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
polyOverXnsubC n c : ('X^n - c%:P \is a polyOver S) = (c \in S).	Proof. by rewrite rpredBl ?polyOverXn// ?polyOverC. Qed.	Lemma	polyOverXnsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "polyOverXn", "rpredBl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deriv_is_linear : linear (@deriv R).	Proof. exact: linearP. Qed.	Fact	deriv_is_linear	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "deriv", "linear", "linearP" ]	Single derivative.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivN : {morph deriv : p / - p}.	Proof. exact: linearN. Qed.	Lemma	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", "linearN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivB : {morph deriv : p q / p - q}.	Proof. exact: linearB. Qed.	Lemma	derivB	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "linearB" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivXsubC (a : R) : ('X - a%:P)^`() = 1.	Proof. by rewrite derivB derivX derivC subr0. Qed.	Lemma	derivXsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "derivB", "derivC", "derivX", "subr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivMNn n p : (p - n)^`() = p^`() - n.	Proof. exact: linearMNn. Qed.	Lemma	derivMNn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linearMNn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivE	:= Eval lazy beta delta [morphism_2 morphism_1] in (derivZ, deriv_mulC, derivC, derivX, derivMXaddC, derivXsubC, derivM, derivB, derivD, derivN, derivXn, derivM, derivMn).	Definition	derivE	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "delta", "derivB", "derivC", "derivD", "derivM", "derivMXaddC", "derivMn", "derivN", "derivX", "derivXn", "derivXsubC", "derivZ", "deriv_mulC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivn_is_linear n : linear (@derivn R n).	Proof. exact: linearP. Qed.	Fact	derivn_is_linear	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "derivn", "linear", "linearP" ]	Iterated derivative.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivn_is_linear n : linear (@nderivn R n).	Proof. exact: linearP. Qed.	Fact	nderivn_is_linear	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linear", "linearP", "nderivn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivnB n : {morph derivn n : p q / p - q}.	Proof. exact: linearB. Qed.	Lemma	derivnB	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "linearB" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivnMNn n m p : (p - m)^`(n) = p^`(n) - m.	Proof. exact: linearMNn. Qed.	Lemma	derivnMNn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linearMNn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivnN n : {morph derivn n : p / - p}.	Proof. exact: linearN. Qed.	Lemma	derivnN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "linearN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnB n : {morph nderivn n : p q / p - q}.	Proof. exact: linearB. Qed.	Lemma	nderivnB	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linearB", "nderivn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnMNn n m p : (p - m)^`N(n) = p^`N(n) - m.	Proof. exact: linearMNn. Qed.	Lemma	nderivnMNn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linearMNn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnN n : {morph nderivn n : p / - p}.	Proof. exact: linearN. Qed.	Lemma	nderivnN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linearN", "nderivn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicXsubC c : 'X - c%:P \is monic.	Proof. exact/eqP/lead_coefXsubC. Qed.	Lemma	monicXsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_coefXsubC", "monic" ]	Monic predicate	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_prod_XsubC I rI (P : pred I) (F : I -> R) : \prod_(i <- rI \| P i) ('X - (F i)%:P) \is monic.	Proof. by apply: monic_prod => i _; apply: monicXsubC. Qed.	Lemma	monic_prod_XsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "monic", "monicXsubC", "monic_prod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_prod_XsubC I rI (P : pred I) (F : I -> R) : lead_coef (\prod_(i <- rI \| P i) ('X - (F i)%:P)) = 1.	Proof. exact/eqP/monic_prod_XsubC. Qed.	Lemma	lead_coef_prod_XsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "lead_coef", "monic_prod_XsubC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_prod_XsubC I rI (F : I -> R) : size (\prod_(i <- rI) ('X - (F i)%:P)) = (size rI).+1.	Proof. elim: rI => [\|i r /= <-]; rewrite ?big_nil ?size_poly1 // big_cons. rewrite size_monicM ?monicXsubC ?monic_neq0 ?monic_prod_XsubC //. by rewrite size_XsubC. Qed.	Lemma	size_prod_XsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "big_cons", "big_nil", "monicXsubC", "monic_neq0", "monic_prod_XsubC", "size", "size_XsubC", "size_monicM", "size_poly1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_exp_XsubC n a : size (('X - a%:P) ^+ n) = n.+1.	Proof. rewrite -[n]card_ord -prodr_const -big_filter size_prod_XsubC. by have [e _ _ [_ ->]] := big_enumP. Qed.	Lemma	size_exp_XsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "big_enumP", "big_filter", "card_ord", "prodr_const", "size", "size_prod_XsubC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicXnsubC n c : 0 < n -> 'X^n - c%:P \is monic.	Proof. by move=> n_gt0; rewrite monicE lead_coefXnsubC. Qed.	Lemma	monicXnsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_coefXnsubC", "monic", "monicE", "n_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_Xn_sub_1 n : n > 0 -> 'X^n - 1 \is @monic R.	Proof. exact/monicXnsubC. Qed.	Lemma	monic_Xn_sub_1	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "monic", "monicXnsubC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lreg_lead p : GRing.lreg (lead_coef p) -> GRing.lreg p.	Proof. move/mulrI_eq0=> reg_p; apply: mulrI0_lreg => q; apply/contra_eq => nz_q. by rewrite -lead_coef_eq0 lead_coef_proper_mul reg_p lead_coef_eq0. Qed.	Lemma	lreg_lead	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "contra_eq", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "lreg", "mulrI0_lreg", "mulrI_eq0" ]	Some facts about regular elements.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rreg_lead p : GRing.rreg (lead_coef p) -> GRing.rreg p.	Proof. move/mulIr_eq0=> reg_p; apply: mulIr0_rreg => q; apply/contra_eq => nz_q. by rewrite -lead_coef_eq0 lead_coef_proper_mul reg_p lead_coef_eq0. Qed.	Lemma	rreg_lead	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "contra_eq", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "mulIr0_rreg", "mulIr_eq0", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_lreg p : p \is monic -> GRing.lreg p.	Proof. by move=> /eqP lp1; apply/lreg_lead; rewrite lp1; apply/lreg1. Qed.	Lemma	monic_lreg	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "lreg", "lreg1", "lreg_lead", "monic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_rreg p : p \is monic -> GRing.rreg p.	Proof. by move=> /eqP lp1; apply/rreg_lead; rewrite lp1; apply/rreg1. Qed.	Lemma	monic_rreg	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "monic", "rreg", "rreg1", "rreg_lead" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rootN p x : root (- p) x = root p x.	Proof. by rewrite rootE hornerN oppr_eq0. Qed.	Lemma	rootN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "hornerN", "oppr_eq0", "root", "rootE" ]	Roots of polynomials	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_XsubC a x : root ('X - a%:P) x = (x == a).	Proof. by rewrite rootE hornerXsubC subr_eq0. Qed.	Lemma	root_XsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "hornerXsubC", "root", "rootE", "subr_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_XaddC a x : root ('X + a%:P) x = (x == - a).	Proof. by rewrite -root_XsubC rmorphN opprK. Qed.	Lemma	root_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" ]	[ "opprK", "rmorphN", "root", "root_XsubC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
