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
derivnZ n : scalable (derivn n).	Proof. exact: linearZZ. Qed.	Lemma	derivnZ	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "linearZZ", "scalable" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivnXn m n : ('X^m)^`(n) = 'X^(m - n) *+ m ^_ n.	Proof. apply/polyP=>i; rewrite coef_derivn coefMn !coefXn. case: (ltnP m n) => [lt_m_n \| le_m_n]. by rewrite eqn_leq leqNgt ltn_addr // mul0rn ffact_small. by rewrite -{1 3}(subnKC le_m_n) eqn_add2l; case: eqP => [->\|]; rewrite ?mul0rn. Qed.	Lemma	derivnXn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coefMn", "coefXn", "coef_derivn", "eqn_add2l", "eqn_leq", "ffact_small", "leqNgt", "ltnP", "ltn_addr", "mul0rn", "polyP", "subnKC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivnMXaddC n p c : (p * 'X + c%:P)^`(n.+1) = p^`(n) + n.+1 + p^`(n.+1) 'X.	Proof. elim: n => [\|n IHn]; first by rewrite derivn1 derivMXaddC. rewrite derivnS IHn derivD derivM derivX mulr1 derivMn -!derivnS. by rewrite addrA addrAC -mulrSr. Qed.	Lemma	derivnMXaddC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addrA", "addrAC", "derivD", "derivM", "derivMXaddC", "derivMn", "derivX", "derivn1", "derivnS", "mulr1", "mulrSr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivn_poly0 p n : size p <= n -> p^`(n) = 0.	Proof. move=> le_p_n; apply/polyP=> i; rewrite coef_derivn. rewrite nth_default; last by rewrite mul0rn coef0. exact/(leq_trans le_p_n)/leq_addr. Qed.	Lemma	derivn_poly0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coef0", "coef_derivn", "last", "leq_addr", "leq_trans", "mul0rn", "nth_default", "polyP", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt_size_deriv (p : {poly R}) : p != 0 -> size p^`() < size p.	Proof. by move=> /polySpred->; apply: size_poly. Qed.	Lemma	lt_size_deriv	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "poly", "polySpred", "size", "size_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivn n p	:= \poly_(i < size p - n) (p`_(n + i) *+ 'C(n + i, n)).	Definition	nderivn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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" ]	A normalising version of derivation to get the division by n! in Taylor	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"a ^`N ( n )"	:= (nderivn n a) : ring_scope.	Notation	a ^`N ( n )	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "nderivn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_nderivn n p i : p^`N(n)`_i = p`_(n + i) *+ 'C(n + i, n).	Proof. rewrite coef_poly ltn_subRL; case: leqP => // le_p_ni. by rewrite nth_default ?mul0rn. Qed.	Lemma	coef_nderivn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "coef_poly", "leqP", "ltn_subRL", "mul0rn", "nth_default" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivn_def n p : p^`(n) = p^`N(n) *+ n`!.	Proof. by apply/polyP=> i; rewrite coefMn coef_nderivn coef_derivn -mulrnA bin_ffact. Qed.	Lemma	nderivn_def	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "bin_ffact", "coefMn", "coef_derivn", "coef_nderivn", "mulrnA", "polyP" ]	Here is the division by n!	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOver_nderivn (ringS : semiringClosed R) : {in polyOver ringS, forall p n, p^`N(n) \in polyOver ringS}.	Proof. move=> p /polyOverP Sp /= n; apply/polyOverP=> i. by rewrite coef_nderivn rpredMn. Qed.	Lemma	polyOver_nderivn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_nderivn", "polyOver", "polyOverP", "rpredMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivn0 p : p^`N(0) = p.	Proof. by rewrite -[p^`N(0)](nderivn_def 0). Qed.	Lemma	nderivn0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "nderivn_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivn1 p : p^`N(1) = p^`().	Proof. by rewrite -[p^`N(1)](nderivn_def 1). Qed.	Lemma	nderivn1	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "nderivn_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnC c n : (c%:P)^`N(n) = if n == 0 then c%:P else 0.	Proof. apply/polyP=> i; rewrite coef_nderivn. by case: n => [\|n]; rewrite ?bin0 // coef0 coefC mul0rn. Qed.	Lemma	nderivnC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "bin0", "coef0", "coefC", "coef_nderivn", "mul0rn", "polyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnXn m n : ('X^m)^`N(n) = 'X^(m - n) *+ 'C(m, n).	Proof. apply/polyP=> i; rewrite coef_nderivn coefMn !coefXn. have [lt_m_n \| le_n_m] := ltnP m n. by rewrite eqn_leq leqNgt ltn_addr // mul0rn bin_small. by rewrite -{1 3}(subnKC le_n_m) eqn_add2l; case: eqP => [->\|]; rewrite ?mul0rn. Qed.	Lemma	nderivnXn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "bin_small", "coefMn", "coefXn", "coef_nderivn", "eqn_add2l", "eqn_leq", "leqNgt", "ltnP", "ltn_addr", "mul0rn", "polyP", "subnKC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivn_is_semilinear n : semilinear (nderivn n).	Proof. split=> [k p\|p q]; apply/polyP => i. by rewrite !(coef_nderivn, coefZ) mulrnAr. by rewrite !(coef_nderivn, coefD) mulrnDl. Qed.	Fact	nderivn_is_semilinear	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coefD", "coefZ", "coef_nderivn", "mulrnAr", "mulrnDl", "nderivn", "polyP", "semilinear", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnD n : {morph nderivn n : p q / p + q}.	Proof. exact: linearD. Qed.	Lemma	nderivnD	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linearD", "nderivn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnMn n m p : (p + m)^`N(n) = p^`N(n) + m.	Proof. exact: linearMn. Qed.	Lemma	nderivnMn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linearMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnZ n : scalable (nderivn n).	Proof. exact: linearZZ. Qed.	Lemma	nderivnZ	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "linearZZ", "nderivn", "scalable" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivnMXaddC n p c : (p * 'X + c%:P)^`N(n.+1) = p^`N(n) + p^`N(n.+1) * 'X.	Proof. apply/polyP=> i; rewrite coef_nderivn !coefD !coefMX coefC. rewrite !addSn /= !coef_nderivn addr0 binS mulrnDr addrC; congr (_ + _). by rewrite addSnnS; case: i; rewrite // addn0 bin_small. Qed.	Lemma	nderivnMXaddC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addSn", "addSnnS", "addn0", "addr0", "addrC", "apply", "binS", "bin_small", "coefC", "coefD", "coefMX", "coef_nderivn", "mulrnDr", "polyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderivn_poly0 p n : size p <= n -> p^`N(n) = 0.	Proof. move=> le_p_n; apply/polyP=> i; rewrite coef_nderivn. rewrite nth_default; last by rewrite mul0rn coef0. exact/(leq_trans le_p_n)/leq_addr. Qed.	Lemma	nderivn_poly0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "coef0", "coef_nderivn", "last", "leq_addr", "leq_trans", "mul0rn", "nth_default", "polyP", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderiv_taylor p x h : GRing.comm x h -> p.[x + h] = \sum_(i < size p) p^`N(i).[x] * h ^+ i.	