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
a	:= p`_2.	Let	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" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
b	:= p`_1.	Let	b	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
c	:= p`_0.	Let	c	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
pneq0 : p != 0.	Proof. by rewrite -size_poly_gt0 degp. Qed.	Let	pneq0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "degp", "size_poly_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aneq0 : a != 0.	Proof. by move: pneq0; rewrite -lead_coef_eq0 lead_coefE degp. Qed.	Let	aneq0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "degp", "lead_coefE", "lead_coef_eq0", "pneq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
a2neq0 : 2 * a != 0.	Proof. by rewrite mulf_neq0. Qed.	Let	a2neq0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "mulf_neq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sqa2neq0 : (2 * a) ^+ 2 != 0.	Proof. exact: expf_neq0. Qed.	Let	sqa2neq0	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "expf_neq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
aa4 : 4 * a * a = (2 * a)^+2.	Proof. by rewrite expr2 mulrACA mulrA -natrM. Qed.	Let	aa4	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "expr2", "mulrA", "mulrACA", "natrM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
splitr (x : F) : x = x / 2 + x / 2.	Proof. by rewrite -mulr2n -[RHS]mulr_natr divfK. Qed.	Let	splitr	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "divfK", "mulr2n", "mulr_natr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pE : p = a : 'X^2 + b : 'X + c%:P.	Proof. apply/polyP => + /[!coefE] => -[\|[\|[\|i]]] /=; rewrite !Monoid.simpm//. by rewrite nth_default// degp. Qed.	Let	pE	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "coefE", "degp", "nth_default", "polyP", "simpm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
delta	:= b ^+ 2 - 4 * a * c.	Let	delta	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
deg2_poly_canonical : p = a : (('X + (b / (2 a))%:P)^+2 - (delta / (4 * a ^+ 2))%:P).	Proof. rewrite pE sqrrD -!addrA scalerDr; congr +%R; rewrite addrA scalerDr; congr +%R. - rewrite -mulrDr -polyCD -!mul_polyC mulrA mulrAC -polyCM. by rewrite [a * _]mulrC mulrDl invfM -!mulrA mulVf// mulr1 -splitr. - rewrite [a ^+ 2]expr2 mulrA aa4 -polyC_exp -polyCB expr_div_n -mulrBl subKr. by rewrite scale_poly...	Lemma	deg2_poly_canonical	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "aa4", "addrA", "delta", "expr2", "expr_div_n", "invfM", "mulVf", "mul_polyC", "mulfV", "mulr1", "mulrA", "mulrAC", "mulrACA", "mulrBl", "mulrC", "mulrCA", "mulrDl", "mulrDr", "pE", "polyCB", "polyCD", "polyCM", "polyC_exp", "scale_polyC", "scalerDr", "splitr", "s...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r_sqrt_delta : r ^+ 2 = delta.		Hypothesis	r_sqrt_delta	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "delta" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r1	:= (- b - r) / (2 * a).	Let	r1	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
r2	:= (- b + r) / (2 * a).	Let	r2	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
deg2_poly_factor : p = a : ('X - r1%:P) ('X - r2%:P).	Proof. rewrite [p]deg2_poly_canonical//= -/a -/b -/c -/delta /r1 /r2. rewrite ![(- b + _) * _]mulrDl 2!polyCD 2!opprD 2!addrA !mulNr !polyCN !opprK. rewrite -scalerAl [in RHS]mulrC -subr_sqr -polyC_exp -[4]/(2 * 2)%:R natrM. by rewrite -expr2 -exprMn [in RHS]exprMn exprVn r_sqrt_delta. Qed.	Lemma	deg2_poly_factor	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "deg2_poly_canonical", "delta", "expr2", "exprMn", "exprVn", "mulNr", "mulrC", "mulrDl", "natrM", "opprD", "opprK", "polyCD", "polyCN", "polyC_exp", "r1", "r2", "r_sqrt_delta", "scalerAl", "subr_sqr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deg2_poly_root1 : root p r1.	