefXaddC	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_coefE", "polyseqXaddC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_MXaddC p c : size (p * 'X + c%:P) = (if (p == 0) && (c == 0) then 0 else (size p).+1).	Proof. by rewrite -cons_poly_def size_cons_poly nil_poly. Qed.	Lemma	size_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" ]	[ "cons_poly_def", "nil_poly", "size", "size_cons_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyseqMX p : p != 0 -> p * 'X = 0 :: p :> seq R.	Proof. by move=> nz_p; rewrite -[p * _]addr0 -cons_poly_def polyseq_cons nil_poly nz_p. Qed.	Lemma	polyseqMX	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", "cons_poly_def", "nil_poly", "nz_p", "polyseq_cons", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_mulX p : p != 0 -> size (p * 'X) = (size p).+1.	Proof. by move/polyseqMX->. Qed.	Lemma	size_mulX	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" ]	[ "polyseqMX", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefMX p : lead_coef (p * 'X) = lead_coef p.	Proof. have [-> \| nzp] := eqVneq p 0; first by rewrite mul0r. by rewrite /lead_coef !nth_last polyseqMX. Qed.	Lemma	lead_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" ]	[ "eqVneq", "lead_coef", "mul0r", "nth_last", "polyseqMX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_XmulC a : a != 0 -> size ('X * a%:P) = 2.	Proof. by move=> nz_a; rewrite -commr_polyX size_mulX ?polyC_eq0 ?size_polyC nz_a. Qed.	Lemma	size_XmulC	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" ]	[ "commr_polyX", "polyC_eq0", "size", "size_mulX", "size_polyC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''X^' n"	:= ('X ^+ n).	Notation	''X^' n	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
coefXn n i : 'X^n`_i = (i == n)%:R.	Proof. by elim: n i => [\|n IHn] [\|i]; rewrite ?coef1 // exprS coefXM ?IHn. Qed.	Lemma	coefXn	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" ]	[ "coef1", "coefXM", "exprS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyseqXn n : 'X^n = rcons (nseq n 0) 1 :> seq R.	Proof. elim: n => [\|n IHn]; rewrite ?polyseq1 // exprSr. by rewrite polyseqMX -?size_poly_eq0 IHn ?size_rcons. Qed.	Lemma	polyseqXn	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" ]	[ "exprSr", "nseq", "polyseq1", "polyseqMX", "rcons", "seq", "size_poly_eq0", "size_rcons" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_polyXn n : size 'X^n = n.+1.	Proof. by rewrite polyseqXn size_rcons size_nseq. Qed.	Lemma	size_polyXn	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" ]	[ "polyseqXn", "size", "size_nseq", "size_rcons" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_polyXn p n : GRing.comm p 'X^n.	Proof. exact/commrX/commr_polyX. Qed.	Lemma	commr_polyXn	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", "commrX", "commr_polyX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefXn n : lead_coef 'X^n = 1.	Proof. by rewrite /lead_coef nth_last polyseqXn last_rcons. Qed.	Lemma	lead_coefXn	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" ]	[ "last_rcons", "lead_coef", "nth_last", "polyseqXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefXnaddC n c : 0 < n -> lead_coef ('X^n + c%:P) = 1.	Proof. move=> n_gt0; rewrite lead_coefDl ?lead_coefXn//. by rewrite size_polyC size_polyXn ltnS (leq_trans (leq_b1 _)). Qed.	Lemma	lead_coefXnaddC	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_coefDl", "lead_coefXn", "leq_b1", "leq_trans", "ltnS", "n_gt0", "size_polyC", "size_polyXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_XnaddC n c : 0 < n -> size ('X^n + c%:P) = n.+1.	Proof. by move=> *; rewrite size_polyDl ?size_polyXn// size_polyC; case: eqP. Qed.	Lemma	size_XnaddC	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", "size_polyDl", "size_polyXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyseqMXn n p : p != 0 -> p * 'X^n = ncons n 0 p :> seq R.	Proof. case: n => [\|n] nz_p; first by rewrite mulr1. elim: n => [\|n IHn]; first exact: polyseqMX. by rewrite exprSr mulrA polyseqMX -?nil_poly IHn. Qed.	Lemma	polyseqMXn	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" ]	[ "exprSr", "mulr1", "mulrA", "ncons", "nil_poly", "nz_p", "polyseqMX", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefMXn n p i : (p * 'X^n)`_i = if i < n then 0 else p`_(i - n).	Proof. have [-> \| /polyseqMXn->] := eqVneq p 0; last exact: nth_ncons. by rewrite mul0r !coef0 if_same. Qed.	Lemma	coefMXn	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", "eqVneq", "last", "mul0r", "nth_ncons", "polyseqMXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_mulXn n p : p != 0 -> size (p * 'X^n) = (n + size p)%N.	Proof. elim: n p => [p p_neq0\| n IH p p_neq0]; first by rewrite mulr1. by rewrite exprS mulrA IH -?size_poly_eq0 size_mulX // addnS. Qed.	Lemma	size_mulXn	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", "mulr1", "mulrA", "size", "size_mulX", "size_poly_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefXnM n p i : ('X^n * p)`_i = if i < n then 0 else p`_(i - n).	Proof. by rewrite -commr_polyXn coefMXn. Qed.	Lemma	coefXnM	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" ]	[ "coefMXn", "commr_polyXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_sumMXn I (r : seq I) (P : pred I) (p : I -> R) (n : I -> nat) k : (\sum_(i <- r \| P i) p i *: 'X^(n i))`_k = \sum_(i <- r \| P i && (n i == k)) p i.	Proof. rewrite coef_sum big_mkcondr; apply: eq_bigr => i Pi. by rewrite coefZ coefXn mulr_natr mulrb eq_sym. Qed.	Lemma	coef_sumMXn	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_mkcondr", "coefXn", "coefZ", "coef_sum", "eq_bigr", "eq_sym", "mulr_natr", "mulrb", "nat", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
poly_def n E : \poly_(i < n) E i = \sum_(i < n) E i *: 'X^i.	Proof. by apply/polyP => i; rewrite coef_sumMXn coef_poly big_ord1_eq. Qed.	Lemma	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" ]	[ "apply", "big_ord1_eq", "coef_poly", "coef_sumMXn", "polyP" ]	Expansion of a polynomial as an indexed sum	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_poly n E1 E2 : (forall i, i < n -> E1 i = E2 i) -> poly n E1 = poly n E2 :> {poly R}.	Proof. by move=> E; rewrite !poly_def; apply: eq_bigr => i _; rewrite E. Qed.	Lemma	eq_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", "eq_bigr", "poly", "poly_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_rec s x	:= if s is a :: s' then horner_rec s' x * x + a else 0.	Fixpoint	horner_rec	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