Proof. move/commrX=> cxh; elim/poly_ind: p => [\|p c IHp]. by rewrite size_poly0 big_ord0 horner0. rewrite hornerMXaddC size_MXaddC. have [-> \| nz_p] := eqVneq p 0. rewrite horner0 !simp; have [-> \| _] := c =P 0; first by rewrite big_ord0. by rewrite size_poly0 big_ord_recl big_ord0 nderivn0 hornerC !simp. rewrite...	Lemma	nderiv_taylor	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addn", "addrA", "addrAC", "addrC", "apply", "big_distrl", "big_ord0", "big_ord_recl", "big_ord_recr", "big_split", "bump", "comm", "comm_polyX", "commrX", "eqVneq", "eq_bigr", "exprSr", "horner0", "hornerC", "hornerD", "hornerMXaddC", "hornerM_comm", "hornerX", "last",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nderiv_taylor_wide n p x h : GRing.comm x h -> size p <= n -> p.[x + h] = \sum_(i < n) p^`N(i).[x] * h ^+ i.	Proof. move/nderiv_taylor=> -> le_p_n. rewrite (big_ord_widen n (fun i => p^`N(i).[x] * h ^+ i)) // big_mkcond. apply: eq_bigr => i _; case: leqP => // /nderivn_poly0->. by rewrite horner0 simp. Qed.	Lemma	nderiv_taylor_wide	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "big_mkcond", "big_ord_widen", "comm", "eq_bigr", "horner0", "leqP", "nderiv_taylor", "nderivn_poly0", "simp", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_pred	:= fun p => lead_coef p == 1.	Definition	monic_pred	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "lead_coef" ]	Monic predicate	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic	:= [qualify p \| monic_pred p].	Definition	monic	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_pred" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicE p : (p \is monic) = (lead_coef p == 1).	Proof. by []. Qed.	Lemma	monicE	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicP p : reflect (lead_coef p = 1) (p \is monic).	Proof. exact: eqP. Qed.	Lemma	monicP	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic1 : 1 \is monic.	Proof. exact/eqP/lead_coef1. Qed.	Lemma	monic1	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_coef1", "monic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicX : 'X \is monic.	Proof. exact/eqP/lead_coefX. Qed.	Lemma	monicX	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_coefX", "monic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicXn n : 'X^n \is monic.	Proof. exact/eqP/lead_coefXn. Qed.	Lemma	monicXn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_coefXn", "monic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_neq0 p : p \is monic -> p != 0.	Proof. by rewrite -lead_coef_eq0 => /eqP->; apply: oner_neq0. Qed.	Lemma	monic_neq0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "lead_coef_eq0", "monic", "oner_neq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_monicM p q : p \is monic -> lead_coef (p * q) = lead_coef q.	Proof. have [-> \| nz_q] := eqVneq q 0; first by rewrite mulr0. by move/monicP=> mon_p; rewrite lead_coef_proper_mul mon_p mul1r ?lead_coef_eq0. Qed.	Lemma	lead_coef_monicM	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "eqVneq", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "monic", "monicP", "mul1r", "mulr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_Mmonic p q : q \is monic -> lead_coef (p * q) = lead_coef p.	Proof. have [-> \| nz_p] := eqVneq p 0; first by rewrite mul0r. by move/monicP=> mon_q; rewrite lead_coef_proper_mul mon_q mulr1 ?lead_coef_eq0. Qed.	Lemma	lead_coef_Mmonic	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "eqVneq", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "monic", "monicP", "mul0r", "mulr1", "nz_p" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_monicM p q : p \is monic -> q != 0 -> size (p * q) = (size p + size q).-1.	Proof. move/monicP=> mon_p nz_q. by rewrite size_proper_mul // mon_p mul1r lead_coef_eq0. Qed.	Lemma	size_monicM	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_eq0", "monic", "monicP", "mul1r", "size", "size_proper_mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_Mmonic p q : p != 0 -> q \is monic -> size (p * q) = (size p + size q).-1.	Proof. move=> nz_p /monicP mon_q. by rewrite size_proper_mul // mon_q mulr1 lead_coef_eq0. Qed.	Lemma	size_Mmonic	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_eq0", "monic", "monicP", "mulr1", "nz_p", "size", "size_proper_mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicMl p q : p \is monic -> (p * q \is monic) = (q \is monic).	Proof. by move=> mon_p; rewrite !monicE lead_coef_monicM. Qed.	Lemma	monicMl	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_monicM", "monic", "monicE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicMr p q : q \is monic -> (p * q \is monic) = (p \is monic).	Proof. by move=> mon_q; rewrite !monicE lead_coef_Mmonic. Qed.	Lemma	monicMr	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_Mmonic", "monic", "monicE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_mulr_closed : mulr_closed monic.	Proof. by split=> [\|p q mon_p]; rewrite (monic1, monicMl). Qed.	Fact	monic_mulr_closed	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "monic", "monic1", "monicMl", "mulr_closed", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_exp p n : p \is monic -> p ^+ n \is monic.	Proof. exact: rpredX. Qed.	Lemma	monic_exp	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "monic", "rpredX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_prod I rI (P : pred I) (F : I -> {poly R}): (forall i, P i -> F i \is monic) -> \prod_(i <- rI \| P i) F i \is monic.	Proof. exact: rpred_prod. Qed.	Lemma	monic_prod	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "monic", "poly", "rpred_prod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicXaddC c : 'X + c%:P \is monic.	Proof. exact/eqP/lead_coefXaddC. Qed.	Lemma	monicXaddC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_coefXaddC", "monic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicXnaddC n c : 0 < n -> 'X^n + c%:P \is monic.	Proof. by move=> n_gt0; rewrite monicE lead_coefXnaddC. Qed.	Lemma	monicXnaddC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_coefXnaddC", "monic", "monicE", "n_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lreg_lead0 p : GRing.lreg (lead_coef p) -> p != 0.	Proof. by move/lreg_neq0; rewrite lead_coef_eq0. Qed.	Lemma	lreg_lead0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_eq0", "lreg", "lreg_neq0" ]	Some facts about regular elements.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rreg_lead0 p : GRing.rreg (lead_coef p) -> p != 0.	Proof. by move/rreg_neq0; rewrite lead_coef_eq0. Qed.	Lemma	rreg_lead0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_eq0", "rreg", "rreg_neq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lreg_size c p : GRing.lreg c -> size (c *: p) = size p.	Proof. move=> reg_c; have [-> \| nz_p] := eqVneq p 0; first by rewrite scaler0. rewrite -mul_polyC size_proper_mul; last by rewrite size_polyC lreg_neq0. by rewrite lead_coefC mulrI_eq0 ?lead_coef_eq0. Qed.	