Proof. apply/factor_theorem. by exists (a *: ('X - r2%:P)); rewrite deg2_poly_factor -!scalerAl mulrC. Qed.	Lemma	deg2_poly_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" ]	[ "apply", "deg2_poly_factor", "factor_theorem", "mulrC", "r1", "r2", "root", "scalerAl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deg2_poly_root2 : root p r2.	Proof. apply/factor_theorem. by exists (a *: ('X - r1%:P)); rewrite deg2_poly_factor -!scalerAl. Qed.	Lemma	deg2_poly_root2	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "deg2_poly_factor", "factor_theorem", "r1", "r2", "root", "scalerAl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monicp : p \is monic.		Hypothesis	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" ]	[ "monic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
a1 : a = 1.	Proof. by move: (monicP monicp); rewrite lead_coefE degp. Qed.	Let	a1	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "degp", "lead_coefE", "monicP", "monicp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
delta	:= b ^+ 2 - 4 * c.	Let	delta	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
deg2_poly_canonical : p = (('X + (b / 2)%:P)^+2 - (delta / 4)%:P).	Proof. by rewrite [p]deg2_poly_canonical// -/a a1 scale1r expr1n !mulr1. Qed.	Lemma	deg2_poly_canonical	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "a1", "delta", "expr1n", "mulr1", "scale1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r1	:= (- b - r) / 2.	Let	r1	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
r2	:= (- b + r) / 2.	Let	r2	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
deg2_poly_factor : p = ('X - r1%:P) * ('X - r2%:P).	Proof. by rewrite [p](@deg2_poly_factor _ _ _ _ r)// -/a a1 !mulr1 ?scale1r. Qed.	Lemma	deg2_poly_factor	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "a1", "mulr1", "r1", "r2", "scale1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deg2_poly_root1 : root p r1.	Proof. rewrite /r1 -[2]mulr1 -[X in 2 * X]a1. by apply: deg2_poly_root1; rewrite // -/a a1 mulr1. Qed.	Lemma	deg2_poly_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" ]	[ "a1", "apply", "mulr1", "r1", "root" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deg2_poly_root2 : root p r2.	Proof. rewrite /r2 -[2]mulr1 -[X in 2 * X]a1. by apply: deg2_poly_root2; rewrite // -/a a1 mulr1. Qed.	Lemma	deg2_poly_root2	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "a1", "apply", "mulr1", "r2", "root" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dec_factor_theorem (p : {poly F}) : {s : seq F & {q : {poly F} \| p = q * \prod_(x <- s) ('X - x%:P) /\ (q != 0 -> forall x, ~~ root q x)}}.	Proof. pose polyT (p : seq F) := (foldr (fun c f => f * 'X_0 + c%:T) (0%R)%:T p)%T. have eval_polyT (q : {poly F}) x : GRing.eval [:: x] (polyT q) = q.[x]. by rewrite /horner; elim: (val q) => //= ? ? ->. have [n] := ubnP (size p); elim: n => // n IHn in p *. have /decPcases /= := @satP F [::] ('exists 'X_0, polyT p ...	Lemma	dec_factor_theorem	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "addn2", "apply", "big_cat", "big_nil", "big_seq1", "cats1", "eqVneq", "eqxx", "eval", "factor_theorem", "foldr", "horner", "last", "monicXsubC", "mul0r", "mulr1", "mulrA", "poly", "rcons", "root", "rootP", "satP", "seq", "sig_eqW", "size", "size_Mmonic", "size_Xs...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
closedF : GRing.closed_field_axiom F.		Hypothesis	closedF	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "closed_field_axiom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