horner p	:= horner_rec p.	Definition	horner	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_rec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p .[ x ]"	:= (horner p x) : ring_scope.	Notation	p .[ 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" ]	[ "horner" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner0 x : (0 : {poly R}).[x] = 0.	Proof. by rewrite /horner polyseq0. Qed.	Lemma	horner0	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", "poly", "polyseq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerC c x : (c%:P).[x] = c.	Proof. by rewrite /horner polyseqC; case: eqP; rewrite /= ?simp. Qed.	Lemma	hornerC	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", "polyseqC", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerX x : 'X.[x] = x.	Proof. by rewrite /horner polyseqX /= !simp. Qed.	Lemma	hornerX	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", "polyseqX", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_cons p c x : (cons_poly c p).[x] = p.[x] * x + c.	Proof. rewrite /horner polyseq_cons; case: nilP => //= ->. by rewrite !simp -/(_.[x]) hornerC. Qed.	Lemma	horner_cons	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" ]	[ "cons_poly", "horner", "hornerC", "nilP", "polyseq_cons", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_coef0 p : p.[0] = p`_0.	Proof. by rewrite /horner; case: (p : seq R) => //= c p'; rewrite !simp. Qed.	Lemma	horner_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" ]	[ "horner", "seq", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerMXaddC p c x : (p * 'X + c%:P).[x] = p.[x] * x + c.	Proof. by rewrite -cons_poly_def horner_cons. Qed.	Lemma	hornerMXaddC	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" ]	[ "cons_poly_def", "horner_cons" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerMX p x : (p * 'X).[x] = p.[x] * x.	Proof. by rewrite -[p * 'X]addr0 hornerMXaddC addr0. Qed.	Lemma	hornerMX	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", "hornerMXaddC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_Poly s x : (Poly s).[x] = horner_rec s x.	Proof. by elim: s => [\|a s /= <-]; rewrite (horner0, horner_cons). Qed.	Lemma	horner_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", "horner0", "horner_cons", "horner_rec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_coef p x : p.[x] = \sum_(i < size p) p`_i * x ^+ i.	Proof. rewrite /horner. elim: {p}(p : seq R) => /= [\|a s ->]; first by rewrite big_ord0. rewrite big_ord_recl simp addrC big_distrl /=. by congr (_ + _); apply: eq_bigr => i _; rewrite -mulrA exprSr. Qed.	Lemma	horner_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" ]	[ "addrC", "apply", "big_distrl", "big_ord0", "big_ord_recl", "eq_bigr", "exprSr", "horner", "mulrA", "seq", "simp", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_coef_wide n p x : size p <= n -> p.[x] = \sum_(i < n) p`_i * x ^+ i.	Proof. move=> le_p_n. rewrite horner_coef (big_ord_widen n (fun i => p`_i * x ^+ i)) // big_mkcond. by apply: eq_bigr => i _; case: ltnP => // le_p_i; rewrite nth_default ?simp. Qed.	Lemma	horner_coef_wide	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_mkcond", "big_ord_widen", "eq_bigr", "horner_coef", "ltnP", "nth_default", "simp", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_poly n E x : (\poly_(i < n) E i).[x] = \sum_(i < n) E i * x ^+ i.	Proof. rewrite (@horner_coef_wide n) ?size_poly //. by apply: eq_bigr => i _; rewrite coef_poly ltn_ord. Qed.	Lemma	horner_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", "coef_poly", "eq_bigr", "horner_coef_wide", "ltn_ord", "size_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerD p q x : (p + q).[x] = p.[x] + q.[x].	Proof. rewrite [in LHS]/+%R /= unlock horner_poly; set m := maxn _ _. rewrite !(@horner_coef_wide m) ?leq_max ?leqnn ?orbT // -big_split /=. by apply: eq_bigr => i _; rewrite -mulrDl. Qed.	Lemma	hornerD	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", "eq_bigr", "horner_coef_wide", "horner_poly", "leq_max", "leqnn", "maxn", "mulrDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerCM a p x : (a%:P * p).[x] = a * p.[x].	Proof. elim/poly_ind: p => [\|p c IHp]; first by rewrite !(mulr0, horner0). by rewrite mulrDr mulrA -polyCM !hornerMXaddC IHp mulrDr mulrA. Qed.	Lemma	hornerCM	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" ]	[ "horner0", "hornerMXaddC", "mulr0", "mulrA", "mulrDr", "polyCM", "poly_ind" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerZ c p x : (c : p).[x] = c p.[x].	Proof. by rewrite -mul_polyC hornerCM. Qed.	Lemma	hornerZ	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" ]	[ "hornerCM", "mul_polyC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_eval (x : R)	:= horner^~ x.	Definition	horner_eval	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_evalE x p : horner_eval x p = p.[x].	Proof. by []. Qed.	Lemma	horner_evalE	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_eval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_sum I (r : seq I) (P : pred I) F x : (\sum_(i <- r \| P i) F i).[x] = \sum_(i <- r \| P i) (F i).[x].	Proof. exact: (raddf_sum (horner_eval _)). Qed.	Lemma	horner_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" ]	[ "horner_eval", "raddf_sum", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerMn n p x : (p + n).[x] = p.[x] + n.	Proof. exact: (raddfMn (horner_eval _)). Qed.	Lemma	hornerMn	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_eval", "raddfMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_coef p x	:= forall i, p`_i * x = x * p`_i.	Definition	comm_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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_poly p x	:= x * p.[x] = p.[x] * x.	Definition	comm_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
comm_coef_poly p x : comm_coef p x -> comm_poly p x.	Proof. move=> cpx; rewrite /comm_poly !horner_coef big_distrl big_distrr /=. by apply: eq_bigr => i _; rewrite /= mulrA -cpx -!mulrA commrX. Qed.	Lemma	comm_coef_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_distrl", "big_distrr", "comm_coef", "comm_poly", "commrX", "eq_bigr", "horner_coef", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_poly0 x : comm_poly 0 x.	Proof. by rewrite /comm_poly !horner0 !simp. Qed.	Lemma	comm_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" ]	[ "comm_poly", "horner0", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_poly1 x : comm_poly 1 x.	Proof. by rewrite /comm_poly !hornerC !simp. Qed.	Lemma	comm_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" ]	[ "comm_poly", "hornerC", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_polyX x : comm_poly 'X x.	Proof. by rewrite /comm_poly !hornerX. Qed.	Lemma	comm_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" ]	[ "comm_poly", "hornerX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_polyD p q x: comm_poly p x -> comm_poly q x -> comm_poly (p + q) x.	