Lemma	lreg_size	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "last", "lead_coefC", "lead_coef_eq0", "lreg", "lreg_neq0", "mul_polyC", "mulrI_eq0", "nz_p", "scaler0", "size", "size_polyC", "size_proper_mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lreg_polyZ_eq0 c p : GRing.lreg c -> (c *: p == 0) = (p == 0).	Proof. by rewrite -!size_poly_eq0 => /lreg_size->. Qed.	Lemma	lreg_polyZ_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" ]	[ "lreg", "lreg_size", "size_poly_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coef_lreg c p : GRing.lreg c -> lead_coef (c : p) = c lead_coef p.	Proof. by move=> reg_c; rewrite !lead_coefE coefZ lreg_size. Qed.	Lemma	lead_coef_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" ]	[ "coefZ", "lead_coef", "lead_coefE", "lreg", "lreg_size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rreg_size c p : GRing.rreg c -> size (p * c%:P) = size p.	Proof. move=> reg_c; have [-> \| nz_p] := eqVneq p 0; first by rewrite mul0r. rewrite size_proper_mul; last by rewrite size_polyC rreg_neq0 ?addn1. by rewrite lead_coefC mulIr_eq0 ?lead_coef_eq0. Qed.	Lemma	rreg_size	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addn1", "eqVneq", "last", "lead_coefC", "lead_coef_eq0", "mul0r", "mulIr_eq0", "nz_p", "rreg", "rreg_neq0", "size", "size_polyC", "size_proper_mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rreg_polyMC_eq0 c p : GRing.rreg c -> (p * c%:P == 0) = (p == 0).	Proof. by rewrite -!size_poly_eq0 => /rreg_size->. Qed.	Lemma	rreg_polyMC_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" ]	[ "rreg", "rreg_size", "size_poly_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rreg_div0 q r d : GRing.rreg (lead_coef d) -> size r < size d -> (q * d + r == 0) = (q == 0) && (r == 0).	Proof. move=> /mulIr_eq0 reg_d lt_r_d; rewrite addrC. have [-> \| nz_q] := eqVneq q 0; first by rewrite mul0r addr0. have qd0: lead_coef q * lead_coef d != 0 by rewrite reg_d lead_coef_eq0. apply/negbTE; rewrite -size_poly_eq0 addrC size_polyDl; last first. by rewrite size_poly_eq0 -lead_coef_eq0 lead_coef_proper_mul....	Lemma	rreg_div0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addSn", "addr0", "addrC", "apply", "eqVneq", "last", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "leq_addl", "leq_trans", "mul0r", "mulIr_eq0", "rreg", "size", "size_polyDl", "size_poly_eq0", "size_proper_mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monic_comreg p : p \is monic -> GRing.comm p (lead_coef p)%:P /\ GRing.rreg (lead_coef p).	Proof. by move/monicP->; split; [apply: commr1 \| apply: rreg1]. Qed.	Lemma	monic_comreg	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "comm", "commr1", "lead_coef", "monic", "monicP", "rreg", "rreg1", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root p : pred R	:= fun x => p.[x] == 0.	Definition	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" ]	[]	Roots of polynomials	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_root p x : (x \in root p) = (p.[x] == 0).	Proof. by []. Qed.	Lemma	mem_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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rootE p x : (root p x = (p.[x] == 0)) * ((x \in root p) = (p.[x] == 0)).	Proof. by []. Qed.	Lemma	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" ]	[ "root" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rootP p x : reflect (p.[x] = 0) (root p x).	Proof. exact: eqP. Qed.	Lemma	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" ]	[ "root" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rootPt p x : reflect (p.[x] == 0) (root p x).	Proof. exact: idP. Qed.	Lemma	rootPt	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rootPf p x : reflect ((p.[x] == 0) = false) (~~ root p x).	Proof. exact: negPf. Qed.	Lemma	rootPf	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rootC a x : root a%:P x = (a == 0).	Proof. by rewrite rootE hornerC. Qed.	Lemma	rootC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "root", "rootE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root0 x : root 0 x.	Proof. by rewrite rootC. Qed.	Lemma	root0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "rootC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root1 x : ~~ root 1 x.	Proof. by rewrite rootC oner_eq0. Qed.	Lemma	root1	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "oner_eq0", "root", "rootC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rootX x : root 'X x = (x == 0).	Proof. by rewrite rootE hornerX. Qed.	Lemma	rootX	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "hornerX", "root", "rootE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
root_size_gt1 a p : p != 0 -> root p a -> 1 < size p.	Proof. rewrite ltnNge => nz_p; apply: contraL => /size1_polyC Dp. by rewrite Dp rootC -polyC_eq0 -Dp. Qed.	Lemma	root_size_gt1	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "ltnNge", "nz_p", "polyC_eq0", "root", "rootC", "size", "size1_polyC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\poly_ ( i < n ) E"	:= (poly n (fun i => E)) : ring_scope.	Notation	\poly_ ( i < n ) E	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"c %:P"	:= (polyC c) : ring_scope.	Notation	c %:P	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "polyC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'X"	:= (polyX _) : ring_scope.	Notation	'X	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "polyX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''X^' n"	:= ('X ^+ n) : ring_scope.	Notation	''X^' n	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"a ^` ()"	:= (deriv a) : ring_scope.	Notation	a ^` ()	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "deriv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_add	:= size_polyD (only parsing).	Notation	size_add	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_polyD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_addl	:= size_polyDl (only parsing).	Notation	size_addl	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_polyDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_mul_leq	:= size_polyMleq (only parsing).	Notation	size_mul_leq	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "size_polyMleq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_prod_leq	:= size_poly_prod_leq (only parsing).	Notation	size_prod_leq	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "size_poly_prod_leq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_exp_leq	:= size_poly_exp_leq (only parsing).	Notation	size_exp_leq	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "size_poly_exp_leq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