factor_theorem p a : reflect (exists q, p = q * ('X - a%:P)) (root p a).	Proof. apply: (iffP eqP) => [pa0 \| [q ->]]; last first. by rewrite hornerM_comm /comm_poly hornerXsubC subrr ?simp. exists (\poly_(i < size p) horner_rec (drop i.+1 p) a). apply/polyP=> i; rewrite mulrBr coefB coefMX coefMC !coef_poly. apply: canRL (addrK _) _; rewrite addrC; have [le_p_i \| lt_i_p] := leqP. rewrite...	Theorem	factor_theorem	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "addrK", "apply", "coefB", "coefMC", "coefMX", "coef_poly", "comm_poly", "drop", "drop1", "drop_nth", "drop_oversize", "horner", "hornerM_comm", "hornerXsubC", "horner_rec", "last", "leqP", "leqSpred", "leq_trans", "ltnW", "mulrBr", "nth_default", "polyP", "r...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
multiplicity_XsubC p a : {m \| exists2 q, (p != 0) ==> ~~ root q a & p = q * ('X - a%:P) ^+ m}.	Proof. have [n le_p_n] := ubnP (size p); elim: n => // n IHn in p le_p_n *. have [-> \| nz_p /=] := eqVneq p 0; first by exists 0, 0; rewrite ?mul0r. have [/sig_eqW[p1 Dp] \| nz_pa] := altP (factor_theorem p a); last first. by exists 0%N, p; rewrite ?mulr1. have nz_p1: p1 != 0 by apply: contraNneq nz_p => p1_0; rewrite...	Lemma	multiplicity_XsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addn2", "apply", "contraNneq", "eqVneq", "exprSr", "factor_theorem", "last", "monicXsubC", "mul0r", "mulr1", "mulrA", "nz_p", "root", "sig2_eqW", "sig_eqW", "size", "size_Mmonic", "size_XsubC", "ubnP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_of_unity n : pred R	:= root ('X^n - 1).	Definition	root_of_unity	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "root" ]	Roots of unity.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"n .-unity_root"	:= (root_of_unity n) : ring_scope.	Notation	n .-unity_root	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "root_of_unity" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unity_rootE n z : n.-unity_root z = (z ^+ n == 1).	Proof. by rewrite /root_of_unity rootE hornerD hornerN hornerXn hornerC subr_eq0. Qed.	Lemma	unity_rootE	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "hornerC", "hornerD", "hornerN", "hornerXn", "rootE", "root_of_unity", "subr_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unity_rootP n z : reflect (z ^+ n = 1) (n.-unity_root z).	Proof. by rewrite unity_rootE; apply: eqP. Qed.	Lemma	unity_rootP	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "unity_rootE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primitive_root_of_unity n z	:= (n > 0) && [forall i : 'I_n, i.+1.-unity_root z == (i.+1 == n)].	Definition	primitive_root_of_unity	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
"n .-primitive_root"	:= (primitive_root_of_unity n) : ring_scope.	Notation	n .-primitive_root	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "primitive_root_of_unity" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_order_exists n z : n > 0 -> z ^+ n = 1 -> {m \| m.-primitive_root z & (m %\| n)}.	Proof. move=> n_gt0 zn1. have: exists m, (m > 0) && (z ^+ m == 1) by exists n; rewrite n_gt0 /= zn1. case/ex_minnP=> m /andP[m_gt0 /eqP zm1] m_min. exists m. apply/andP; split=> //; apply/eqfunP=> [[i]] /=. rewrite leq_eqVlt unity_rootE. case: eqP => [-> _ \| _]; first by rewrite zm1 eqxx. by apply: contraTF => ...	Lemma	prim_order_exists	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "eqfunP", "eqxx", "ex_minnP", "expr_mod", "leqNgt", "leq_eqVlt", "lt0n", "ltn_mod", "n_gt0", "split", "unity_rootE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_z : n.-primitive_root z.		Hypothesis	prim_z	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
prim_order_gt0 : n > 0.	Proof. by case/andP: prim_z. Qed.	Lemma	prim_order_gt0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "prim_z" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
n_gt0	:= prim_order_gt0.	Let	n_gt0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "prim_order_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_expr_order : z ^+ n = 1.	Proof. case/andP: prim_z => _; rewrite -(prednK n_gt0) => /forallP/(_ ord_max). by rewrite unity_rootE eqxx eqb_id => /eqP. Qed.	Lemma	prim_expr_order	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "eqb_id", "eqxx", "forallP", "n_gt0", "ord_max", "prednK", "prim_z", "unity_rootE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_expr_mod i : z ^+ (i %% n) = z ^+ i.	Proof. exact: expr_mod prim_expr_order. Qed.	Lemma	prim_expr_mod	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "expr_mod", "prim_expr_order" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_order_dvd i : (n %\| i) = (z ^+ i == 1).	Proof. move: n_gt0; rewrite -prim_expr_mod /dvdn -(ltn_mod i). case: {i}(i %% n)%N => [\|i] lt_i; first by rewrite !eqxx. case/andP: prim_z => _ /forallP/(_ (Ordinal (ltnW lt_i)))/eqP. by rewrite unity_rootE eqn_leq andbC leqNgt lt_i. Qed.	Lemma	prim_order_dvd	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "dvdn", "eqn_leq", "eqxx", "forallP", "leqNgt", "ltnW", "ltn_mod", "n_gt0", "prim_expr_mod", "prim_z", "unity_rootE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_prim_root_expr i j : (z ^+ i == z ^+ j) = (i == j %[mod n]).	Proof. wlog le_ji: i j / j <= i. move=> IH; case: (leqP j i) => [\|/ltnW] /IH //. by rewrite eq_sym (eq_sym (j %% n)%N). rewrite -{1}(subnKC le_ji) exprD -prim_expr_mod eqn_mod_dvd //. rewrite prim_order_dvd; apply/eqP/eqP=> [\|->]; last by rewrite mulr1. move/(congr1 ( *%R (z ^+ (n - j %% n)))); rewrite mulrA -exprD...	Lemma	eq_prim_root_expr	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_sym", "eqn_mod_dvd", "exprD", "last", "leqP", "ltnW", "ltn_mod", "mul1r", "mulr1", "mulrA", "prim_expr_mod", "prim_expr_order", "prim_order_dvd", "subnK", "subnKC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exp_prim_root k : (n %/ gcdn k n).-primitive_root (z ^+ k).	Proof. set d := gcdn k n; have d_gt0: (0 < d)%N by rewrite gcdn_gt0 orbC n_gt0. have [d_dv_k d_dv_n]: (d %\| k /\ d %\| n)%N by rewrite dvdn_gcdl dvdn_gcdr. set q := (n %/ d)%N; rewrite /q.-primitive_root ltn_divRL // n_gt0. apply/forallP=> i; rewrite unity_rootE -exprM -prim_order_dvd. rewrite -(divnK d_dv_n) -/q -(divn...	Lemma	exp_prim_root	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "Gauss_dvdr", "apply", "coprime", "d_gt0", "divnK", "dvdn_gcdl", "dvdn_gcdr", "dvdn_leq", "dvdn_mull", "dvdn_pmul2r", "eqn_leq", "eqn_pmul2r", "exprM", "forallP", "gcdn", "gcdnC", "gcdn_gt0", "last", "ltn_divRL", "ltn_ord", "mul1n", "mulnAC", "muln_gcdl", "n_gt0", "pr...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_prim_root m : (m %\| n)%N -> m.-primitive_root (z ^+ (n %/ m)).	Proof. set k := (n %/ m)%N => m_dv_n; rewrite -{1}(mulKn m n_gt0) -divnA // -/k. by rewrite -{1}(@gcdn_idPl k n _) ?exp_prim_root // -(divnK m_dv_n) dvdn_mulr. Qed.	Lemma	dvdn_prim_root	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "divnA", "divnK", "dvdn_mulr", "exp_prim_root", "gcdn_idPl", "mulKn", "n_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_root_eq0 : (z == 0) = (n == 0%N).	Proof. rewrite gtn_eqF//; apply/eqP => z0; have /esym/eqP := prim_expr_order. by rewrite z0 expr0n gtn_eqF//= oner_eq0. Qed.	Lemma	prim_root_eq0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "expr0n", "gtn_eqF", "oner_eq0", "prim_expr_order", "z0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prim_root_exp_coprime n z k : n.-primitive_root z -> n.-primitive_root (z ^+ k) = coprime k n.	