Proof. by rewrite /comm_poly hornerD mulrDr mulrDl => -> ->. Qed.	Lemma	comm_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" ]	[ "comm_poly", "hornerD", "mulrDl", "mulrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_horner a b p : GRing.comm a b -> comm_coef p a -> GRing.comm a p.[b].	Proof. move=> cab cpa; rewrite horner_coef; apply: commr_sum => i _. by apply: commrM => //; apply: commrX. Qed.	Lemma	commr_horner	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", "comm", "comm_coef", "commrM", "commrX", "commr_sum", "horner_coef" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerM_comm p q x : comm_poly q x -> (p * q).[x] = p.[x] * q.[x].	Proof. move=> comm_qx. elim/poly_ind: p => [\|p c IHp]; first by rewrite !(simp, horner0). rewrite mulrDl hornerD hornerCM -mulrA -commr_polyX mulrA hornerMX. by rewrite {}IHp -mulrA -comm_qx mulrA -mulrDl hornerMXaddC. Qed.	Lemma	hornerM_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_poly", "commr_polyX", "horner0", "hornerCM", "hornerD", "hornerMX", "hornerMXaddC", "mulrA", "mulrDl", "poly_ind", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_polyM p q x: comm_poly p x -> comm_poly q x -> comm_poly (p * q) x.	Proof. by move=> px qx; rewrite /comm_poly hornerM_comm// mulrA px -mulrA qx mulrA. Qed.	Lemma	comm_polyM	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", "hornerM_comm", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_exp_comm p x n : comm_poly p x -> (p ^+ n).[x] = p.[x] ^+ n.	Proof. move=> comm_px; elim: n => [\|n IHn]; first by rewrite hornerC. by rewrite !exprSr -IHn hornerM_comm. Qed.	Lemma	horner_exp_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_poly", "exprSr", "hornerC", "hornerM_comm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_poly_exp p n x: comm_poly p x -> comm_poly (p ^+ n) x.	Proof. by move=> px; rewrite /comm_poly !horner_exp_comm// commrX. Qed.	Lemma	comm_poly_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" ]	[ "comm_poly", "commrX", "horner_exp_comm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerXn x n : ('X^n).[x] = x ^+ n.	Proof. by rewrite horner_exp_comm /comm_poly hornerX. Qed.	Lemma	hornerXn	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", "hornerX", "horner_exp_comm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_pred S	:= fun p : {poly R} => all (mem S) p.	Definition	polyOver_pred	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" ]	[ "all", "poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver S	:= [qualify a p \| polyOver_pred S p].	Definition	polyOver	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" ]	[ "polyOver_pred" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverS (S1 S2 : {pred R}) : {subset S1 <= S2} -> {subset polyOver S1 <= polyOver S2}.	Proof. by move=> sS12 p /(all_nthP 0)S1p; apply/(all_nthP 0)=> i /S1p; apply: sS12. Qed.	Lemma	polyOverS	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" ]	[ "S1", "S2", "all_nthP", "apply", "polyOver" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver0 S : 0 \is a polyOver S.	Proof. by rewrite qualifE /= polyseq0. Qed.	Lemma	polyOver0	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" ]	[ "polyOver", "polyseq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_poly S n E : (forall i, i < n -> E i \in S) -> \poly_(i < n) E i \is a polyOver S.	Proof. move=> S_E; apply/(all_nthP 0)=> i lt_i_p /=; rewrite coef_poly. by case: ifP => [/S_E// \| /idP[]]; apply: leq_trans lt_i_p (size_poly n E). Qed.	Lemma	polyOver_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" ]	[ "all_nthP", "apply", "coef_poly", "leq_trans", "polyOver", "size_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverP {p} : reflect (forall i, p`_i \in S) (p \in polyOver S).	Proof. apply: (iffP (all_nthP 0)) => [Sp i \| Sp i _]; last exact: Sp. by have [/Sp // \| /(nth_default 0)->] := ltnP i (size p); apply: rpred0. Qed.	Lemma	polyOverP	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" ]	[ "all_nthP", "apply", "last", "ltnP", "nth_default", "polyOver", "rpred0", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverC c : (c%:P \in polyOver S) = (c \in S).	Proof. by rewrite qualifE /= polyseqC; case: eqP => [->\|] /=; rewrite ?andbT ?rpred0. Qed.	Lemma	polyOverC	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" ]	[ "polyOver", "polyseqC", "rpred0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_nmod_closed : nmod_closed (polyOver S).	Proof. split=> [\|p q Sp Sq]; first exact: polyOver0. by apply/polyOverP=> i; rewrite coefD rpredD ?(polyOverP _). Qed.	Fact	polyOver_nmod_closed	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", "nmod_closed", "polyOver", "polyOver0", "polyOverP", "rpredD", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_mulr_2closed : GRing.mulr_2closed (polyOver S).	Proof. move=> p q /polyOverP Sp /polyOverP Sq; apply/polyOverP=> i. by rewrite coefM rpred_sum // => j _; rewrite rpredM. Qed.	Lemma	polyOver_mulr_2closed	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", "mulr_2closed", "polyOver", "polyOverP", "rpredM", "rpred_sum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_mul1_closed : 1 \in polyOver S.	Proof. by rewrite polyOverC rpred1. Qed.	Fact	polyOver_mul1_closed	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" ]	[ "polyOver", "polyOverC", "rpred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverZ : {in S & polyOver S, forall c p, c *: p \is a polyOver S}.	Proof. by move=> c p Sc /polyOverP Sp; apply/polyOverP=> i; rewrite coefZ rpredM ?Sp. Qed.	Lemma	polyOverZ	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", "polyOver", "polyOverP", "rpredM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverX : 'X \in polyOver S.	Proof. by rewrite qualifE /= polyseqX /= rpred0 rpred1. Qed.	Lemma	polyOverX	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" ]	[ "polyOver", "polyseqX", "rpred0", "rpred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverXn n : 'X^n \in polyOver S.	Proof. by rewrite rpredX// polyOverX. Qed.	Lemma	polyOverXn	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" ]	[ "polyOver", "polyOverX", "rpredX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpred_horner : {in polyOver S & S, forall p x, p.[x] \in S}.	Proof. move=> p x /polyOverP Sp Sx; rewrite horner_coef rpred_sum // => i _. by rewrite rpredM ?rpredX. Qed.	Lemma	rpred_horner	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_coef", "polyOver", "polyOverP", "rpredM", "rpredX", "rpred_sum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deriv p	:= \poly_(i < (size p).-1) (p`_i.+1 *+ i.+1).	Definition	deriv	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" ]	Single derivative.