char_poly	:= pchar_poly (only parsing).	Notation	char_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" ]	[ "pchar_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_poly_def p	:= \poly_(i < size p) - p`_i.	Definition	opp_poly_def	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "size" ]	Zmodule structure for polynomial	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_poly_key : unit.	Proof. by []. Qed.	Fact	opp_poly_key	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "unit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_poly	:= locked_with opp_poly_key opp_poly_def.	Definition	opp_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" ]	[ "opp_poly_def", "opp_poly_key" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_poly_unlockable	:= [unlockable fun opp_poly].	Canonical	opp_poly_unlockable	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "opp_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coef_opp_poly p i : (opp_poly p)`_i = - p`_i.	Proof. rewrite unlock coef_poly /=. by case: leqP => // le_p_i; rewrite nth_default ?oppr0. Qed.	Fact	coef_opp_poly	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "coef_poly", "leqP", "nth_default", "opp_poly", "oppr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_polyN : left_inverse 0%:P opp_poly (@add_poly _).	Proof. by move=> p; apply/polyP => i; rewrite coefD coef_opp_poly coef0 addNr. Qed.	Fact	add_polyN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addNr", "add_poly", "apply", "coef0", "coefD", "coef_opp_poly", "opp_poly", "polyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefN p i : (- p)`_i = - p`_i.	Proof. exact: coef_opp_poly. Qed.	Lemma	coefN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_opp_poly" ]	Size, leading coef, morphism properties of coef	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefB p q i : (p - q)`_i = p`_i - q`_i.	Proof. by rewrite coefD coefN. Qed.	Lemma	coefB	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "coefD", "coefN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coefMNn p n i : (p - n)`_i = p`_i - n.	Proof. by rewrite coefN coefMn. Qed.	Lemma	coefMNn	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "coefMn", "coefN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyCN : {morph (@polyC R) : c / - c}.	Proof. exact: raddfN. Qed.	Lemma	polyCN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "polyC", "raddfN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyCB : {morph (@polyC R) : a b / a - b}.	Proof. exact: raddfB. Qed.	Lemma	polyCB	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "polyC", "raddfB" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_polyN p : size (- p) = size p.	Proof. by apply/eqP; rewrite eqn_leq -{3}(opprK p) -[-%R]/opp_poly unlock !size_poly. Qed.	Lemma	size_polyN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "eqn_leq", "opp_poly", "opprK", "size", "size_poly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefN p : lead_coef (- p) = - lead_coef p.	Proof. by rewrite /lead_coef size_polyN coefN. Qed.	Lemma	lead_coefN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "coefN", "lead_coef", "size_polyN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_Msign p n : size ((-1) ^+ n * p) = size p.	Proof. by rewrite -signr_odd; case: (odd n); rewrite ?mul1r // mulN1r size_polyN. Qed.	Lemma	size_Msign	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "mul1r", "mulN1r", "odd", "signr_odd", "size", "size_polyN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyseqXsubC a : 'X - a%:P = [:: - a; 1] :> seq R.	Proof. by rewrite -polyCN polyseqXaddC. Qed.	Lemma	polyseqXsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "polyCN", "polyseqXaddC", "seq" ]	The indeterminate	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_XsubC a : size ('X - a%:P) = 2.	Proof. by rewrite polyseqXsubC. Qed.	Lemma	size_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" ]	[ "polyseqXsubC", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefXsubC a : lead_coef ('X - a%:P) = 1.	Proof. by rewrite lead_coefE polyseqXsubC. Qed.	Lemma	lead_coefXsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "lead_coef", "lead_coefE", "polyseqXsubC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyXsubC_eq0 a : ('X - a%:P == 0) = false.	Proof. by rewrite -nil_poly polyseqXsubC. Qed.	Lemma	polyXsubC_eq0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "nil_poly", "polyseqXsubC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lead_coefXnsubC n c : 0 < n -> lead_coef ('X^n - c%:P) = 1.	Proof. by move=> n_gt0; rewrite -polyCN lead_coefXnaddC. Qed.	Lemma	lead_coefXnsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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_coefXnaddC", "n_gt0", "polyCN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_XnsubC n c : 0 < n -> size ('X^n - c%:P) = n.+1.	Proof. by move=> *; rewrite -polyCN size_XnaddC. Qed.	Lemma	size_XnsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "polyCN", "size", "size_XnaddC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_Xn_sub_1 n : n > 0 -> size ('X^n - 1 : {poly R}) = n.+1.	Proof. exact/size_XnsubC. Qed.	Lemma	size_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" ]	[ "poly", "size", "size_XnsubC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerN p x : (- p).[x] = - p.[x].	Proof. by apply/esym/addr0_eq; rewrite -hornerD subrr horner0. Qed.	Lemma	hornerN	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addr0_eq", "apply", "horner0", "hornerD", "subrr" ]	Horner evaluation of polynomials	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerXsubC a x : ('X - a%:P).[x] = x - a.	Proof. by rewrite hornerD hornerN hornerC hornerX. Qed.	Lemma	hornerXsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "hornerX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hornerE_comm	:= (hornerD, hornerN, hornerX, hornerC, horner_cons, simp, hornerCM, hornerZ, (fun p x => hornerM_comm p (comm_polyX x))).	Definition	hornerE_comm	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "comm_polyX", "hornerC", "hornerCM", "hornerD", "hornerM_comm", "hornerN", "hornerX", "hornerZ", "horner_cons", "simp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverNr (zmodS : zmodClosed R) : oppr_closed (polyOver zmodS).	Proof. by move=> p /polyOverP Sp; apply/polyOverP=> i; rewrite coefN rpredN. Qed.	Fact	polyOverNr	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "coefN", "oppr_closed", "polyOver", "polyOverP", "rpredN" ]	Lifting a ring predicate to polynomials.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverXaddC c : ('X + c%:P \in polyOver S) = (c \in S).	Proof. by rewrite rpredDl ?polyOverX ?polyOverC. Qed.	Lemma	polyOverXaddC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "polyOverX", "rpredDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverXnaddC n c : ('X^n + c%:P \is a polyOver S) = (c \in S).	Proof. by rewrite rpredDl ?polyOverXn// ?polyOverC. Qed.	Lemma	polyOverXnaddC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "rpredDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
polyOverXsubC c : ('X - c%:P \in polyOver S) = (c \in S).	Proof. by rewrite rpredBl ?polyOverX ?polyOverC. Qed.	Lemma	polyOverXsubC	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "polyOverX", "rpredBl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnZ n : scalable (derivn n).