closed_rootP p : reflect (exists x, root p x) (size p != 1).	Proof. have [-> \| nz_p] := eqVneq p 0. by rewrite size_poly0; left; exists 0; rewrite root0. rewrite neq_ltn [in _ < 1]polySpred //=. apply: (iffP idP) => [p_gt1 \| [a]]; last exact: root_size_gt1. pose n := (size p).-1; have n_gt0: n > 0 by rewrite -ltnS -polySpred. have [a Dan] := closedF (fun i => - p`_i / lead_coe...	Lemma	closed_rootP	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "apply", "big1", "big_ord_recr", "big_split", "closedF", "eqVneq", "horner_coef", "last", "lead_coef", "lead_coef_eq0", "ltnS", "mulNr", "mulVKf", "mulrA", "mulrCA", "mulr_sumr", "n_gt0", "neq_ltn", "nz_p", "p_gt1", "polySpred", "root", "root0", "rootP", "root_size_gt...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
closed_nonrootP p : reflect (exists x, ~~ root p x) (p != 0).	Proof. apply: (iffP idP) => [nz_p \| [x]]; last first. by apply: contraNneq => ->; apply: root0. have [[x /rootP p1x0]\|] := altP (closed_rootP (p - 1)). by exists x; rewrite -[p](subrK 1) /root hornerD p1x0 add0r hornerC oner_eq0. rewrite negbK => /size_poly1P[c _ /(canRL (subrK 1)) Dp]. by exists 0; rewrite Dp -rad...	Lemma	closed_nonrootP	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "add0r", "apply", "closed_rootP", "contraNneq", "hornerC", "hornerD", "last", "nz_p", "oner_eq0", "polyC_eq0", "raddfD", "root", "root0", "rootC", "rootP", "size_poly1P", "subrK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
closedF	:= @solve_monicpoly F.	Let	closedF	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "finset", "tuple", "div", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "prime" ]	[ "solve_monicpoly" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
closed_rootP p : reflect (exists x, root p x) (size p != 1).	Proof. exact: PreClosedField.closed_rootP. Qed.	Lemma	closed_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", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
closed_nonrootP p : reflect (exists x, ~~ root p x) (p != 0).	Proof. exact: PreClosedField.closed_nonrootP. Qed.	Lemma	closed_nonrootP	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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
closed_field_poly_normal p : {r : seq F \| p = lead_coef p *: \prod_(z <- r) ('X - z%:P)}.	Proof. apply: sig_eqW; have [r [q [->]]] /= := dec_factor_theorem p. have [->\|] := eqVneq; first by exists [::]; rewrite mul0r lead_coef0 scale0r. have [[x rqx ? /(_ isT x) /negP /(_ rqx)] //\|] := altP (closed_rootP q). rewrite negbK => /size_poly1P [c c_neq0-> _ _]; exists r. rewrite mul_polyC lead_coefZ (monicP _) ?m...	Lemma	closed_field_poly_normal	algebra	algebra/poly.v	[ "HB", "structures", "mathcomp", "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", "closed_rootP", "dec_factor_theorem", "eqVneq", "lead_coef", "lead_coef0", "lead_coefZ", "monicP", "monicXsubC", "monic_prod", "mul0r", "mul_polyC", "mulr1", "scale0r", "seq", "sig_eqW", "size_poly1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
redivp_rec (q : {poly R})	:= let sq := size q in let cq := lead_coef q in fix loop (k : nat) (qq r : {poly R})(n : nat) {struct n} := if size r < sq then (k, qq, r) else let m := (lead_coef r) : 'X^(size r - sq) in let qq1 := qq cq%:P + m in let r1 := r * cq%:P - m * q in if n is n1.+1 then loop k.+1 qq1 r1 n1 e...	Definition	redivp_rec	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "lead_coef", "nat", "poly", "r1", "size", "sq" ]	Pseudo division, defined on an arbitrary ring	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
redivp_expanded_def p q	:= if q == 0 then (0, 0, p) else redivp_rec q 0 0 p (size p).	Definition	redivp_expanded_def	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "redivp_rec", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
redivp_key : unit.	Proof. by []. Qed.	Fact	redivp_key	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "unit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