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"a ^` ()"	:= (deriv a).	Notation	a ^` ()	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" ]	[ "deriv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_deriv p i : p^`()`_i = p`_i.+1 *+ i.+1.	Proof. rewrite coef_poly -subn1 ltn_subRL. by case: leqP => // /(nth_default 0) ->; rewrite mul0rn. Qed.	Lemma	coef_deriv	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", "ltn_subRL", "mul0rn", "nth_default", "subn1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_deriv (ringS : semiringClosed R) : {in polyOver ringS, forall p, p^`() \is a polyOver ringS}.	Proof. by move=> p /polyOverP Kp; apply/polyOverP=> i; rewrite coef_deriv rpredMn ?Kp. Qed.	Lemma	polyOver_deriv	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_deriv", "polyOver", "polyOverP", "rpredMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivC c : c%:P^`() = 0.	Proof. by apply/polyP=> i; rewrite coef_deriv coef0 coefC mul0rn. Qed.	Lemma	derivC	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", "coef0", "coefC", "coef_deriv", "mul0rn", "polyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivX : ('X)^`() = 1.	Proof. by apply/polyP=> [[\|i]]; rewrite coef_deriv coef1 coefX ?mul0rn. Qed.	Lemma	derivX	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", "coef1", "coefX", "coef_deriv", "mul0rn", "polyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivXn n : ('X^n)^`() = 'X^(n.-1) *+ n.	Proof. case: n => [\|n]; first exact: derivC. apply/polyP=> i; rewrite coef_deriv coefMn !coefXn eqSS. by case: eqP => [-> // \| _]; rewrite !mul0rn. Qed.	Lemma	derivXn	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", "coefMn", "coefXn", "coef_deriv", "derivC", "eqSS", "mul0rn", "polyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deriv_is_semilinear : semilinear deriv.	Proof. split=> [k p\|p q]; apply/polyP => i. by rewrite !(coef_deriv, coefZ) mulrnAr. by rewrite !(coef_deriv, coefD) mulrnDl. Qed.	Fact	deriv_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" ]	[ "apply", "coefD", "coefZ", "coef_deriv", "deriv", "mulrnAr", "mulrnDl", "polyP", "semilinear", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deriv0 : 0^`() = 0.	Proof. exact: linear0. Qed.	Lemma	deriv0	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivD : {morph deriv : p q / p + q}.	Proof. exact: linearD. Qed.	Lemma	derivD	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" ]	[ "deriv", "linearD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivMn n p : (p + n)^`() = p^`() + n.	Proof. exact: linearMn. Qed.	Lemma	derivMn	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" ]	[ "linearMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivZ c p : (c : p)^`() = c : p^`().	Proof. exact: linearZ. Qed.	Lemma	derivZ	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" ]	[ "linearZ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deriv_mulC c p : (c%:P * p)^`() = c%:P * p^`().	Proof. by rewrite !mul_polyC derivZ. Qed.	Lemma	deriv_mulC	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" ]	[ "derivZ", "mul_polyC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivMXaddC p c : (p * 'X + c%:P)^`() = p + p^`() * 'X.	Proof. apply/polyP=> i; rewrite raddfD /= derivC addr0 coefD !(coefMX, coef_deriv). by case: i; rewrite ?addr0. Qed.	Lemma	derivMXaddC	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", "coefD", "coefMX", "coef_deriv", "derivC", "polyP", "raddfD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivM p q : (p * q)^`() = p^`() * q + p * q^`().	Proof. elim/poly_ind: p => [\|p b IHp]; first by rewrite !(mul0r, add0r, derivC). rewrite mulrDl -mulrA -commr_polyX mulrA -[_ * 'X]addr0 raddfD /= !derivMXaddC. by rewrite deriv_mulC IHp !mulrDl -!mulrA !commr_polyX !addrA. Qed.	Lemma	derivM	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", "addrA", "commr_polyX", "derivC", "derivMXaddC", "deriv_mulC", "mul0r", "mulrA", "mulrDl", "poly_ind", "raddfD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivn n p	:= iter n deriv p.	Definition	derivn	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" ]	[ "deriv", "iter" ]	Iterated derivative.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"a ^` ( n )"	:= (derivn n a) : ring_scope.	Notation	a ^` ( n )	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" ]	[ "derivn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivn0 p : p^`(0) = p.	Proof. by []. Qed.	Lemma	derivn0	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
derivn1 p : p^`(1) = p^`().	Proof. by []. Qed.	Lemma	derivn1	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
derivnS p n : p^`(n.+1) = p^`(n)^`().	Proof. by []. Qed.	Lemma	derivnS	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
derivSn p n : p^`(n.+1) = p^`()^`(n).	Proof. exact: iterSr. Qed.	Lemma	derivSn	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" ]	[ "iterSr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_derivn n p i : p^`(n)`_i = p`_(n + i) *+ (n + i) ^_ n.	Proof. elim: n i => [\|n IHn] i; first by rewrite ffactn0 mulr1n. by rewrite derivnS coef_deriv IHn -mulrnA ffactnSr addSnnS addKn. Qed.	Lemma	coef_derivn	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", "addSnnS", "coef_deriv", "derivnS", "ffactn0", "ffactnSr", "mulr1n", "mulrnA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_derivn (ringS : semiringClosed R) : {in polyOver ringS, forall p n, p^`(n) \is a polyOver ringS}.	Proof. move=> p /polyOverP Kp /= n; apply/polyOverP=> i. by rewrite coef_derivn rpredMn. Qed.	Lemma	polyOver_derivn	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_derivn", "polyOver", "polyOverP", "rpredMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivn_is_semilinear n : semilinear (derivn n).	Proof. by elim: n => // n IHn; split=> [a p\|p q]; rewrite derivnS IHn semilinearPZ. Qed.	Fact	derivn_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" ]	[ "derivn", "derivnS", "semilinear", "semilinearPZ", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivnC c n : c%:P^`(n) = if n == 0 then c%:P else 0.	Proof. by case: n => // n; rewrite derivSn derivC linear0. Qed.	Lemma	derivnC	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" ]	[ "derivC", "derivSn", "linear0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivnD n : {morph derivn n : p q / p + q}.	Proof. exact: linearD. Qed.	Lemma	derivnD	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" ]	[ "derivn", "linearD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivnMn n m p : (p + m)^`(n) = p^`(n) + m.	Proof. exact: linearMn. Qed.	Lemma	derivnMn	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" ]	[ "linearMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