Proof. exact: linearZZ. Qed.

Lemma

derivnZ

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "linearZZ", "scalable" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnXn m n : ('X^m)^`(n) = 'X^(m - n) *+ m ^_ n.

Proof. apply/polyP=>i; rewrite coef_derivn coefMn !coefXn. case: (ltnP m n) => [lt_m_n | le_m_n]. by rewrite eqn_leq leqNgt ltn_addr // mul0rn ffact_small. by rewrite -{1 3}(subnKC le_m_n) eqn_add2l; case: eqP => [->|]; rewrite ?mul0rn. Qed.

Lemma

derivnXn

algebra

algebra/poly.v

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

[ "apply", "coefMn", "coefXn", "coef_derivn", "eqn_add2l", "eqn_leq", "ffact_small", "leqNgt", "ltnP", "ltn_addr", "mul0rn", "polyP", "subnKC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivnMXaddC n p c : (p * 'X + c%:P)^`(n.+1) = p^`(n) *+ n.+1 + p^`(n.+1) * 'X.

Proof. elim: n => [|n IHn]; first by rewrite derivn1 derivMXaddC. rewrite derivnS IHn derivD derivM derivX mulr1 derivMn -!derivnS. by rewrite addrA addrAC -mulrSr. Qed.

Lemma

derivnMXaddC

algebra

algebra/poly.v

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

[ "addrA", "addrAC", "derivD", "derivM", "derivMXaddC", "derivMn", "derivX", "derivn1", "derivnS", "mulr1", "mulrSr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

derivn_poly0 p n : size p <= n -> p^`(n) = 0.

Proof. move=> le_p_n; apply/polyP=> i; rewrite coef_derivn. rewrite nth_default; last by rewrite mul0rn coef0. exact/(leq_trans le_p_n)/leq_addr. Qed.

Lemma

derivn_poly0

algebra

algebra/poly.v

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

[ "apply", "coef0", "coef_derivn", "last", "leq_addr", "leq_trans", "mul0rn", "nth_default", "polyP", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt_size_deriv (p : {poly R}) : p != 0 -> size p^`() < size p.

Proof. by move=> /polySpred->; apply: size_poly. Qed.

Lemma

lt_size_deriv

algebra

algebra/poly.v

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

[ "apply", "poly", "polySpred", "size", "size_poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivn n p

:= \poly_(i < size p - n) (p`_(n + i) *+ 'C(n + i, n)).

Definition

nderivn

algebra

algebra/poly.v

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

A normalising version of derivation to get the division by n! in Taylor

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"a ^`N ( n )"

:= (nderivn n a) : ring_scope.

Notation

a ^`N ( n )

algebra

algebra/poly.v

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

[ "nderivn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coef_nderivn n p i : p^`N(n)`_i = p`_(n + i) *+ 'C(n + i, n).

Proof. rewrite coef_poly ltn_subRL; case: leqP => // le_p_ni. by rewrite nth_default ?mul0rn. Qed.

Lemma

coef_nderivn

algebra

algebra/poly.v

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

[ "coef_poly", "leqP", "ltn_subRL", "mul0rn", "nth_default" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivn_def n p : p^`(n) = p^`N(n) *+ n`!.

Proof. by apply/polyP=> i; rewrite coefMn coef_nderivn coef_derivn -mulrnA bin_ffact. Qed.

Lemma

nderivn_def

algebra

algebra/poly.v

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

[ "apply", "bin_ffact", "coefMn", "coef_derivn", "coef_nderivn", "mulrnA", "polyP" ]

Here is the division by n!

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOver_nderivn (ringS : semiringClosed R) : {in polyOver ringS, forall p n, p^`N(n) \in polyOver ringS}.

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

Lemma

polyOver_nderivn

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_nderivn", "polyOver", "polyOverP", "rpredMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivn0 p : p^`N(0) = p.

Proof. by rewrite -[p^`N(0)](nderivn_def 0). Qed.

Lemma

nderivn0

algebra

algebra/poly.v

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

[ "nderivn_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivn1 p : p^`N(1) = p^`().

Proof. by rewrite -[p^`N(1)](nderivn_def 1). Qed.

Lemma

nderivn1

algebra

algebra/poly.v

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

[ "nderivn_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. apply/polyP=> i; rewrite coef_nderivn. by case: n => [|n]; rewrite ?bin0 // coef0 coefC mul0rn. Qed.

Lemma

nderivnC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "bin0", "coef0", "coefC", "coef_nderivn", "mul0rn", "polyP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivnXn m n : ('X^m)^`N(n) = 'X^(m - n) *+ 'C(m, n).

Proof. apply/polyP=> i; rewrite coef_nderivn coefMn !coefXn. have [lt_m_n | le_n_m] := ltnP m n. by rewrite eqn_leq leqNgt ltn_addr // mul0rn bin_small. by rewrite -{1 3}(subnKC le_n_m) eqn_add2l; case: eqP => [->|]; rewrite ?mul0rn. Qed.

Lemma

nderivnXn

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "bin_small", "coefMn", "coefXn", "coef_nderivn", "eqn_add2l", "eqn_leq", "leqNgt", "ltnP", "ltn_addr", "mul0rn", "polyP", "subnKC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivn_is_semilinear n : semilinear (nderivn n).

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

Fact

nderivn_is_semilinear

algebra

algebra/poly.v

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

[ "apply", "coefD", "coefZ", "coef_nderivn", "mulrnAr", "mulrnDl", "nderivn", "polyP", "semilinear", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivnD n : {morph nderivn n : p q / p + q}.

Proof. exact: linearD. Qed.

Lemma

nderivnD

algebra

algebra/poly.v

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

[ "linearD", "nderivn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. exact: linearMn. Qed.

Lemma

nderivnMn

algebra

algebra/poly.v

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

[ "linearMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivnZ n : scalable (nderivn n).

Proof. exact: linearZZ. Qed.

Lemma

nderivnZ

algebra

algebra/poly.v

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

[ "linearZZ", "nderivn", "scalable" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivnMXaddC n p c : (p * 'X + c%:P)^`N(n.+1) = p^`N(n) + p^`N(n.+1) * 'X.

Proof. apply/polyP=> i; rewrite coef_nderivn !coefD !coefMX coefC. rewrite !addSn /= !coef_nderivn addr0 binS mulrnDr addrC; congr (_ + _). by rewrite addSnnS; case: i; rewrite // addn0 bin_small. Qed.

Lemma

nderivnMXaddC

algebra

algebra/poly.v

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

[ "addSn", "addSnnS", "addn0", "addr0", "addrC", "apply", "binS", "bin_small", "coefC", "coefD", "coefMX", "coef_nderivn", "mulrnDr", "polyP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderivn_poly0 p n : size p <= n -> p^`N(n) = 0.

Proof. move=> le_p_n; apply/polyP=> i; rewrite coef_nderivn. rewrite nth_default; last by rewrite mul0rn coef0. exact/(leq_trans le_p_n)/leq_addr. Qed.