Proof. move=> prim_z; have n_gt0 := prim_order_gt0 prim_z. apply/idP/idP=> [prim_zk \| co_k_n]. set d := gcdn k n; have dv_d_n: (d %\| n)%N := dvdn_gcdr _ _. rewrite /coprime -/d -(eqn_pmul2r n_gt0) mul1n -{2}(gcdnMl n d). rewrite -{2}(divnK dv_d_n) (mulnC _ d) -muln_gcdr (gcdn_idPr _) //. rewrite (prim_order_dvd...	Lemma	prim_root_exp_coprime	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "Gauss_dvdr", "apply", "coprime", "coprime_sym", "divnK", "dvdn_gcdr", "dvdn_mulr", "eqn_dvd", "eqn_pmul2r", "expr1n", "exprAC", "exprM", "gcdn", "gcdnMl", "gcdn_idPr", "mul1n", "mulnC", "muln_divCA_gcd", "muln_gcdr", "n_gt0", "prim_expr_order", "prim_order_dvd", "prim_or...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_opp	:= size_polyN (only parsing).	Notation	size_opp	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_polyN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly (p : {poly aR})	:= \poly_(i < size p) f p`_i.	Definition	map_poly	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "poly", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyE p : map_poly p = Poly (map f p).	Proof. rewrite /map_poly unlock; congr Poly. apply: (@eq_from_nth _ 0); rewrite size_mkseq ?size_map // => i lt_i_p. by rewrite [RHS](nth_map 0) ?nth_mkseq. Qed.	Lemma	map_polyE	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "Poly", "apply", "eq_from_nth", "map", "map_poly", "nth_map", "nth_mkseq", "size_map", "size_mkseq" ]	instance of size_poly.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_rmorph u	:= forall x, GRing.comm u (f x).	Definition	commr_rmorph	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_morph u & commr_rmorph u	:= fun p => (map_poly p).[u].	Definition	horner_morph	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_rmorph", "map_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p ^f"	:= (map_poly f p) : ring_scope.	Notation	p ^f	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "map_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly0 : 0^f = 0.	Proof. by rewrite map_polyE polyseq0. Qed.	Lemma	map_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" ]	[ "map_polyE", "polyseq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_map_poly (g : aR -> rR) : f =1 g -> map_poly f =1 map_poly g.	Proof. by move=> eq_fg p; rewrite !map_polyE (eq_map eq_fg). Qed.	Lemma	eq_map_poly	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "eq_map", "map_poly", "map_polyE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_id g (p : {poly iR}) : {in (p : seq iR), g =1 id} -> map_poly g p = p.	Proof. by move=> g_id; rewrite map_polyE map_id_in ?polyseqK. Qed.	Lemma	map_poly_id	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "id", "map_id_in", "map_poly", "map_polyE", "poly", "polyseqK", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_map_id0 p i : f 0 = 0 -> (p^f)`_i = f p`_i.	Proof. by move=> f0; rewrite coef_poly; case: ltnP => // le_p_i; rewrite nth_default. Qed.	Lemma	coef_map_id0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "ltnP", "nth_default" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_Poly_id0 s : f 0 = 0 -> (Poly s)^f = Poly (map f s).	Proof. move=> f0; apply/polyP=> j; rewrite coef_map_id0 ?coef_Poly //. have [/(nth_map 0 0)->// \| le_s_j] := ltnP j (size s). by rewrite !nth_default ?size_map. Qed.	Lemma	map_Poly_id0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "apply", "coef_Poly", "coef_map_id0", "ltnP", "map", "nth_default", "nth_map", "polyP", "size", "size_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_comp_id0 (g : iR -> aR) p : f 0 = 0 -> map_poly (f \o g) p = (map_poly g p)^f.	Proof. by move=> f0; rewrite map_polyE map_comp -map_Poly_id0 -?map_polyE. Qed.	Lemma	map_poly_comp_id0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "map_Poly_id0", "map_comp", "map_poly", "map_polyE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_map_poly_id0 p : f (lead_coef p) != 0 -> size p^f = size p.	Proof. by move=> nz_fp; apply: size_poly_eq. Qed.	Lemma	size_map_poly_id0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "lead_coef", "size", "size_poly_eq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_eq0_id0 p : f (lead_coef p) != 0 -> (p^f == 0) = (p == 0).	Proof. by rewrite -!size_poly_eq0 => /size_map_poly_id0->. Qed.	Lemma	map_poly_eq0_id0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "size_map_poly_id0", "size_poly_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_map_id0 p : f 0 = 0 -> f (lead_coef p) != 0 -> lead_coef p^f = f (lead_coef p).	Proof. by move=> f0 nz_fp; rewrite lead_coefE coef_map_id0 ?size_map_poly_id0. Qed.	Lemma	lead_coef_map_id0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_map_id0", "lead_coef", "lead_coefE", "size_map_poly_id0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(inj_f : injective f) (f_0 : f 0 = 0).		Hypotheses	inj_f	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
size_map_inj_poly p : size p^f = size p.	Proof. have [-> \| nz_p] := eqVneq p 0; first by rewrite map_poly0 !size_poly0. by rewrite size_map_poly_id0 // -f_0 (inj_eq inj_f) lead_coef_eq0. Qed.	Lemma	size_map_inj_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" ]	[ "eqVneq", "inj_eq", "inj_f", "lead_coef_eq0", "map_poly0", "nz_p", "size", "size_map_poly_id0", "size_poly0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_inj_poly : injective (map_poly f).	Proof. move=> p q /polyP eq_pq; apply/polyP=> i; apply: inj_f. by rewrite -!coef_map_id0 ?eq_pq. Qed.	Lemma	map_inj_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_map_id0", "inj_f", "map_poly", "polyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_map_inj p : lead_coef p^f = f (lead_coef p).	Proof. by rewrite !lead_coefE size_map_inj_poly coef_map_id0. Qed.	Lemma	lead_coef_map_inj	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_map_id0", "lead_coef", "lead_coefE", "size_map_inj_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyK (f : aR -> rR) g : cancel g f -> f 0 = 0 -> cancel (map_poly g) (map_poly f).	Proof. by move=> gK f_0 p; rewrite /= -map_poly_comp_id0 ?map_poly_id // => x _ //=. Qed.	Lemma	map_polyK	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "gK", "map_poly", "map_poly_comp_id0", "map_poly_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_map_poly_id0 (f g : aR -> rR) (S : addrClosed aR) : f 0 = 0 -> g 0 = 0 -> {in S, f =1 g} -> {in polyOver S, map_poly f =1 map_poly g}.	Proof. move=> f0 g0 eq_fg p pP; apply/polyP => i. by rewrite !coef_map_id0// eq_fg// (polyOverP _). Qed.	Lemma	eq_in_map_poly_id0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addrClosed", "apply", "coef_map_id0", "map_poly", "pP", "polyOver", "polyOverP", "polyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_map_poly (f g : {additive aR -> rR}) (S : addrClosed aR) : {in S, f =1 g} -> {in polyOver S, map_poly f =1 map_poly g}.	Proof. by move=> /eq_in_map_poly_id0; apply; rewrite //?raddf0. Qed.	Lemma	eq_in_map_poly	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "additive", "addrClosed", "apply", "eq_in_map_poly_id0", "map_poly", "polyOver", "raddf0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_map p i : p^f`_i = f p`_i.	Proof. exact: coef_map_id0 (raddf0 f). Qed.	Lemma	coef_map	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_map_id0", "raddf0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_Poly s : (Poly s)^f = Poly (map f s).	Proof. exact: map_Poly_id0 (raddf0 f). Qed.	Lemma	map_Poly	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "Poly", "map", "map_Poly_id0", "raddf0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_comp (g : iR -> aR) p : map_poly (f \o g) p = map_poly f (map_poly g p).	Proof. exact: map_poly_comp_id0 (raddf0 f). Qed.	