redivp : {poly R} -> {poly R} -> nat * {poly R} * {poly R}	:= locked_with redivp_key redivp_expanded_def.	Definition	redivp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "nat", "poly", "redivp_expanded_def", "redivp_key" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
redivp_unlockable	:= [unlockable fun redivp].	Canonical	redivp_unlockable	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "redivp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivp p q	:= ((redivp p q).1).2.	Definition	rdivp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "redivp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodp p q	:= (redivp p q).2.	Definition	rmodp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "redivp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rscalp p q	:= ((redivp p q).1).1.	Definition	rscalp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "redivp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdp p q	:= rmodp q p == 0.	Definition	rdvdp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rmodp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
redivp_def p q : redivp p q = (rscalp p q, rdivp p q, rmodp p q).	Proof. by rewrite /rscalp /rdivp /rmodp; case: (redivp p q) => [[]] /=. Qed.	Lemma	redivp_def	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdivp", "redivp", "rmodp", "rscalp" ]	Definition rmultp := [rel m d \| rdvdp d m].	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdiv0p p : rdivp 0 p = 0.	Proof. rewrite /rdivp unlock; case: ifP => // Hp; rewrite /redivp_rec !size_poly0. by rewrite polySpred ?Hp. Qed.	Lemma	rdiv0p	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "polySpred", "rdivp", "redivp_rec", "size_poly0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivp0 p : rdivp p 0 = 0.	Proof. by rewrite /rdivp unlock eqxx. Qed.	Lemma	rdivp0	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "eqxx", "rdivp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivp_small p q : size p < size q -> rdivp p q = 0.	Proof. rewrite /rdivp unlock; have [-> \| _ ltpq] := eqP; first by rewrite size_poly0. by case: (size p) => [\|s]; rewrite /= ltpq. Qed.	Lemma	rdivp_small	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdivp", "size", "size_poly0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_rdivp p q : size (rdivp p q) <= size p.	Proof. have [/rdivp_small->\|] := ltnP (size p) (size q); first by rewrite size_poly0. rewrite /rdivp /rmodp /rscalp unlock. have [->\|q0] //= := eqVneq q 0. have: size (0 : {poly R}) <= size p by rewrite size_poly0. move: {2 3 4 6}(size p) (leqnn (size p)) => A. elim: (size p) 0%N (0 : {poly R}) {1 3 4}p (leqnn (size p)...	Lemma	leq_rdivp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "add1n", "addn0", "addn1", "addnA", "addnS", "apply", "coefB", "coefMC", "coefXnM", "coefZ", "eqVneq", "geq_max", "lead_coefE", "lead_coef_eq0", "leqLHS", "leq_add", "leq_add2r", "leq_eqVlt", "leq_ltn_trans", "leq_psubRL", "leq_sizeP", "leq_subLR", "leq_trans", "leqnn",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmod0p p : rmodp 0 p = 0.	Proof. rewrite /rmodp unlock; case: ifP => // Hp; rewrite /redivp_rec !size_poly0. by rewrite polySpred ?Hp. Qed.	Lemma	rmod0p	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "polySpred", "redivp_rec", "rmodp", "size_poly0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodp0 p : rmodp p 0 = p.	Proof. by rewrite /rmodp unlock eqxx. Qed.	Lemma	rmodp0	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "eqxx", "rmodp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rscalp_small p q : size p < size q -> rscalp p q = 0.	Proof. rewrite /rscalp unlock; case: eqP => _ // spq. by case sp: (size p) => [\| s] /=; rewrite spq. Qed.	Lemma	rscalp_small	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rscalp", "size", "sp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltn_rmodp p q : (size (rmodp p q) < size q) = (q != 0).	