Lemma

nderivn_poly0

algebra

algebra/poly.v

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

[ "apply", "coef0", "coef_nderivn", "last", "leq_addr", "leq_trans", "mul0rn", "nth_default", "polyP", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderiv_taylor p x h : GRing.comm x h -> p.[x + h] = \sum_(i < size p) p^`N(i).[x] * h ^+ i.

Proof. move/commrX=> cxh; elim/poly_ind: p => [|p c IHp]. by rewrite size_poly0 big_ord0 horner0. rewrite hornerMXaddC size_MXaddC. have [-> | nz_p] := eqVneq p 0. rewrite horner0 !simp; have [-> | _] := c =P 0; first by rewrite big_ord0. by rewrite size_poly0 big_ord_recl big_ord0 nderivn0 hornerC !simp. rewrite...

Lemma

nderiv_taylor

algebra

algebra/poly.v

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

[ "addn", "addrA", "addrAC", "addrC", "apply", "big_distrl", "big_ord0", "big_ord_recl", "big_ord_recr", "big_split", "bump", "comm", "comm_polyX", "commrX", "eqVneq", "eq_bigr", "exprSr", "horner0", "hornerC", "hornerD", "hornerMXaddC", "hornerM_comm", "hornerX", "last",...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nderiv_taylor_wide n p x h : GRing.comm x h -> size p <= n -> p.[x + h] = \sum_(i < n) p^`N(i).[x] * h ^+ i.

Proof. move/nderiv_taylor=> -> le_p_n. rewrite (big_ord_widen n (fun i => p^`N(i).[x] * h ^+ i)) // big_mkcond. apply: eq_bigr => i _; case: leqP => // /nderivn_poly0->. by rewrite horner0 simp. Qed.

Lemma

nderiv_taylor_wide

algebra

algebra/poly.v

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

[ "apply", "big_mkcond", "big_ord_widen", "comm", "eq_bigr", "horner0", "leqP", "nderiv_taylor", "nderivn_poly0", "simp", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_pred

:= fun p => lead_coef p == 1.

Definition

monic_pred

algebra

algebra/poly.v

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

[ "lead_coef" ]

Monic predicate

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic

:= [qualify p | monic_pred p].

Definition

monic

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicE p : (p \is monic) = (lead_coef p == 1).

Proof. by []. Qed.

Lemma

monicE

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicP p : reflect (lead_coef p = 1) (p \is monic).

Proof. exact: eqP. Qed.

Lemma

monicP

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic1 : 1 \is monic.

Proof. exact/eqP/lead_coef1. Qed.

Lemma

monic1

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_coef1", "monic" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicX : 'X \is monic.

Proof. exact/eqP/lead_coefX. Qed.

Lemma

monicX

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicXn n : 'X^n \is monic.

Proof. exact/eqP/lead_coefXn. Qed.

Lemma

monicXn

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_neq0 p : p \is monic -> p != 0.

Proof. by rewrite -lead_coef_eq0 => /eqP->; apply: oner_neq0. Qed.

Lemma

monic_neq0

algebra

algebra/poly.v

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

[ "apply", "lead_coef_eq0", "monic", "oner_neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coef_monicM p q : p \is monic -> lead_coef (p * q) = lead_coef q.

Proof. have [-> | nz_q] := eqVneq q 0; first by rewrite mulr0. by move/monicP=> mon_p; rewrite lead_coef_proper_mul mon_p mul1r ?lead_coef_eq0. Qed.

Lemma

lead_coef_monicM

algebra

algebra/poly.v

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

[ "eqVneq", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "monic", "monicP", "mul1r", "mulr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coef_Mmonic p q : q \is monic -> lead_coef (p * q) = lead_coef p.

Proof. have [-> | nz_p] := eqVneq p 0; first by rewrite mul0r. by move/monicP=> mon_q; rewrite lead_coef_proper_mul mon_q mulr1 ?lead_coef_eq0. Qed.

Lemma

lead_coef_Mmonic

algebra

algebra/poly.v

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

[ "eqVneq", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "monic", "monicP", "mul0r", "mulr1", "nz_p" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_monicM p q : p \is monic -> q != 0 -> size (p * q) = (size p + size q).-1.

Proof. move/monicP=> mon_p nz_q. by rewrite size_proper_mul // mon_p mul1r lead_coef_eq0. Qed.

Lemma

size_monicM

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_eq0", "monic", "monicP", "mul1r", "size", "size_proper_mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_Mmonic p q : p != 0 -> q \is monic -> size (p * q) = (size p + size q).-1.

Proof. move=> nz_p /monicP mon_q. by rewrite size_proper_mul // mon_q mulr1 lead_coef_eq0. Qed.

Lemma

size_Mmonic

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_eq0", "monic", "monicP", "mulr1", "nz_p", "size", "size_proper_mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicMl p q : p \is monic -> (p * q \is monic) = (q \is monic).

Proof. by move=> mon_p; rewrite !monicE lead_coef_monicM. Qed.

Lemma

monicMl

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_monicM", "monic", "monicE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicMr p q : q \is monic -> (p * q \is monic) = (p \is monic).

Proof. by move=> mon_q; rewrite !monicE lead_coef_Mmonic. Qed.

Lemma

monicMr

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_Mmonic", "monic", "monicE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_mulr_closed : mulr_closed monic.

Proof. by split=> [|p q mon_p]; rewrite (monic1, monicMl). Qed.

Fact

monic_mulr_closed

algebra

algebra/poly.v

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

[ "monic", "monic1", "monicMl", "mulr_closed", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_exp p n : p \is monic -> p ^+ n \is monic.

Proof. exact: rpredX. Qed.

Lemma

monic_exp

algebra

algebra/poly.v

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

[ "monic", "rpredX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_prod I rI (P : pred I) (F : I -> {poly R}): (forall i, P i -> F i \is monic) -> \prod_(i <- rI | P i) F i \is monic.

Proof. exact: rpred_prod. Qed.

Lemma

monic_prod

algebra

algebra/poly.v

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

[ "monic", "poly", "rpred_prod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicXaddC c : 'X + c%:P \is monic.

Proof. exact/eqP/lead_coefXaddC. Qed.

Lemma

monicXaddC

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Lemma

monicXnaddC

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lreg_lead0 p : GRing.lreg (lead_coef p) -> p != 0.

Proof. by move/lreg_neq0; rewrite lead_coef_eq0. Qed.

Lemma

lreg_lead0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_eq0", "lreg", "lreg_neq0" ]

Some facts about regular elements.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rreg_lead0 p : GRing.rreg (lead_coef p) -> p != 0.

Proof. by move/rreg_neq0; rewrite lead_coef_eq0. Qed.

Lemma

rreg_lead0

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_eq0", "rreg", "rreg_neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lreg_size c p : GRing.lreg c -> size (c *: p) = size p.