Lemma	map_poly_comp	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "map_poly", "map_poly_comp_id0", "raddf0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_is_nmod_morphism : nmod_morphism (map_poly f).	Proof. split=> [\|p q]; apply/polyP => i; first by rewrite coef_map !coef0 raddf0. by rewrite !(coef_map, coefD) raddfD. Qed.	Fact	map_poly_is_nmod_morphism	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coef0", "coefD", "coef_map", "map_poly", "nmod_morphism", "polyP", "raddf0", "raddfD", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyC a : (a%:P)^f = (f a)%:P.	Proof. by apply/polyP=> i; rewrite !(coef_map, coefC) -!mulrb raddfMn. Qed.	Lemma	map_polyC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coefC", "coef_map", "mulrb", "polyP", "raddfMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_map_eq p : f (lead_coef p) != 0 -> lead_coef p^f = f (lead_coef p).	Proof. exact: lead_coef_map_id0 (raddf0 f). Qed.	Lemma	lead_coef_map_eq	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "lead_coef", "lead_coef_map_id0", "raddf0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_is_monoid_morphism : monoid_morphism (map_poly f).	Proof. split=> [\|p q]; apply/polyP=> i. by rewrite !(coef_map, coef1) /= rmorph_nat. rewrite coef_map /= !coefM /= !rmorph_sum; apply: eq_bigr => j _. by rewrite !coef_map rmorphM. Qed.	Fact	map_poly_is_monoid_morphism	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coef1", "coefM", "coef_map", "eq_bigr", "map_poly", "monoid_morphism", "polyP", "rmorphM", "rmorph_nat", "rmorph_sum", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_is_multiplicative	:= (fun g => (g.2, g.1)) map_poly_is_monoid_morphism.	Definition	map_poly_is_multiplicative	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "map_poly_is_monoid_morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyZ c p : (c : p)^f = f c : p^f.	Proof. by apply/polyP=> i; rewrite !(coef_map, coefZ) /= rmorphM. Qed.	Lemma	map_polyZ	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coefZ", "coef_map", "polyP", "rmorphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyX : ('X)^f = 'X.	Proof. by apply/polyP=> i; rewrite coef_map !coefX /= rmorph_nat. Qed.	Lemma	map_polyX	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coefX", "coef_map", "polyP", "rmorph_nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyXn n : ('X^n)^f = 'X^n.	Proof. by rewrite rmorphXn /= map_polyX. Qed.	Lemma	map_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" ]	[ "map_polyX", "rmorphXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyXaddC x : ('X + x%:P)^f = 'X + (f x)%:P.	Proof. by rewrite raddfD/= map_polyX map_polyC. Qed.	Lemma	map_polyXaddC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "map_polyC", "map_polyX", "raddfD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_map p : p \is monic -> p^f \is monic.	Proof. move/monicP=> mon_p; rewrite monicE. by rewrite lead_coef_map_eq mon_p /= rmorph1 ?oner_neq0. Qed.	Lemma	monic_map	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_map_eq", "monic", "monicE", "monicP", "oner_neq0", "rmorph1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_map p x : p^f.[f x] = f p.[x].	Proof. elim/poly_ind: p => [\|p c IHp]; first by rewrite !(rmorph0, horner0). rewrite hornerMXaddC !rmorphD !rmorphM /=. by rewrite map_polyX map_polyC hornerMXaddC IHp. Qed.	Lemma	horner_map	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "map_polyC", "map_polyX", "poly_ind", "rmorph0", "rmorphD", "rmorphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_comm_poly p x : comm_poly p x -> comm_poly p^f (f x).	Proof. by rewrite /comm_poly horner_map -!rmorphM // => ->. Qed.	Lemma	map_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" ]	[ "comm_poly", "horner_map", "rmorphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_comm_coef p x : comm_coef p x -> comm_coef p^f (f x).	Proof. by move=> cpx i; rewrite coef_map -!rmorphM ?cpx. Qed.	Lemma	map_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" ]	[ "coef_map", "comm_coef", "rmorphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_root p x : root p x -> root p^f (f x).	Proof. by move/eqP=> px0; rewrite rootE horner_map px0 rmorph0. Qed.	Lemma	rmorph_root	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "horner_map", "rmorph0", "root", "rootE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cfu : commr_rmorph f u.		Hypothesis	cfu	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_rmorph" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_morphC a : horner_morph cfu a%:P = f a.	Proof. by rewrite /horner_morph map_polyC hornerC. Qed.	Lemma	horner_morphC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "cfu", "hornerC", "horner_morph", "map_polyC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_morphX : horner_morph cfu 'X = u.	Proof. by rewrite /horner_morph map_polyX hornerX. Qed.	Lemma	horner_morphX	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "cfu", "hornerX", "horner_morph", "map_polyX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_is_semilinear : semilinear_for (f \; *%R) (horner_morph cfu).	Proof. split=> [c p\|p q]; rewrite /horner_morph; first by rewrite linearZ hornerZ. by rewrite linearD hornerD. Qed.	Fact	horner_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" ]	[ "cfu", "hornerD", "hornerZ", "horner_morph", "linearD", "linearZ", "semilinear_for", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_is_monoid_morphism : monoid_morphism (horner_morph cfu).	Proof. split=> [\|p q]; first by rewrite /horner_morph rmorph1 hornerC. rewrite /horner_morph rmorphM /= hornerM_comm //. by apply: comm_coef_poly => i; rewrite coef_map cfu. Qed.	Fact	horner_is_monoid_morphism	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "cfu", "coef_map", "comm_coef_poly", "hornerC", "hornerM_comm", "horner_morph", "monoid_morphism", "rmorph1", "rmorphM", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
horner_is_multiplicative	:= (fun g => (g.2, g.1)) horner_is_monoid_morphism.	Definition	horner_is_multiplicative	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "horner_is_monoid_morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deriv_map p : p^f^`() = (p^`())^f.	Proof. by apply/polyP => i; rewrite !(coef_map, coef_deriv) //= rmorphMn. Qed.	Lemma	deriv_map	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "coef_map", "polyP", "rmorphMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivn_map p n : p^f^`(n) = (p^`(n))^f.	Proof. by apply/polyP => i; rewrite !(coef_map, coef_derivn) //= rmorphMn. Qed.	Lemma	derivn_map	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "coef_map", "polyP", "rmorphMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivn_map p n : p^f^`N(n) = (p^`N(n))^f.	Proof. by apply/polyP => i; rewrite !(coef_map, coef_nderivn) //= rmorphMn. Qed.	Lemma	nderivn_map	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_map", "coef_nderivn", "polyP", "rmorphMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_is_zmod_morphism : zmod_morphism (map_poly f).	Proof. exact: raddfB. Qed.	Fact	map_poly_is_zmod_morphism	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "map_poly", "raddfB", "zmod_morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_poly_is_additive : zmod_morphism (map_poly f).	Proof. exact: raddfB. Qed.	Fact	map_poly_is_additive	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "map_poly", "raddfB", "zmod_morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_polyXsubC x : ('X - x%:P)^f = 'X - (f x)%:P.	Proof. by rewrite raddfB/= map_polyX map_polyC. Qed.	Lemma	map_polyXsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "map_polyC", "map_polyX", "raddfB" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverXnsubC n c : ('X^n - c%:P \is a polyOver S) = (c \in S).