Proof. rewrite /rdivp /rmodp /rscalp unlock; have [->\|q0] := eqVneq q 0. by rewrite /= size_poly0 ltn0. elim: (size p) 0%N 0 {1 3}p (leqnn (size p)) => [\|n ihn] k q1 r. move/size_poly_leq0P->. by rewrite /= size_poly0 size_poly_gt0 q0 size_poly0 size_poly_gt0. move=> hr /=; case: (ltnP (size r)) => // hsrq; apply...	Lemma	ltn_rmodp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "add1n", "apply", "coefB", "coefMC", "coefXnM", "coefZ", "eqVneq", "lead_coefE", "leq_add", "leq_add2r", "leq_addr", "leq_eqVlt", "leq_sizeP", "leq_subRL", "leq_trans", "leqnn", "ltn0", "ltnNge", "ltnP", "ltnS", "ltn_subRL", "mul0r", "mulr0", "predU1P", "prednK", "p...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltn_rmodpN0 p q : q != 0 -> size (rmodp p q) < size q.	Proof. by rewrite ltn_rmodp. Qed.	Lemma	ltn_rmodpN0	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "ltn_rmodp", "rmodp", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodp1 p : rmodp p 1 = 0.	Proof. apply/eqP; have := ltn_rmodp p 1. by rewrite !oner_neq0 -size_poly_eq0 size_poly1 ltnS leqn0. Qed.	Lemma	rmodp1	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "leqn0", "ltnS", "ltn_rmodp", "oner_neq0", "rmodp", "size_poly1", "size_poly_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodp_small p q : size p < size q -> rmodp p q = p.	Proof. rewrite /rmodp unlock; have [->\|_] := eqP; first by rewrite size_poly0. by case sp: (size p) => [\| s] Hs /=; rewrite sp Hs /=. Qed.	Lemma	rmodp_small	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rmodp", "size", "size_poly0", "sp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_rmodp m d : size (rmodp m d) <= size m.	Proof. have [/rmodp_small -> //\|h] := ltnP (size m) (size d). have [->\|d0] := eqVneq d 0; first by rewrite rmodp0. by apply: leq_trans h; apply: ltnW; rewrite ltn_rmodp. Qed.	Lemma	leq_rmodp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "eqVneq", "leq_trans", "ltnP", "ltnW", "ltn_rmodp", "rmodp", "rmodp0", "rmodp_small", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodpC p c : c != 0 -> rmodp p c%:P = 0.	Proof. move=> Hc; apply/eqP; rewrite -size_poly_leq0 -ltnS. have -> : 1%N = nat_of_bool (c != 0) by rewrite Hc. by rewrite -size_polyC ltn_rmodp polyC_eq0. Qed.	Lemma	rmodpC	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "ltnS", "ltn_rmodp", "nat_of_bool", "polyC_eq0", "rmodp", "size_polyC", "size_poly_leq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdp0 d : rdvdp d 0.	Proof. by rewrite /rdvdp rmod0p. Qed.	Lemma	rdvdp0	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdvdp", "rmod0p" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvd0p n : rdvdp 0 n = (n == 0).	Proof. by rewrite /rdvdp rmodp0. Qed.	Lemma	rdvd0p	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdvdp", "rmodp0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvd0pP n : reflect (n = 0) (rdvdp 0 n).	Proof. by apply: (iffP idP); rewrite rdvd0p; move/eqP. Qed.	Lemma	rdvd0pP	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "rdvd0p", "rdvdp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdpN0 p q : rdvdp p q -> q != 0 -> p != 0.	Proof. by move=> pq hq; apply: contraTneq pq => ->; rewrite rdvd0p. Qed.	Lemma	rdvdpN0	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "contraTneq", "rdvd0p", "rdvdp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdp1 d : rdvdp d 1 = (size d == 1).	Proof. rewrite /rdvdp; have [->\|] := eqVneq d 0. by rewrite rmodp0 size_poly0 (negPf (oner_neq0 _)). rewrite -size_poly_leq0 -ltnS; case: ltngtP => // [\|/eqP] hd _. by rewrite rmodp_small ?size_poly1 // oner_eq0. have [c cn0 ->] := size_poly1P _ hd. rewrite /rmodp unlock -size_poly_eq0 size_poly1 /= size_poly1 size...	