cons_poly_def p a : cons_poly a p = p * 'X + a%:P.

Proof. apply/polyP=> i; rewrite coef_cons coefD coefMX coefC. by case: ifP; rewrite !simp. Qed.

Lemma

cons_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" ]

[ "apply", "coefC", "coefD", "coefMX", "coef_cons", "cons_poly", "polyP", "simp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

poly_ind (K : {poly R} -> Type) : K 0 -> (forall p c, K p -> K (p * 'X + c%:P)) -> (forall p, K p).

Proof. move=> K0 Kcons p; rewrite -[p]polyseqK. by elim: {p}(p : seq R) => //= p c IHp; rewrite cons_poly_def; apply: Kcons. Qed.

Lemma

poly_ind

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", "cons_poly_def", "poly", "polyseqK", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyseqXaddC a : 'X + a%:P = [:: a; 1] :> seq R.

Proof. by rewrite -['X]mul1r -cons_poly_def polyseq_cons polyseq1. Qed.

Lemma

polyseqXaddC

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" ]

[ "cons_poly_def", "mul1r", "polyseq1", "polyseq_cons", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_XaddC b : size ('X + b%:P) = 2.

Proof. by rewrite polyseqXaddC. Qed.

Lemma

size_XaddC

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" ]

[ "polyseqXaddC", "size" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coefXaddC a : lead_coef ('X + a%:P) = 1.

Proof. by rewrite lead_coefE polyseqXaddC. Qed.

Lemma

lead_coefXaddC

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_coefE", "polyseqXaddC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_MXaddC p c : size (p * 'X + c%:P) = (if (p == 0) && (c == 0) then 0 else (size p).+1).

Proof. by rewrite -cons_poly_def size_cons_poly nil_poly. Qed.

Lemma

size_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" ]

[ "cons_poly_def", "nil_poly", "size", "size_cons_poly" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyseqMX p : p != 0 -> p * 'X = 0 :: p :> seq R.

Proof. by move=> nz_p; rewrite -[p * _]addr0 -cons_poly_def polyseq_cons nil_poly nz_p. Qed.

Lemma

polyseqMX

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", "cons_poly_def", "nil_poly", "nz_p", "polyseq_cons", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_mulX p : p != 0 -> size (p * 'X) = (size p).+1.

Proof. by move/polyseqMX->. Qed.

Lemma

size_mulX

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" ]

[ "polyseqMX", "size" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coefMX p : lead_coef (p * 'X) = lead_coef p.

Proof. have [-> | nzp] := eqVneq p 0; first by rewrite mul0r. by rewrite /lead_coef !nth_last polyseqMX. Qed.

Lemma

lead_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" ]

[ "eqVneq", "lead_coef", "mul0r", "nth_last", "polyseqMX" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_XmulC a : a != 0 -> size ('X * a%:P) = 2.

Proof. by move=> nz_a; rewrite -commr_polyX size_mulX ?polyC_eq0 ?size_polyC nz_a. Qed.

Lemma

size_XmulC

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" ]

[ "commr_polyX", "polyC_eq0", "size", "size_mulX", "size_polyC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''X^' n"

:= ('X ^+ n).

Notation

''X^' n

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

coefXn n i : 'X^n`_i = (i == n)%:R.

Proof. by elim: n i => [|n IHn] [|i]; rewrite ?coef1 // exprS coefXM ?IHn. Qed.

Lemma

coefXn

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" ]

[ "coef1", "coefXM", "exprS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyseqXn n : 'X^n = rcons (nseq n 0) 1 :> seq R.

Proof. elim: n => [|n IHn]; rewrite ?polyseq1 // exprSr. by rewrite polyseqMX -?size_poly_eq0 IHn ?size_rcons. Qed.

Lemma

polyseqXn

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" ]

[ "exprSr", "nseq", "polyseq1", "polyseqMX", "rcons", "seq", "size_poly_eq0", "size_rcons" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_polyXn n : size 'X^n = n.+1.

Proof. by rewrite polyseqXn size_rcons size_nseq. Qed.

Lemma

size_polyXn

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" ]

[ "polyseqXn", "size", "size_nseq", "size_rcons" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_polyXn p n : GRing.comm p 'X^n.

Proof. exact/commrX/commr_polyX. Qed.

Lemma

commr_polyXn

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", "commrX", "commr_polyX" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coefXn n : lead_coef 'X^n = 1.

Proof. by rewrite /lead_coef nth_last polyseqXn last_rcons. Qed.

Lemma

lead_coefXn

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" ]

[ "last_rcons", "lead_coef", "nth_last", "polyseqXn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coefXnaddC n c : 0 < n -> lead_coef ('X^n + c%:P) = 1.

Proof. move=> n_gt0; rewrite lead_coefDl ?lead_coefXn//. by rewrite size_polyC size_polyXn ltnS (leq_trans (leq_b1 _)). Qed.

Lemma

lead_coefXnaddC

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_coefDl", "lead_coefXn", "leq_b1", "leq_trans", "ltnS", "n_gt0", "size_polyC", "size_polyXn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_XnaddC n c : 0 < n -> size ('X^n + c%:P) = n.+1.

Proof. by move=> *; rewrite size_polyDl ?size_polyXn// size_polyC; case: eqP. Qed.

Lemma

size_XnaddC

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", "size_polyDl", "size_polyXn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyseqMXn n p : p != 0 -> p * 'X^n = ncons n 0 p :> seq R.

Proof. case: n => [|n] nz_p; first by rewrite mulr1. elim: n => [|n IHn]; first exact: polyseqMX. by rewrite exprSr mulrA polyseqMX -?nil_poly IHn. Qed.

Lemma

polyseqMXn

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" ]

[ "exprSr", "mulr1", "mulrA", "ncons", "nil_poly", "nz_p", "polyseqMX", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coefMXn n p i : (p * 'X^n)`_i = if i < n then 0 else p`_(i - n).

Proof. have [-> | /polyseqMXn->] := eqVneq p 0; last exact: nth_ncons. by rewrite mul0r !coef0 if_same. Qed.

Lemma

coefMXn

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", "eqVneq", "last", "mul0r", "nth_ncons", "polyseqMXn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_mulXn n p : p != 0 -> size (p * 'X^n) = (n + size p)%N.

Proof. elim: n p => [p p_neq0| n IH p p_neq0]; first by rewrite mulr1. by rewrite exprS mulrA IH -?size_poly_eq0 size_mulX // addnS. Qed.

Lemma

size_mulXn

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", "mulr1", "mulrA", "size", "size_mulX", "size_poly_eq0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coefXnM n p i : ('X^n * p)`_i = if i < n then 0 else p`_(i - n).

Proof. by rewrite -commr_polyXn coefMXn. Qed.

Lemma

coefXnM

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" ]

[ "coefMXn", "commr_polyXn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coef_sumMXn I (r : seq I) (P : pred I) (p : I -> R) (n : I -> nat) k : (\sum_(i <- r | P i) p i *: 'X^(n i))`_k = \sum_(i <- r | P i && (n i == k)) p i.

Proof. rewrite coef_sum big_mkcondr; apply: eq_bigr => i Pi. by rewrite coefZ coefXn mulr_natr mulrb eq_sym. Qed.

Lemma

coef_sumMXn

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_mkcondr", "coefXn", "coefZ", "coef_sum", "eq_bigr", "eq_sym", "mulr_natr", "mulrb", "nat", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

poly_def n E : \poly_(i < n) E i = \sum_(i < n) E i *: 'X^i.

Proof. by apply/polyP => i; rewrite coef_sumMXn coef_poly big_ord1_eq. Qed.