Proof. by rewrite rpredBl ?polyOverXn// ?polyOverC. Qed.

Lemma

polyOverXnsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "polyOverXn", "rpredBl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deriv_is_linear : linear (@deriv R).

Proof. exact: linearP. Qed.

Fact

deriv_is_linear

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "deriv", "linear", "linearP" ]

Single derivative.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivN : {morph deriv : p / - p}.

Proof. exact: linearN. Qed.

Lemma

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", "linearN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivB : {morph deriv : p q / p - q}.

Proof. exact: linearB. Qed.

Lemma

derivB

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "linearB" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivXsubC (a : R) : ('X - a%:P)^`() = 1.

Proof. by rewrite derivB derivX derivC subr0. Qed.

Lemma

derivXsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "derivB", "derivC", "derivX", "subr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivMNn n p : (p *- n)^`() = p^`() *- n.

Proof. exact: linearMNn. Qed.

Lemma

derivMNn

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "linearMNn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivE

:= Eval lazy beta delta [morphism_2 morphism_1] in (derivZ, deriv_mulC, derivC, derivX, derivMXaddC, derivXsubC, derivM, derivB, derivD, derivN, derivXn, derivM, derivMn).

Definition

derivE

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "delta", "derivB", "derivC", "derivD", "derivM", "derivMXaddC", "derivMn", "derivN", "derivX", "derivXn", "derivXsubC", "derivZ", "deriv_mulC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivn_is_linear n : linear (@derivn R n).

Proof. exact: linearP. Qed.

Fact

derivn_is_linear

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "derivn", "linear", "linearP" ]

Iterated derivative.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivn_is_linear n : linear (@nderivn R n).

Proof. exact: linearP. Qed.

Fact

nderivn_is_linear

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "linear", "linearP", "nderivn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnB n : {morph derivn n : p q / p - q}.

Proof. exact: linearB. Qed.

Lemma

derivnB

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "linearB" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnMNn n m p : (p *- m)^`(n) = p^`(n) *- m.

Proof. exact: linearMNn. Qed.

Lemma

derivnMNn

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "linearMNn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnN n : {morph derivn n : p / - p}.

Proof. exact: linearN. Qed.

Lemma

derivnN

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "linearN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivnB n : {morph nderivn n : p q / p - q}.

Proof. exact: linearB. Qed.

Lemma

nderivnB

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "linearB", "nderivn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivnMNn n m p : (p *- m)^`N(n) = p^`N(n) *- m.

Proof. exact: linearMNn. Qed.

Lemma

nderivnMNn

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "linearMNn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivnN n : {morph nderivn n : p / - p}.

Proof. exact: linearN. Qed.

Lemma

nderivnN

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "linearN", "nderivn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicXsubC c : 'X - c%:P \is monic.

Proof. exact/eqP/lead_coefXsubC. Qed.

Lemma

monicXsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_coefXsubC", "monic" ]

Monic predicate

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_prod_XsubC I rI (P : pred I) (F : I -> R) : \prod_(i <- rI | P i) ('X - (F i)%:P) \is monic.

Proof. by apply: monic_prod => i _; apply: monicXsubC. Qed.

Lemma

monic_prod_XsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "apply", "monic", "monicXsubC", "monic_prod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coef_prod_XsubC I rI (P : pred I) (F : I -> R) : lead_coef (\prod_(i <- rI | P i) ('X - (F i)%:P)) = 1.

Proof. exact/eqP/monic_prod_XsubC. Qed.

Lemma

lead_coef_prod_XsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "lead_coef", "monic_prod_XsubC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_prod_XsubC I rI (F : I -> R) : size (\prod_(i <- rI) ('X - (F i)%:P)) = (size rI).+1.

Proof. elim: rI => [|i r /= <-]; rewrite ?big_nil ?size_poly1 // big_cons. rewrite size_monicM ?monicXsubC ?monic_neq0 ?monic_prod_XsubC //. by rewrite size_XsubC. Qed.

Lemma

size_prod_XsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "big_cons", "big_nil", "monicXsubC", "monic_neq0", "monic_prod_XsubC", "size", "size_XsubC", "size_monicM", "size_poly1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_exp_XsubC n a : size (('X - a%:P) ^+ n) = n.+1.

Proof. rewrite -[n]card_ord -prodr_const -big_filter size_prod_XsubC. by have [e _ _ [_ ->]] := big_enumP. Qed.

Lemma

size_exp_XsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "big_enumP", "big_filter", "card_ord", "prodr_const", "size", "size_prod_XsubC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicXnsubC n c : 0 < n -> 'X^n - c%:P \is monic.

Proof. by move=> n_gt0; rewrite monicE lead_coefXnsubC. Qed.

Lemma

monicXnsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_coefXnsubC", "monic", "monicE", "n_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_Xn_sub_1 n : n > 0 -> 'X^n - 1 \is @monic R.

Proof. exact/monicXnsubC. Qed.

Lemma

monic_Xn_sub_1

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "monic", "monicXnsubC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lreg_lead p : GRing.lreg (lead_coef p) -> GRing.lreg p.

Proof. move/mulrI_eq0=> reg_p; apply: mulrI0_lreg => q; apply/contra_eq => nz_q. by rewrite -lead_coef_eq0 lead_coef_proper_mul reg_p lead_coef_eq0. Qed.

Lemma

lreg_lead

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "contra_eq", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "lreg", "mulrI0_lreg", "mulrI_eq0" ]

Some facts about regular elements.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rreg_lead p : GRing.rreg (lead_coef p) -> GRing.rreg p.

Proof. move/mulIr_eq0=> reg_p; apply: mulIr0_rreg => q; apply/contra_eq => nz_q. by rewrite -lead_coef_eq0 lead_coef_proper_mul reg_p lead_coef_eq0. Qed.

Lemma

rreg_lead

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "contra_eq", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "mulIr0_rreg", "mulIr_eq0", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_lreg p : p \is monic -> GRing.lreg p.

Proof. by move=> /eqP lp1; apply/lreg_lead; rewrite lp1; apply/lreg1. Qed.

Lemma

monic_lreg

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "lreg", "lreg1", "lreg_lead", "monic" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_rreg p : p \is monic -> GRing.rreg p.

Proof. by move=> /eqP lp1; apply/rreg_lead; rewrite lp1; apply/rreg1. Qed.

Lemma

monic_rreg

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "monic", "rreg", "rreg1", "rreg_lead" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rootN p x : root (- p) x = root p x.

Proof. by rewrite rootE hornerN oppr_eq0. Qed.

Lemma

rootN

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "hornerN", "oppr_eq0", "root", "rootE" ]

Roots of polynomials

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root_XsubC a x : root ('X - a%:P) x = (x == a).

Proof. by rewrite rootE hornerXsubC subr_eq0. Qed.

Lemma

root_XsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "hornerXsubC", "root", "rootE", "subr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root_XaddC a x : root ('X + a%:P) x = (x == - a).

Proof. by rewrite -root_XsubC rmorphN opprK. Qed.

Lemma

root_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" ]

[ "opprK", "rmorphN", "root", "root_XsubC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

factor_theorem p a : reflect (exists q, p = q * ('X - a%:P)) (root p a).

Proof. apply: (iffP eqP) => [pa0 | [q ->]]; last first. by rewrite hornerM_comm /comm_poly hornerXsubC subrr ?simp. exists (\poly_(i < size p) horner_rec (drop i.+1 p) a). apply/polyP=> i; rewrite mulrBr coefB coefMX coefMC !coef_poly. apply: canRL (addrK _) _; rewrite addrC; have [le_p_i | lt_i_p] := leqP. rewrite...