Lemma	rdvdp1	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "eqVneq", "lead_coefC", "ltnS", "ltngtP", "oner_eq0", "oner_neq0", "polyC_eq0", "rdvdp", "rmodp", "rmodp0", "rmodp_small", "scale1r", "size", "size_poly0", "size_poly1", "size_poly1P", "size_polyC", "size_poly_eq0", "size_poly_leq0", "subrr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvd1p m : rdvdp 1 m.	Proof. by rewrite /rdvdp rmodp1. Qed.	Lemma	rdvd1p	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdvdp", "rmodp1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Nrdvdp_small (n d : {poly R}) : n != 0 -> size n < size d -> rdvdp d n = false.	Proof. by move=> nn0 hs; rewrite /rdvdp (rmodp_small hs); apply: negPf. Qed.	Lemma	Nrdvdp_small	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "poly", "rdvdp", "rmodp_small", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodp_eq0P p q : reflect (rmodp p q = 0) (rdvdp q p).	Proof. exact: (iffP eqP). Qed.	Lemma	rmodp_eq0P	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdvdp", "rmodp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodp_eq0 p q : rdvdp q p -> rmodp p q = 0.	Proof. exact: rmodp_eq0P. Qed.	Lemma	rmodp_eq0	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdvdp", "rmodp", "rmodp_eq0P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdp_leq p q : rdvdp p q -> q != 0 -> size p <= size q.	Proof. by move=> dvd_pq; rewrite leqNgt; apply: contra => /rmodp_small <-. Qed.	Lemma	rdvdp_leq	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "leqNgt", "rdvdp", "rmodp_small", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rgcdp p q	:= let: (p1, q1) := if size p < size q then (q, p) else (p, q) in if p1 == 0 then q1 else let fix loop (n : nat) (pp qq : {poly R}) {struct n} := let rr := rmodp pp qq in if rr == 0 then qq else if n is n1.+1 then loop n1 qq rr else rr in loop (size p1) p1 q1.	Definition	rgcdp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "nat", "poly", "rmodp", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rgcd0p : left_id 0 rgcdp.	Proof. move=> p; rewrite /rgcdp size_poly0 size_poly_gt0 if_neg. case: ifP => /= [_ \| nzp]; first by rewrite eqxx. by rewrite polySpred !(rmodp0, nzp) //; case: _.-1 => [\|m]; rewrite rmod0p eqxx. Qed.	Lemma	rgcd0p	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "eqxx", "polySpred", "rgcdp", "rmod0p", "rmodp0", "size_poly0", "size_poly_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rgcdp0 : right_id 0 rgcdp.	Proof. move=> p; have:= rgcd0p p; rewrite /rgcdp size_poly0 size_poly_gt0. by case: eqVneq => p0; rewrite ?(eqxx, p0) //= eqxx. Qed.	Lemma	rgcdp0	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "eqVneq", "eqxx", "p0", "rgcd0p", "rgcdp", "size_poly0", "size_poly_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rgcdpE p q : rgcdp p q = if size p < size q then rgcdp (rmodp q p) p else rgcdp (rmodp p q) q.	Proof. pose rgcdp_rec := fix rgcdp_rec (n : nat) (pp qq : {poly R}) {struct n} := let rr := rmodp pp qq in if rr == 0 then qq else if n is n1.+1 then rgcdp_rec n1 qq rr else rr. have Irec: forall m n p q, size q <= m -> size q <= n -> size q < size p -> rgcdp_rec m p q = rgcdp_rec n p q. + elim=> [\|m H...	Lemma	rgcdpE	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "eqVneq", "eqxx", "last", "leqW", "leq_trans", "ltnP", "ltnS", "ltnW", "ltn_rmodp", "nat", "poly", "polySpred", "rgcd0p", "rgcdp", "rgcdp0", "rmod0p", "rmodp", "rmodp0", "size", "size_poly0", "size_poly_gt0", "size_poly_leq0P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_redivp_spec m d : nat * {poly R} * {poly R} -> Type	:= ComEdivnSpec k (q r : {poly R}) of (GRing.comm d (lead_coef d)%:P -> m * (lead_coef d ^+ k)%:P = q * d + r) & (d != 0 -> size r < size d) : comm_redivp_spec m d (k, q, r).	Variant	comm_redivp_spec	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "comm", "lead_coef", "nat", "poly", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_redivpP m d : comm_redivp_spec m d (redivp m d).	