Lemma

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" ]

[ "apply", "big_ord1_eq", "coef_poly", "coef_sumMXn", "polyP" ]

Expansion of a polynomial as an indexed sum

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_poly n E1 E2 : (forall i, i < n -> E1 i = E2 i) -> poly n E1 = poly n E2 :> {poly R}.

Proof. by move=> E; rewrite !poly_def; apply: eq_bigr => i _; rewrite E. Qed.

Lemma

eq_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", "eq_bigr", "poly", "poly_def" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_rec s x

:= if s is a :: s' then horner_rec s' x * x + a else 0.

Fixpoint

horner_rec

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

horner p

:= horner_rec p.

Definition

horner

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_rec" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"p .[ x ]"

:= (horner p x) : ring_scope.

Notation

p .[ 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" ]

[ "horner" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner0 x : (0 : {poly R}).[x] = 0.

Proof. by rewrite /horner polyseq0. Qed.

Lemma

horner0

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", "poly", "polyseq0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerC c x : (c%:P).[x] = c.

Proof. by rewrite /horner polyseqC; case: eqP; rewrite /= ?simp. Qed.

Lemma

hornerC

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", "polyseqC", "simp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerX x : 'X.[x] = x.

Proof. by rewrite /horner polyseqX /= !simp. Qed.

Lemma

hornerX

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", "polyseqX", "simp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_cons p c x : (cons_poly c p).[x] = p.[x] * x + c.

Proof. rewrite /horner polyseq_cons; case: nilP => //= ->. by rewrite !simp -/(_.[x]) hornerC. Qed.

Lemma

horner_cons

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" ]

[ "cons_poly", "horner", "hornerC", "nilP", "polyseq_cons", "simp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_coef0 p : p.[0] = p`_0.

Proof. by rewrite /horner; case: (p : seq R) => //= c p'; rewrite !simp. Qed.

Lemma

horner_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" ]

[ "horner", "seq", "simp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerMXaddC p c x : (p * 'X + c%:P).[x] = p.[x] * x + c.

Proof. by rewrite -cons_poly_def horner_cons. Qed.

Lemma

hornerMXaddC

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" ]

[ "cons_poly_def", "horner_cons" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerMX p x : (p * 'X).[x] = p.[x] * x.

Proof. by rewrite -[p * 'X]addr0 hornerMXaddC addr0. Qed.

Lemma

hornerMX

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", "hornerMXaddC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_Poly s x : (Poly s).[x] = horner_rec s x.

Proof. by elim: s => [|a s /= <-]; rewrite (horner0, horner_cons). Qed.

Lemma

horner_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", "horner0", "horner_cons", "horner_rec" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_coef p x : p.[x] = \sum_(i < size p) p`_i * x ^+ i.

Proof. rewrite /horner. elim: {p}(p : seq R) => /= [|a s ->]; first by rewrite big_ord0. rewrite big_ord_recl simp addrC big_distrl /=. by congr (_ + _); apply: eq_bigr => i _; rewrite -mulrA exprSr. Qed.

Lemma

horner_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" ]

[ "addrC", "apply", "big_distrl", "big_ord0", "big_ord_recl", "eq_bigr", "exprSr", "horner", "mulrA", "seq", "simp", "size" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_coef_wide n p x : size p <= n -> p.[x] = \sum_(i < n) p`_i * x ^+ i.

Proof. move=> le_p_n. rewrite horner_coef (big_ord_widen n (fun i => p`_i * x ^+ i)) // big_mkcond. by apply: eq_bigr => i _; case: ltnP => // le_p_i; rewrite nth_default ?simp. Qed.

Lemma

horner_coef_wide

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_mkcond", "big_ord_widen", "eq_bigr", "horner_coef", "ltnP", "nth_default", "simp", "size" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_poly n E x : (\poly_(i < n) E i).[x] = \sum_(i < n) E i * x ^+ i.

Proof. rewrite (@horner_coef_wide n) ?size_poly //. by apply: eq_bigr => i _; rewrite coef_poly ltn_ord. Qed.

Lemma

horner_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", "coef_poly", "eq_bigr", "horner_coef_wide", "ltn_ord", "size_poly" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerD p q x : (p + q).[x] = p.[x] + q.[x].

Proof. rewrite [in LHS]/+%R /= unlock horner_poly; set m := maxn _ _. rewrite !(@horner_coef_wide m) ?leq_max ?leqnn ?orbT // -big_split /=. by apply: eq_bigr => i _; rewrite -mulrDl. Qed.

Lemma

hornerD

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", "eq_bigr", "horner_coef_wide", "horner_poly", "leq_max", "leqnn", "maxn", "mulrDl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerCM a p x : (a%:P * p).[x] = a * p.[x].

Proof. elim/poly_ind: p => [|p c IHp]; first by rewrite !(mulr0, horner0). by rewrite mulrDr mulrA -polyCM !hornerMXaddC IHp mulrDr mulrA. Qed.

Lemma

hornerCM

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" ]

[ "horner0", "hornerMXaddC", "mulr0", "mulrA", "mulrDr", "polyCM", "poly_ind" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerZ c p x : (c *: p).[x] = c * p.[x].

Proof. by rewrite -mul_polyC hornerCM. Qed.

Lemma

hornerZ

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" ]

[ "hornerCM", "mul_polyC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_eval (x : R)

:= horner^~ x.

Definition

horner_eval

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" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_evalE x p : horner_eval x p = p.[x].

Proof. by []. Qed.

Lemma

horner_evalE

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_eval" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_sum I (r : seq I) (P : pred I) F x : (\sum_(i <- r | P i) F i).[x] = \sum_(i <- r | P i) (F i).[x].

Proof. exact: (raddf_sum (horner_eval _)). Qed.

Lemma

horner_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" ]

[ "horner_eval", "raddf_sum", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerMn n p x : (p *+ n).[x] = p.[x] *+ n.

Proof. exact: (raddfMn (horner_eval _)). Qed.

Lemma

hornerMn

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_eval", "raddfMn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_coef p x

:= forall i, p`_i * x = x * p`_i.

Definition

comm_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" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_poly p x

:= x * p.[x] = p.[x] * x.

Definition

comm_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

comm_coef_poly p x : comm_coef p x -> comm_poly p x.

Proof. move=> cpx; rewrite /comm_poly !horner_coef big_distrl big_distrr /=. by apply: eq_bigr => i _; rewrite /= mulrA -cpx -!mulrA commrX. Qed.

Lemma

comm_coef_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_distrl", "big_distrr", "comm_coef", "comm_poly", "commrX", "eq_bigr", "horner_coef", "mulrA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_poly0 x : comm_poly 0 x.

Proof. by rewrite /comm_poly !horner0 !simp. Qed.

Lemma

comm_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" ]

[ "comm_poly", "horner0", "simp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_poly1 x : comm_poly 1 x.