Theorem

factor_theorem

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "addrK", "apply", "coefB", "coefMC", "coefMX", "coef_poly", "comm_poly", "drop", "drop1", "drop_nth", "drop_oversize", "horner", "hornerM_comm", "hornerXsubC", "horner_rec", "last", "leqP", "leqSpred", "leq_trans", "ltnW", "mulrBr", "nth_default", "polyP", "r...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

multiplicity_XsubC p a : {m | exists2 q, (p != 0) ==> ~~ root q a & p = q * ('X - a%:P) ^+ m}.

Proof. have [n le_p_n] := ubnP (size p); elim: n => // n IHn in p le_p_n *. have [-> | nz_p /=] := eqVneq p 0; first by exists 0, 0; rewrite ?mul0r. have [/sig_eqW[p1 Dp] | nz_pa] := altP (factor_theorem p a); last first. by exists 0%N, p; rewrite ?mulr1. have nz_p1: p1 != 0 by apply: contraNneq nz_p => p1_0; rewrite...

Lemma

multiplicity_XsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "addn2", "apply", "contraNneq", "eqVneq", "exprSr", "factor_theorem", "last", "monicXsubC", "mul0r", "mulr1", "mulrA", "nz_p", "root", "sig2_eqW", "sig_eqW", "size", "size_Mmonic", "size_XsubC", "ubnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root_of_unity n : pred R

:= root ('X^n - 1).

Definition

root_of_unity

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "root" ]

Roots of unity.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"n .-unity_root"

:= (root_of_unity n) : ring_scope.

Notation

n .-unity_root

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "root_of_unity" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unity_rootE n z : n.-unity_root z = (z ^+ n == 1).

Proof. by rewrite /root_of_unity rootE hornerD hornerN hornerXn hornerC subr_eq0. Qed.

Lemma

unity_rootE

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "hornerC", "hornerD", "hornerN", "hornerXn", "rootE", "root_of_unity", "subr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unity_rootP n z : reflect (z ^+ n = 1) (n.-unity_root z).

Proof. by rewrite unity_rootE; apply: eqP. Qed.

Lemma

unity_rootP

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "unity_rootE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primitive_root_of_unity n z

:= (n > 0) && [forall i : 'I_n, i.+1.-unity_root z == (i.+1 == n)].

Definition

primitive_root_of_unity

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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

"n .-primitive_root"

:= (primitive_root_of_unity n) : ring_scope.

Notation

n .-primitive_root

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "primitive_root_of_unity" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prim_order_exists n z : n > 0 -> z ^+ n = 1 -> {m | m.-primitive_root z & (m %| n)}.

Proof. move=> n_gt0 zn1. have: exists m, (m > 0) && (z ^+ m == 1) by exists n; rewrite n_gt0 /= zn1. case/ex_minnP=> m /andP[m_gt0 /eqP zm1] m_min. exists m. apply/andP; split=> //; apply/eqfunP=> [[i]] /=. rewrite leq_eqVlt unity_rootE. case: eqP => [-> _ | _]; first by rewrite zm1 eqxx. by apply: contraTF => ...

Lemma

prim_order_exists

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "eqfunP", "eqxx", "ex_minnP", "expr_mod", "leqNgt", "leq_eqVlt", "lt0n", "ltn_mod", "n_gt0", "split", "unity_rootE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prim_z : n.-primitive_root z.

Hypothesis

prim_z

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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

prim_order_gt0 : n > 0.

Proof. by case/andP: prim_z. Qed.

Lemma

prim_order_gt0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "prim_z" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

n_gt0

:= prim_order_gt0.

Let

n_gt0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "prim_order_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prim_expr_order : z ^+ n = 1.

Proof. case/andP: prim_z => _; rewrite -(prednK n_gt0) => /forallP/(_ ord_max). by rewrite unity_rootE eqxx eqb_id => /eqP. Qed.

Lemma

prim_expr_order

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "eqb_id", "eqxx", "forallP", "n_gt0", "ord_max", "prednK", "prim_z", "unity_rootE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prim_expr_mod i : z ^+ (i %% n) = z ^+ i.

Proof. exact: expr_mod prim_expr_order. Qed.

Lemma

prim_expr_mod

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "expr_mod", "prim_expr_order" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prim_order_dvd i : (n %| i) = (z ^+ i == 1).

Proof. move: n_gt0; rewrite -prim_expr_mod /dvdn -(ltn_mod i). case: {i}(i %% n)%N => [|i] lt_i; first by rewrite !eqxx. case/andP: prim_z => _ /forallP/(_ (Ordinal (ltnW lt_i)))/eqP. by rewrite unity_rootE eqn_leq andbC leqNgt lt_i. Qed.

Lemma

prim_order_dvd

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "dvdn", "eqn_leq", "eqxx", "forallP", "leqNgt", "ltnW", "ltn_mod", "n_gt0", "prim_expr_mod", "prim_z", "unity_rootE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_prim_root_expr i j : (z ^+ i == z ^+ j) = (i == j %[mod n]).

Proof. wlog le_ji: i j / j <= i. move=> IH; case: (leqP j i) => [|/ltnW] /IH //. by rewrite eq_sym (eq_sym (j %% n)%N). rewrite -{1}(subnKC le_ji) exprD -prim_expr_mod eqn_mod_dvd //. rewrite prim_order_dvd; apply/eqP/eqP=> [|->]; last by rewrite mulr1. move/(congr1 ( *%R (z ^+ (n - j %% n)))); rewrite mulrA -exprD...

Lemma

eq_prim_root_expr

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_sym", "eqn_mod_dvd", "exprD", "last", "leqP", "ltnW", "ltn_mod", "mul1r", "mulr1", "mulrA", "prim_expr_mod", "prim_expr_order", "prim_order_dvd", "subnK", "subnKC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exp_prim_root k : (n %/ gcdn k n).-primitive_root (z ^+ k).

Proof. set d := gcdn k n; have d_gt0: (0 < d)%N by rewrite gcdn_gt0 orbC n_gt0. have [d_dv_k d_dv_n]: (d %| k /\ d %| n)%N by rewrite dvdn_gcdl dvdn_gcdr. set q := (n %/ d)%N; rewrite /q.-primitive_root ltn_divRL // n_gt0. apply/forallP=> i; rewrite unity_rootE -exprM -prim_order_dvd. rewrite -(divnK d_dv_n) -/q -(divn...

Lemma

exp_prim_root

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "Gauss_dvdr", "apply", "coprime", "d_gt0", "divnK", "dvdn_gcdl", "dvdn_gcdr", "dvdn_leq", "dvdn_mull", "dvdn_pmul2r", "eqn_leq", "eqn_pmul2r", "exprM", "forallP", "gcdn", "gcdnC", "gcdn_gt0", "last", "ltn_divRL", "ltn_ord", "mul1n", "mulnAC", "muln_gcdl", "n_gt0", "pr...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_prim_root m : (m %| n)%N -> m.-primitive_root (z ^+ (n %/ m)).

Proof. set k := (n %/ m)%N => m_dv_n; rewrite -{1}(mulKn m n_gt0) -divnA // -/k. by rewrite -{1}(@gcdn_idPl k n _) ?exp_prim_root // -(divnK m_dv_n) dvdn_mulr. Qed.

Lemma

dvdn_prim_root

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "divnA", "divnK", "dvdn_mulr", "exp_prim_root", "gcdn_idPl", "mulKn", "n_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prim_root_eq0 : (z == 0) = (n == 0%N).

Proof. rewrite gtn_eqF//; apply/eqP => z0; have /esym/eqP := prim_expr_order. by rewrite z0 expr0n gtn_eqF//= oner_eq0. Qed.