Proof. move=> reg_c; have [-> | nz_p] := eqVneq p 0; first by rewrite scaler0. rewrite -mul_polyC size_proper_mul; last by rewrite size_polyC lreg_neq0. by rewrite lead_coefC mulrI_eq0 ?lead_coef_eq0. Qed.

Lemma

lreg_size

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "last", "lead_coefC", "lead_coef_eq0", "lreg", "lreg_neq0", "mul_polyC", "mulrI_eq0", "nz_p", "scaler0", "size", "size_polyC", "size_proper_mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lreg_polyZ_eq0 c p : GRing.lreg c -> (c *: p == 0) = (p == 0).

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

Lemma

lreg_polyZ_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" ]

[ "lreg", "lreg_size", "size_poly_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coef_lreg c p : GRing.lreg c -> lead_coef (c *: p) = c * lead_coef p.

Proof. by move=> reg_c; rewrite !lead_coefE coefZ lreg_size. Qed.

Lemma

lead_coef_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" ]

[ "coefZ", "lead_coef", "lead_coefE", "lreg", "lreg_size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rreg_size c p : GRing.rreg c -> size (p * c%:P) = size p.

Proof. move=> reg_c; have [-> | nz_p] := eqVneq p 0; first by rewrite mul0r. rewrite size_proper_mul; last by rewrite size_polyC rreg_neq0 ?addn1. by rewrite lead_coefC mulIr_eq0 ?lead_coef_eq0. Qed.

Lemma

rreg_size

algebra

algebra/poly.v

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

[ "addn1", "eqVneq", "last", "lead_coefC", "lead_coef_eq0", "mul0r", "mulIr_eq0", "nz_p", "rreg", "rreg_neq0", "size", "size_polyC", "size_proper_mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rreg_polyMC_eq0 c p : GRing.rreg c -> (p * c%:P == 0) = (p == 0).

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

Lemma

rreg_polyMC_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" ]

[ "rreg", "rreg_size", "size_poly_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rreg_div0 q r d : GRing.rreg (lead_coef d) -> size r < size d -> (q * d + r == 0) = (q == 0) && (r == 0).

Proof. move=> /mulIr_eq0 reg_d lt_r_d; rewrite addrC. have [-> | nz_q] := eqVneq q 0; first by rewrite mul0r addr0. have qd0: lead_coef q * lead_coef d != 0 by rewrite reg_d lead_coef_eq0. apply/negbTE; rewrite -size_poly_eq0 addrC size_polyDl; last first. by rewrite size_poly_eq0 -lead_coef_eq0 lead_coef_proper_mul....

Lemma

rreg_div0

algebra

algebra/poly.v

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

[ "addSn", "addr0", "addrC", "apply", "eqVneq", "last", "lead_coef", "lead_coef_eq0", "lead_coef_proper_mul", "leq_addl", "leq_trans", "mul0r", "mulIr_eq0", "rreg", "size", "size_polyDl", "size_poly_eq0", "size_proper_mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monic_comreg p : p \is monic -> GRing.comm p (lead_coef p)%:P /\ GRing.rreg (lead_coef p).

Proof. by move/monicP->; split; [apply: commr1 | apply: rreg1]. Qed.

Lemma

monic_comreg

algebra

algebra/poly.v

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

[ "apply", "comm", "commr1", "lead_coef", "monic", "monicP", "rreg", "rreg1", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root p : pred R

:= fun x => p.[x] == 0.

Definition

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" ]

[]

Roots of polynomials

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_root p x : (x \in root p) = (p.[x] == 0).

Proof. by []. Qed.

Lemma

mem_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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rootE p x : (root p x = (p.[x] == 0)) * ((x \in root p) = (p.[x] == 0)).

Proof. by []. Qed.

Lemma

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" ]

[ "root" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rootP p x : reflect (p.[x] = 0) (root p x).

Proof. exact: eqP. Qed.

Lemma

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" ]

[ "root" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rootPt p x : reflect (p.[x] == 0) (root p x).

Proof. exact: idP. Qed.

Lemma

rootPt

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rootPf p x : reflect ((p.[x] == 0) = false) (~~ root p x).

Proof. exact: negPf. Qed.

Lemma

rootPf

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rootC a x : root a%:P x = (a == 0).

Proof. by rewrite rootE hornerC. Qed.

Lemma

rootC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "root", "rootE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root0 x : root 0 x.

Proof. by rewrite rootC. Qed.

Lemma

root0

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root1 x : ~~ root 1 x.

Proof. by rewrite rootC oner_eq0. Qed.

Lemma

root1

algebra

algebra/poly.v

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

[ "oner_eq0", "root", "rootC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rootX x : root 'X x = (x == 0).

Proof. by rewrite rootE hornerX. Qed.

Lemma

rootX

algebra

algebra/poly.v

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

[ "hornerX", "root", "rootE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

root_size_gt1 a p : p != 0 -> root p a -> 1 < size p.

Proof. rewrite ltnNge => nz_p; apply: contraL => /size1_polyC Dp. by rewrite Dp rootC -polyC_eq0 -Dp. Qed.

Lemma

root_size_gt1

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "ltnNge", "nz_p", "polyC_eq0", "root", "rootC", "size", "size1_polyC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\poly_ ( i < n ) E"

:= (poly n (fun i => E)) : ring_scope.

Notation

\poly_ ( i < n ) E

algebra

algebra/poly.v

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

[ "poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"c %:P"

:= (polyC c) : ring_scope.

Notation

c %:P

algebra

algebra/poly.v

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

[ "polyC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'X"

:= (polyX _) : ring_scope.

Notation

'X

algebra

algebra/poly.v

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

[ "polyX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''X^' n"

:= ('X ^+ n) : ring_scope.

Notation

''X^' n

algebra

algebra/poly.v

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

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"a ^` ()"

:= (deriv a) : ring_scope.

Notation

a ^` ()

algebra

algebra/poly.v

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

[ "deriv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_add

:= size_polyD (only parsing).

Notation

size_add

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_addl

:= size_polyDl (only parsing).

Notation

size_addl

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_mul_leq

:= size_polyMleq (only parsing).

Notation

size_mul_leq

algebra

algebra/poly.v

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

[ "size_polyMleq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_prod_leq

:= size_poly_prod_leq (only parsing).

Notation

size_prod_leq

algebra

algebra/poly.v

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

[ "size_poly_prod_leq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_exp_leq

:= size_poly_exp_leq (only parsing).

Notation

size_exp_leq

algebra

algebra/poly.v

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

[ "size_poly_exp_leq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

char_poly

:= pchar_poly (only parsing).