Proof. by rewrite /comm_poly !hornerC !simp. Qed.

Lemma

comm_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" ]

[ "comm_poly", "hornerC", "simp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_polyX x : comm_poly 'X x.

Proof. by rewrite /comm_poly !hornerX. Qed.

Lemma

comm_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" ]

[ "comm_poly", "hornerX" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_polyD p q x: comm_poly p x -> comm_poly q x -> comm_poly (p + q) x.

Proof. by rewrite /comm_poly hornerD mulrDr mulrDl => -> ->. Qed.

Lemma

comm_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" ]

[ "comm_poly", "hornerD", "mulrDl", "mulrDr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_horner a b p : GRing.comm a b -> comm_coef p a -> GRing.comm a p.[b].

Proof. move=> cab cpa; rewrite horner_coef; apply: commr_sum => i _. by apply: commrM => //; apply: commrX. Qed.

Lemma

commr_horner

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", "comm", "comm_coef", "commrM", "commrX", "commr_sum", "horner_coef" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerM_comm p q x : comm_poly q x -> (p * q).[x] = p.[x] * q.[x].

Proof. move=> comm_qx. elim/poly_ind: p => [|p c IHp]; first by rewrite !(simp, horner0). rewrite mulrDl hornerD hornerCM -mulrA -commr_polyX mulrA hornerMX. by rewrite {}IHp -mulrA -comm_qx mulrA -mulrDl hornerMXaddC. Qed.

Lemma

hornerM_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_poly", "commr_polyX", "horner0", "hornerCM", "hornerD", "hornerMX", "hornerMXaddC", "mulrA", "mulrDl", "poly_ind", "simp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_polyM p q x: comm_poly p x -> comm_poly q x -> comm_poly (p * q) x.

Proof. by move=> px qx; rewrite /comm_poly hornerM_comm// mulrA px -mulrA qx mulrA. Qed.

Lemma

comm_polyM

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", "hornerM_comm", "mulrA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_exp_comm p x n : comm_poly p x -> (p ^+ n).[x] = p.[x] ^+ n.

Proof. move=> comm_px; elim: n => [|n IHn]; first by rewrite hornerC. by rewrite !exprSr -IHn hornerM_comm. Qed.

Lemma

horner_exp_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_poly", "exprSr", "hornerC", "hornerM_comm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_poly_exp p n x: comm_poly p x -> comm_poly (p ^+ n) x.

Proof. by move=> px; rewrite /comm_poly !horner_exp_comm// commrX. Qed.

Lemma

comm_poly_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" ]

[ "comm_poly", "commrX", "horner_exp_comm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerXn x n : ('X^n).[x] = x ^+ n.

Proof. by rewrite horner_exp_comm /comm_poly hornerX. Qed.

Lemma

hornerXn

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", "hornerX", "horner_exp_comm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_pred S

:= fun p : {poly R} => all (mem S) p.

Definition

polyOver_pred

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" ]

[ "all", "poly" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver S

:= [qualify a p | polyOver_pred S p].

Definition

polyOver

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" ]

[ "polyOver_pred" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverS (S1 S2 : {pred R}) : {subset S1 <= S2} -> {subset polyOver S1 <= polyOver S2}.

Proof. by move=> sS12 p /(all_nthP 0)S1p; apply/(all_nthP 0)=> i /S1p; apply: sS12. Qed.

Lemma

polyOverS

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" ]

[ "S1", "S2", "all_nthP", "apply", "polyOver" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver0 S : 0 \is a polyOver S.

Proof. by rewrite qualifE /= polyseq0. Qed.

Lemma

polyOver0

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" ]

[ "polyOver", "polyseq0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_poly S n E : (forall i, i < n -> E i \in S) -> \poly_(i < n) E i \is a polyOver S.

Proof. move=> S_E; apply/(all_nthP 0)=> i lt_i_p /=; rewrite coef_poly. by case: ifP => [/S_E// | /idP[]]; apply: leq_trans lt_i_p (size_poly n E). Qed.

Lemma

polyOver_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" ]

[ "all_nthP", "apply", "coef_poly", "leq_trans", "polyOver", "size_poly" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverP {p} : reflect (forall i, p`_i \in S) (p \in polyOver S).

Proof. apply: (iffP (all_nthP 0)) => [Sp i | Sp i _]; last exact: Sp. by have [/Sp // | /(nth_default 0)->] := ltnP i (size p); apply: rpred0. Qed.

Lemma

polyOverP

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" ]

[ "all_nthP", "apply", "last", "ltnP", "nth_default", "polyOver", "rpred0", "size" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverC c : (c%:P \in polyOver S) = (c \in S).

Proof. by rewrite qualifE /= polyseqC; case: eqP => [->|] /=; rewrite ?andbT ?rpred0. Qed.

Lemma

polyOverC

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" ]

[ "polyOver", "polyseqC", "rpred0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_nmod_closed : nmod_closed (polyOver S).

Proof. split=> [|p q Sp Sq]; first exact: polyOver0. by apply/polyOverP=> i; rewrite coefD rpredD ?(polyOverP _). Qed.

Fact

polyOver_nmod_closed

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", "nmod_closed", "polyOver", "polyOver0", "polyOverP", "rpredD", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_mulr_2closed : GRing.mulr_2closed (polyOver S).

Proof. move=> p q /polyOverP Sp /polyOverP Sq; apply/polyOverP=> i. by rewrite coefM rpred_sum // => j _; rewrite rpredM. Qed.

Lemma

polyOver_mulr_2closed

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", "mulr_2closed", "polyOver", "polyOverP", "rpredM", "rpred_sum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_mul1_closed : 1 \in polyOver S.

Proof. by rewrite polyOverC rpred1. Qed.

Fact

polyOver_mul1_closed

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" ]

[ "polyOver", "polyOverC", "rpred1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverZ : {in S & polyOver S, forall c p, c *: p \is a polyOver S}.

Proof. by move=> c p Sc /polyOverP Sp; apply/polyOverP=> i; rewrite coefZ rpredM ?Sp. Qed.

Lemma

polyOverZ

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", "polyOver", "polyOverP", "rpredM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverX : 'X \in polyOver S.

Proof. by rewrite qualifE /= polyseqX /= rpred0 rpred1. Qed.

Lemma

polyOverX

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" ]

[ "polyOver", "polyseqX", "rpred0", "rpred1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverXn n : 'X^n \in polyOver S.

Proof. by rewrite rpredX// polyOverX. Qed.

Lemma

polyOverXn

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" ]

[ "polyOver", "polyOverX", "rpredX" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rpred_horner : {in polyOver S & S, forall p x, p.[x] \in S}.

Proof. move=> p x /polyOverP Sp Sx; rewrite horner_coef rpred_sum // => i _. by rewrite rpredM ?rpredX. Qed.