Lemma

prim_root_eq0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "apply", "expr0n", "gtn_eqF", "oner_eq0", "prim_expr_order", "z0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prim_root_exp_coprime n z k : n.-primitive_root z -> n.-primitive_root (z ^+ k) = coprime k n.

Proof. move=> prim_z; have n_gt0 := prim_order_gt0 prim_z. apply/idP/idP=> [prim_zk | co_k_n]. set d := gcdn k n; have dv_d_n: (d %| n)%N := dvdn_gcdr _ _. rewrite /coprime -/d -(eqn_pmul2r n_gt0) mul1n -{2}(gcdnMl n d). rewrite -{2}(divnK dv_d_n) (mulnC _ d) -muln_gcdr (gcdn_idPr _) //. rewrite (prim_order_dvd...

Lemma

prim_root_exp_coprime

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "Gauss_dvdr", "apply", "coprime", "coprime_sym", "divnK", "dvdn_gcdr", "dvdn_mulr", "eqn_dvd", "eqn_pmul2r", "expr1n", "exprAC", "exprM", "gcdn", "gcdnMl", "gcdn_idPr", "mul1n", "mulnC", "muln_divCA_gcd", "muln_gcdr", "n_gt0", "prim_expr_order", "prim_order_dvd", "prim_or...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_opp

:= size_polyN (only parsing).

Notation

size_opp

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_polyN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly (p : {poly aR})

:= \poly_(i < size p) f p`_i.

Definition

map_poly

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "poly", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_polyE p : map_poly p = Poly (map f p).

Proof. rewrite /map_poly unlock; congr Poly. apply: (@eq_from_nth _ 0); rewrite size_mkseq ?size_map // => i lt_i_p. by rewrite [RHS](nth_map 0) ?nth_mkseq. Qed.

Lemma

map_polyE

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "Poly", "apply", "eq_from_nth", "map", "map_poly", "nth_map", "nth_mkseq", "size_map", "size_mkseq" ]

instance of size_poly.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_rmorph u

:= forall x, GRing.comm u (f x).

Definition

commr_rmorph

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_morph u & commr_rmorph u

:= fun p => (map_poly p).[u].

Definition

horner_morph

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_rmorph", "map_poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"p ^f"

:= (map_poly f p) : ring_scope.

Notation

p ^f

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "map_poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly0 : 0^f = 0.

Proof. by rewrite map_polyE polyseq0. Qed.

Lemma

map_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" ]

[ "map_polyE", "polyseq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_map_poly (g : aR -> rR) : f =1 g -> map_poly f =1 map_poly g.

Proof. by move=> eq_fg p; rewrite !map_polyE (eq_map eq_fg). Qed.

Lemma

eq_map_poly

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "eq_map", "map_poly", "map_polyE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_id g (p : {poly iR}) : {in (p : seq iR), g =1 id} -> map_poly g p = p.

Proof. by move=> g_id; rewrite map_polyE map_id_in ?polyseqK. Qed.

Lemma

map_poly_id

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "id", "map_id_in", "map_poly", "map_polyE", "poly", "polyseqK", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coef_map_id0 p i : f 0 = 0 -> (p^f)`_i = f p`_i.

Proof. by move=> f0; rewrite coef_poly; case: ltnP => // le_p_i; rewrite nth_default. Qed.

Lemma

coef_map_id0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "ltnP", "nth_default" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_Poly_id0 s : f 0 = 0 -> (Poly s)^f = Poly (map f s).

Proof. move=> f0; apply/polyP=> j; rewrite coef_map_id0 ?coef_Poly //. have [/(nth_map 0 0)->// | le_s_j] := ltnP j (size s). by rewrite !nth_default ?size_map. Qed.

Lemma

map_Poly_id0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "apply", "coef_Poly", "coef_map_id0", "ltnP", "map", "nth_default", "nth_map", "polyP", "size", "size_map" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_comp_id0 (g : iR -> aR) p : f 0 = 0 -> map_poly (f \o g) p = (map_poly g p)^f.

Proof. by move=> f0; rewrite map_polyE map_comp -map_Poly_id0 -?map_polyE. Qed.

Lemma

map_poly_comp_id0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "map_Poly_id0", "map_comp", "map_poly", "map_polyE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_map_poly_id0 p : f (lead_coef p) != 0 -> size p^f = size p.

Proof. by move=> nz_fp; apply: size_poly_eq. Qed.

Lemma

size_map_poly_id0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "lead_coef", "size", "size_poly_eq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_eq0_id0 p : f (lead_coef p) != 0 -> (p^f == 0) = (p == 0).

Proof. by rewrite -!size_poly_eq0 => /size_map_poly_id0->. Qed.

Lemma

map_poly_eq0_id0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "size_map_poly_id0", "size_poly_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coef_map_id0 p : f 0 = 0 -> f (lead_coef p) != 0 -> lead_coef p^f = f (lead_coef p).

Proof. by move=> f0 nz_fp; rewrite lead_coefE coef_map_id0 ?size_map_poly_id0. Qed.

Lemma

lead_coef_map_id0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_map_id0", "lead_coef", "lead_coefE", "size_map_poly_id0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(inj_f : injective f) (f_0 : f 0 = 0).

Hypotheses

inj_f

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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

size_map_inj_poly p : size p^f = size p.

Proof. have [-> | nz_p] := eqVneq p 0; first by rewrite map_poly0 !size_poly0. by rewrite size_map_poly_id0 // -f_0 (inj_eq inj_f) lead_coef_eq0. Qed.

Lemma

size_map_inj_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" ]

[ "eqVneq", "inj_eq", "inj_f", "lead_coef_eq0", "map_poly0", "nz_p", "size", "size_map_poly_id0", "size_poly0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_inj_poly : injective (map_poly f).

Proof. move=> p q /polyP eq_pq; apply/polyP=> i; apply: inj_f. by rewrite -!coef_map_id0 ?eq_pq. Qed.

Lemma

map_inj_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_map_id0", "inj_f", "map_poly", "polyP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coef_map_inj p : lead_coef p^f = f (lead_coef p).

Proof. by rewrite !lead_coefE size_map_inj_poly coef_map_id0. Qed.

Lemma

lead_coef_map_inj

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_map_id0", "lead_coef", "lead_coefE", "size_map_inj_poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_polyK (f : aR -> rR) g : cancel g f -> f 0 = 0 -> cancel (map_poly g) (map_poly f).

Proof. by move=> gK f_0 p; rewrite /= -map_poly_comp_id0 ?map_poly_id // => x _ //=. Qed.

Lemma

map_polyK

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "gK", "map_poly", "map_poly_comp_id0", "map_poly_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_map_poly_id0 (f g : aR -> rR) (S : addrClosed aR) : f 0 = 0 -> g 0 = 0 -> {in S, f =1 g} -> {in polyOver S, map_poly f =1 map_poly g}.

Proof. move=> f0 g0 eq_fg p pP; apply/polyP => i. by rewrite !coef_map_id0// eq_fg// (polyOverP _). Qed.

Lemma

eq_in_map_poly_id0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "addrClosed", "apply", "coef_map_id0", "map_poly", "pP", "polyOver", "polyOverP", "polyP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_map_poly (f g : {additive aR -> rR}) (S : addrClosed aR) : {in S, f =1 g} -> {in polyOver S, map_poly f =1 map_poly g}.

Proof. by move=> /eq_in_map_poly_id0; apply; rewrite //?raddf0. Qed.

Lemma

eq_in_map_poly

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "additive", "addrClosed", "apply", "eq_in_map_poly_id0", "map_poly", "polyOver", "raddf0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coef_map p i : p^f`_i = f p`_i.

Proof. exact: coef_map_id0 (raddf0 f). Qed.

Lemma

coef_map

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_map_id0", "raddf0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_Poly s : (Poly s)^f = Poly (map f s).