Notation

char_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" ]

[ "pchar_poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_poly_def p

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

Definition

opp_poly_def

algebra

algebra/poly.v

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

[ "size" ]

Zmodule structure for polynomial

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_poly_key : unit.

Proof. by []. Qed.

Fact

opp_poly_key

algebra

algebra/poly.v

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

[ "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_poly

:= locked_with opp_poly_key opp_poly_def.

Definition

opp_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" ]

[ "opp_poly_def", "opp_poly_key" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp_poly_unlockable

:= [unlockable fun opp_poly].

Canonical

opp_poly_unlockable

algebra

algebra/poly.v

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

[ "opp_poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coef_opp_poly p i : (opp_poly p)`_i = - p`_i.

Proof. rewrite unlock coef_poly /=. by case: leqP => // le_p_i; rewrite nth_default ?oppr0. Qed.

Fact

coef_opp_poly

algebra

algebra/poly.v

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

[ "coef_poly", "leqP", "nth_default", "opp_poly", "oppr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add_polyN : left_inverse 0%:P opp_poly (@add_poly _).

Proof. by move=> p; apply/polyP => i; rewrite coefD coef_opp_poly coef0 addNr. Qed.

Fact

add_polyN

algebra

algebra/poly.v

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

[ "addNr", "add_poly", "apply", "coef0", "coefD", "coef_opp_poly", "opp_poly", "polyP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coefN p i : (- p)`_i = - p`_i.

Proof. exact: coef_opp_poly. Qed.

Lemma

coefN

algebra

algebra/poly.v

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

Size, leading coef, morphism properties of coef

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coefB p q i : (p - q)`_i = p`_i - q`_i.

Proof. by rewrite coefD coefN. Qed.

Lemma

coefB

algebra

algebra/poly.v

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

[ "coefD", "coefN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coefMNn p n i : (p *- n)`_i = p`_i *- n.

Proof. by rewrite coefN coefMn. Qed.

Lemma

coefMNn

algebra

algebra/poly.v

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

[ "coefMn", "coefN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyCN : {morph (@polyC R) : c / - c}.

Proof. exact: raddfN. Qed.

Lemma

polyCN

algebra

algebra/poly.v

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

[ "polyC", "raddfN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyCB : {morph (@polyC R) : a b / a - b}.

Proof. exact: raddfB. Qed.

Lemma

polyCB

algebra

algebra/poly.v

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

[ "polyC", "raddfB" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_polyN p : size (- p) = size p.

Proof. by apply/eqP; rewrite eqn_leq -{3}(opprK p) -[-%R]/opp_poly unlock !size_poly. Qed.

Lemma

size_polyN

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "eqn_leq", "opp_poly", "opprK", "size", "size_poly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coefN p : lead_coef (- p) = - lead_coef p.

Proof. by rewrite /lead_coef size_polyN coefN. Qed.

Lemma

lead_coefN

algebra

algebra/poly.v

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

[ "coefN", "lead_coef", "size_polyN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_Msign p n : size ((-1) ^+ n * p) = size p.

Proof. by rewrite -signr_odd; case: (odd n); rewrite ?mul1r // mulN1r size_polyN. Qed.

Lemma

size_Msign

algebra

algebra/poly.v

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

[ "mul1r", "mulN1r", "odd", "signr_odd", "size", "size_polyN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyseqXsubC a : 'X - a%:P = [:: - a; 1] :> seq R.

Proof. by rewrite -polyCN polyseqXaddC. Qed.

Lemma

polyseqXsubC

algebra

algebra/poly.v

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

[ "polyCN", "polyseqXaddC", "seq" ]

The indeterminate

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_XsubC a : size ('X - a%:P) = 2.

Proof. by rewrite polyseqXsubC. Qed.

Lemma

size_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" ]

[ "polyseqXsubC", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lead_coefXsubC a : lead_coef ('X - a%:P) = 1.

Proof. by rewrite lead_coefE polyseqXsubC. Qed.

Lemma

lead_coefXsubC

algebra

algebra/poly.v

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

[ "lead_coef", "lead_coefE", "polyseqXsubC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyXsubC_eq0 a : ('X - a%:P == 0) = false.

Proof. by rewrite -nil_poly polyseqXsubC. Qed.

Lemma

polyXsubC_eq0

algebra

algebra/poly.v

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

[ "nil_poly", "polyseqXsubC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> n_gt0; rewrite -polyCN lead_coefXnaddC. Qed.

Lemma

lead_coefXnsubC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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_coefXnaddC", "n_gt0", "polyCN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> *; rewrite -polyCN size_XnaddC. Qed.

Lemma

size_XnsubC

algebra

algebra/poly.v

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

[ "polyCN", "size", "size_XnaddC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_Xn_sub_1 n : n > 0 -> size ('X^n - 1 : {poly R}) = n.+1.

Proof. exact/size_XnsubC. Qed.

Lemma

size_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" ]

[ "poly", "size", "size_XnsubC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerN p x : (- p).[x] = - p.[x].

Proof. by apply/esym/addr0_eq; rewrite -hornerD subrr horner0. Qed.

Lemma

hornerN

algebra

algebra/poly.v

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

[ "addr0_eq", "apply", "horner0", "hornerD", "subrr" ]

Horner evaluation of polynomials

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerXsubC a x : ('X - a%:P).[x] = x - a.

Proof. by rewrite hornerD hornerN hornerC hornerX. Qed.

Lemma

hornerXsubC

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hornerE_comm

:= (hornerD, hornerN, hornerX, hornerC, horner_cons, simp, hornerCM, hornerZ, (fun p x => hornerM_comm p (comm_polyX x))).

Definition

hornerE_comm

algebra

algebra/poly.v

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

[ "comm_polyX", "hornerC", "hornerCM", "hornerD", "hornerM_comm", "hornerN", "hornerX", "hornerZ", "horner_cons", "simp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverNr (zmodS : zmodClosed R) : oppr_closed (polyOver zmodS).

Proof. by move=> p /polyOverP Sp; apply/polyOverP=> i; rewrite coefN rpredN. Qed.

Fact

polyOverNr

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "coefN", "oppr_closed", "polyOver", "polyOverP", "rpredN" ]

Lifting a ring predicate to polynomials.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverXaddC c : ('X + c%:P \in polyOver S) = (c \in S).

Proof. by rewrite rpredDl ?polyOverX ?polyOverC. Qed.

Lemma

polyOverXaddC

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "polyOverX", "rpredDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Lemma

polyOverXnaddC

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

polyOverXsubC c : ('X - c%:P \in polyOver S) = (c \in S).

Proof. by rewrite rpredBl ?polyOverX ?polyOverC. Qed.

Lemma

polyOverXsubC

algebra

algebra/poly.v

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d