Lemma

rpred_horner

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_coef", "polyOver", "polyOverP", "rpredM", "rpredX", "rpred_sum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deriv p

:= \poly_(i < (size p).-1) (p`_i.+1 *+ i.+1).

Definition

deriv

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" ]

Single derivative.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"a ^` ()"

:= (deriv a).

Notation

a ^` ()

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" ]

[ "deriv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coef_deriv p i : p^`()`_i = p`_i.+1 *+ i.+1.

Proof. rewrite coef_poly -subn1 ltn_subRL. by case: leqP => // /(nth_default 0) ->; rewrite mul0rn. Qed.

Lemma

coef_deriv

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", "ltn_subRL", "mul0rn", "nth_default", "subn1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_deriv (ringS : semiringClosed R) : {in polyOver ringS, forall p, p^`() \is a polyOver ringS}.

Proof. by move=> p /polyOverP Kp; apply/polyOverP=> i; rewrite coef_deriv rpredMn ?Kp. Qed.

Lemma

polyOver_deriv

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_deriv", "polyOver", "polyOverP", "rpredMn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivC c : c%:P^`() = 0.

Proof. by apply/polyP=> i; rewrite coef_deriv coef0 coefC mul0rn. Qed.

Lemma

derivC

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", "coef0", "coefC", "coef_deriv", "mul0rn", "polyP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivX : ('X)^`() = 1.

Proof. by apply/polyP=> [[|i]]; rewrite coef_deriv coef1 coefX ?mul0rn. Qed.

Lemma

derivX

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", "coef1", "coefX", "coef_deriv", "mul0rn", "polyP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivXn n : ('X^n)^`() = 'X^(n.-1) *+ n.

Proof. case: n => [|n]; first exact: derivC. apply/polyP=> i; rewrite coef_deriv coefMn !coefXn eqSS. by case: eqP => [-> // | _]; rewrite !mul0rn. Qed.

Lemma

derivXn

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", "coefMn", "coefXn", "coef_deriv", "derivC", "eqSS", "mul0rn", "polyP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deriv_is_semilinear : semilinear deriv.

Proof. split=> [k p|p q]; apply/polyP => i. by rewrite !(coef_deriv, coefZ) mulrnAr. by rewrite !(coef_deriv, coefD) mulrnDl. Qed.

Fact

deriv_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" ]

[ "apply", "coefD", "coefZ", "coef_deriv", "deriv", "mulrnAr", "mulrnDl", "polyP", "semilinear", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deriv0 : 0^`() = 0.

Proof. exact: linear0. Qed.

Lemma

deriv0

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" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivD : {morph deriv : p q / p + q}.

Proof. exact: linearD. Qed.

Lemma

derivD

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" ]

[ "deriv", "linearD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivMn n p : (p *+ n)^`() = p^`() *+ n.

Proof. exact: linearMn. Qed.

Lemma

derivMn

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" ]

[ "linearMn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivZ c p : (c *: p)^`() = c *: p^`().

Proof. exact: linearZ. Qed.

Lemma

derivZ

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" ]

[ "linearZ" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deriv_mulC c p : (c%:P * p)^`() = c%:P * p^`().

Proof. by rewrite !mul_polyC derivZ. Qed.

Lemma

deriv_mulC

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" ]

[ "derivZ", "mul_polyC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivMXaddC p c : (p * 'X + c%:P)^`() = p + p^`() * 'X.

Proof. apply/polyP=> i; rewrite raddfD /= derivC addr0 coefD !(coefMX, coef_deriv). by case: i; rewrite ?addr0. Qed.

Lemma

derivMXaddC

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", "coefD", "coefMX", "coef_deriv", "derivC", "polyP", "raddfD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivM p q : (p * q)^`() = p^`() * q + p * q^`().

Proof. elim/poly_ind: p => [|p b IHp]; first by rewrite !(mul0r, add0r, derivC). rewrite mulrDl -mulrA -commr_polyX mulrA -[_ * 'X]addr0 raddfD /= !derivMXaddC. by rewrite deriv_mulC IHp !mulrDl -!mulrA !commr_polyX !addrA. Qed.

Lemma

derivM

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", "addrA", "commr_polyX", "derivC", "derivMXaddC", "deriv_mulC", "mul0r", "mulrA", "mulrDl", "poly_ind", "raddfD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivn n p

:= iter n deriv p.

Definition

derivn

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" ]

[ "deriv", "iter" ]

Iterated derivative.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"a ^` ( n )"

:= (derivn n a) : ring_scope.

Notation

a ^` ( n )

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" ]

[ "derivn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivn0 p : p^`(0) = p.

Proof. by []. Qed.

Lemma

derivn0

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

derivn1 p : p^`(1) = p^`().

Proof. by []. Qed.

Lemma

derivn1

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

derivnS p n : p^`(n.+1) = p^`(n)^`().

Proof. by []. Qed.

Lemma

derivnS

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

derivSn p n : p^`(n.+1) = p^`()^`(n).

Proof. exact: iterSr. Qed.

Lemma

derivSn

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" ]

[ "iterSr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coef_derivn n p i : p^`(n)`_i = p`_(n + i) *+ (n + i) ^_ n.

Proof. elim: n i => [|n IHn] i; first by rewrite ffactn0 mulr1n. by rewrite derivnS coef_deriv IHn -mulrnA ffactnSr addSnnS addKn. Qed.

Lemma

coef_derivn

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", "addSnnS", "coef_deriv", "derivnS", "ffactn0", "ffactnSr", "mulr1n", "mulrnA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_derivn (ringS : semiringClosed R) : {in polyOver ringS, forall p n, p^`(n) \is a polyOver ringS}.

Proof. move=> p /polyOverP Kp /= n; apply/polyOverP=> i. by rewrite coef_derivn rpredMn. Qed.

Lemma

polyOver_derivn

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_derivn", "polyOver", "polyOverP", "rpredMn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivn_is_semilinear n : semilinear (derivn n).

Proof. by elim: n => // n IHn; split=> [a p|p q]; rewrite derivnS IHn semilinearPZ. Qed.

Fact

derivn_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" ]

[ "derivn", "derivnS", "semilinear", "semilinearPZ", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnC c n : c%:P^`(n) = if n == 0 then c%:P else 0.

Proof. by case: n => // n; rewrite derivSn derivC linear0. Qed.

Lemma

derivnC

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" ]

[ "derivC", "derivSn", "linear0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnD n : {morph derivn n : p q / p + q}.

Proof. exact: linearD. Qed.

Lemma

derivnD

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" ]

[ "derivn", "linearD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnMn n m p : (p *+ m)^`(n) = p^`(n) *+ m.

Proof. exact: linearMn. Qed.

Lemma

derivnMn

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" ]

[ "linearMn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d