Proof. exact: map_Poly_id0 (raddf0 f). Qed.

Lemma

map_Poly

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "Poly", "map", "map_Poly_id0", "raddf0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_comp (g : iR -> aR) p : map_poly (f \o g) p = map_poly f (map_poly g p).

Proof. exact: map_poly_comp_id0 (raddf0 f). Qed.

Lemma

map_poly_comp

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "map_poly", "map_poly_comp_id0", "raddf0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_is_nmod_morphism : nmod_morphism (map_poly f).

Proof. split=> [|p q]; apply/polyP => i; first by rewrite coef_map !coef0 raddf0. by rewrite !(coef_map, coefD) raddfD. Qed.

Fact

map_poly_is_nmod_morphism

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "apply", "coef0", "coefD", "coef_map", "map_poly", "nmod_morphism", "polyP", "raddf0", "raddfD", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_polyC a : (a%:P)^f = (f a)%:P.

Proof. by apply/polyP=> i; rewrite !(coef_map, coefC) -!mulrb raddfMn. Qed.

Lemma

map_polyC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "apply", "coefC", "coef_map", "mulrb", "polyP", "raddfMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coef_map_eq p : f (lead_coef p) != 0 -> lead_coef p^f = f (lead_coef p).

Proof. exact: lead_coef_map_id0 (raddf0 f). Qed.

Lemma

lead_coef_map_eq

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "lead_coef", "lead_coef_map_id0", "raddf0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_is_monoid_morphism : monoid_morphism (map_poly f).

Proof. split=> [|p q]; apply/polyP=> i. by rewrite !(coef_map, coef1) /= rmorph_nat. rewrite coef_map /= !coefM /= !rmorph_sum; apply: eq_bigr => j _. by rewrite !coef_map rmorphM. Qed.

Fact

map_poly_is_monoid_morphism

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "apply", "coef1", "coefM", "coef_map", "eq_bigr", "map_poly", "monoid_morphism", "polyP", "rmorphM", "rmorph_nat", "rmorph_sum", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_is_multiplicative

:= (fun g => (g.2, g.1)) map_poly_is_monoid_morphism.

Definition

map_poly_is_multiplicative

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "map_poly_is_monoid_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_polyZ c p : (c *: p)^f = f c *: p^f.

Proof. by apply/polyP=> i; rewrite !(coef_map, coefZ) /= rmorphM. Qed.

Lemma

map_polyZ

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "apply", "coefZ", "coef_map", "polyP", "rmorphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_polyX : ('X)^f = 'X.

Proof. by apply/polyP=> i; rewrite coef_map !coefX /= rmorph_nat. Qed.

Lemma

map_polyX

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "apply", "coefX", "coef_map", "polyP", "rmorph_nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_polyXn n : ('X^n)^f = 'X^n.

Proof. by rewrite rmorphXn /= map_polyX. Qed.

Lemma

map_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" ]

[ "map_polyX", "rmorphXn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_polyXaddC x : ('X + x%:P)^f = 'X + (f x)%:P.

Proof. by rewrite raddfD/= map_polyX map_polyC. Qed.

Lemma

map_polyXaddC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "map_polyC", "map_polyX", "raddfD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_map p : p \is monic -> p^f \is monic.

Proof. move/monicP=> mon_p; rewrite monicE. by rewrite lead_coef_map_eq mon_p /= rmorph1 ?oner_neq0. Qed.

Lemma

monic_map

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_map_eq", "monic", "monicE", "monicP", "oner_neq0", "rmorph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_map p x : p^f.[f x] = f p.[x].

Proof. elim/poly_ind: p => [|p c IHp]; first by rewrite !(rmorph0, horner0). rewrite hornerMXaddC !rmorphD !rmorphM /=. by rewrite map_polyX map_polyC hornerMXaddC IHp. Qed.

Lemma

horner_map

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "map_polyC", "map_polyX", "poly_ind", "rmorph0", "rmorphD", "rmorphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_comm_poly p x : comm_poly p x -> comm_poly p^f (f x).

Proof. by rewrite /comm_poly horner_map -!rmorphM // => ->. Qed.

Lemma

map_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" ]

[ "comm_poly", "horner_map", "rmorphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_comm_coef p x : comm_coef p x -> comm_coef p^f (f x).

Proof. by move=> cpx i; rewrite coef_map -!rmorphM ?cpx. Qed.

Lemma

map_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" ]

[ "coef_map", "comm_coef", "rmorphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmorph_root p x : root p x -> root p^f (f x).

Proof. by move/eqP=> px0; rewrite rootE horner_map px0 rmorph0. Qed.

Lemma

rmorph_root

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "horner_map", "rmorph0", "root", "rootE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cfu : commr_rmorph f u.

Hypothesis

cfu

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_rmorph" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_morphC a : horner_morph cfu a%:P = f a.

Proof. by rewrite /horner_morph map_polyC hornerC. Qed.

Lemma

horner_morphC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "cfu", "hornerC", "horner_morph", "map_polyC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_morphX : horner_morph cfu 'X = u.

Proof. by rewrite /horner_morph map_polyX hornerX. Qed.

Lemma

horner_morphX

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "cfu", "hornerX", "horner_morph", "map_polyX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_is_semilinear : semilinear_for (f \; *%R) (horner_morph cfu).

Proof. split=> [c p|p q]; rewrite /horner_morph; first by rewrite linearZ hornerZ. by rewrite linearD hornerD. Qed.

Fact

horner_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" ]

[ "cfu", "hornerD", "hornerZ", "horner_morph", "linearD", "linearZ", "semilinear_for", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_is_monoid_morphism : monoid_morphism (horner_morph cfu).

Proof. split=> [|p q]; first by rewrite /horner_morph rmorph1 hornerC. rewrite /horner_morph rmorphM /= hornerM_comm //. by apply: comm_coef_poly => i; rewrite coef_map cfu. Qed.

Fact

horner_is_monoid_morphism

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "apply", "cfu", "coef_map", "comm_coef_poly", "hornerC", "hornerM_comm", "horner_morph", "monoid_morphism", "rmorph1", "rmorphM", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

horner_is_multiplicative

:= (fun g => (g.2, g.1)) horner_is_monoid_morphism.

Definition

horner_is_multiplicative

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "horner_is_monoid_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deriv_map p : p^f^`() = (p^`())^f.

Proof. by apply/polyP => i; rewrite !(coef_map, coef_deriv) //= rmorphMn. Qed.

Lemma

deriv_map

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "coef_map", "polyP", "rmorphMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivn_map p n : p^f^`(n) = (p^`(n))^f.

Proof. by apply/polyP => i; rewrite !(coef_map, coef_derivn) //= rmorphMn. Qed.

Lemma

derivn_map

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "coef_map", "polyP", "rmorphMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivn_map p n : p^f^`N(n) = (p^`N(n))^f.

Proof. by apply/polyP => i; rewrite !(coef_map, coef_nderivn) //= rmorphMn. Qed.

Lemma

nderivn_map

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_map", "coef_nderivn", "polyP", "rmorphMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_is_zmod_morphism : zmod_morphism (map_poly f).

Proof. exact: raddfB. Qed.

Fact

map_poly_is_zmod_morphism

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "map_poly", "raddfB", "zmod_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_poly_is_additive : zmod_morphism (map_poly f).

Proof. exact: raddfB. Qed.

Fact

map_poly_is_additive

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "map_poly", "raddfB", "zmod_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_polyXsubC x : ('X - x%:P)^f = 'X - (f x)%:P.

Proof. by rewrite raddfB/= map_polyX map_polyC. Qed.

Lemma

map_polyXsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]

[ "map_polyC", "map_polyX", "raddfB" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d