Proof. rewrite unlock; have [->\|Hd] := eqVneq d 0. by constructor; rewrite !(simp, eqxx). have: GRing.comm d (lead_coef d)%:P -> m * (lead_coef d ^+ 0)%:P = 0 * d + m. by rewrite !simp. elim: (size m) 0%N 0 {1 4 6}m (leqnn (size m)) => [\|n IHn] k q r Hr /=. move/size_poly_leq0P: Hr ->. suff hsd: size (0: {poly ...	Lemma	comm_redivpP	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "add1n", "addrA", "addrN", "apply", "coefB", "coefMC", "coefXnM", "coefZ", "comm", "comm_redivp_spec", "eqVneq", "eqxx", "exprSr", "last", "lead_coef", "leq_add2r", "leq_eqVlt", "leq_ltn_trans", "leq_sizeP", "leq_sub2r", "leq_subLR", "leq_trans", "leqnn", "ltnNge", "l...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodpp p : GRing.comm p (lead_coef p)%:P -> rmodp p p = 0.	Proof. move=> hC; rewrite /rmodp unlock; have [-> //\|] := eqVneq. rewrite -size_poly_eq0 /redivp_rec; case sp: (size p)=> [\|n] // _. rewrite sp ltnn subnn expr0 hC alg_polyC !simp subrr. by case: n sp => [\|n] sp; rewrite size_polyC /= eqxx. Qed.	Lemma	rmodpp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "alg_polyC", "comm", "eqVneq", "eqxx", "expr0", "lead_coef", "ltnn", "redivp_rec", "rmodp", "simp", "size", "size_polyC", "size_poly_eq0", "sp", "subnn", "subrr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoprimep (p q : {poly R})	:= size (rgcdp p q) == 1.	Definition	rcoprimep	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "poly", "rgcdp", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rgdcop_rec q p n	:= if n is m.+1 then if rcoprimep p q then p else rgdcop_rec q (rdivp p (rgcdp p q)) m else (q == 0)%:R.	Fixpoint	rgdcop_rec	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rcoprimep", "rdivp", "rgcdp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rgdcop q p	:= rgdcop_rec q p (size p).	Definition	rgdcop	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rgdcop_rec", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rgdcop0 q : rgdcop q 0 = (q == 0)%:R.	Proof. by rewrite /rgdcop size_poly0. Qed.	Lemma	rgdcop0	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rgdcop", "size_poly0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Cdl : GRing.comm d (lead_coef d)%:P.		Hypothesis	Cdl	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "comm", "lead_coef" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rreg : GRing.rreg (lead_coef d).		Hypothesis	Rreg	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "lead_coef", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
redivp_eq q r : size r < size d -> let k := (redivp (q * d + r) d).1.1 in let c := (lead_coef d ^+ k)%:P in redivp (q * d + r) d = (k, q * c, r * c).	Proof. move=> lt_rd; case: comm_redivpP=> k q1 r1 /(_ Cdl) Heq. have dn0: d != 0 by case: (size d) lt_rd (size_poly_eq0 d) => // n _ <-. move=> /(_ dn0) Hs. have eC : q * d * (lead_coef d ^+ k)%:P = q * (lead_coef d ^+ k)%:P * d. by rewrite -mulrA polyC_exp (commrX k Cdl) mulrA. suff e1 : q1 = q * (lead_coef d ^+ k)%...	Lemma	redivp_eq	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "Cdl", "addn0", "addnS", "addrACA", "addrI", "apply", "comm_redivpP", "commrX", "eC", "gtn_max", "last", "lead_coef", "leq_ltn_trans", "mulNr", "mulrA", "mulrDl", "opprB", "opprD", "polyC_exp", "r1", "redivp", "rreg_div0", "size", "size_polyC", "size_polyD", "size_p...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivp_eq p : p * (lead_coef d ^+ (rscalp p d))%:P = (rdivp p d) * d + (rmodp p d).	Proof. by rewrite /rdivp /rmodp /rscalp; case: comm_redivpP=> k q1 r1 Hc _; apply: Hc. Qed.	Lemma	rdivp_eq	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "comm_redivpP", "lead_coef", "r1", "rdivp", "rmodp", "rscalp" ]	this is a bad name	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_rdvdp k q1 p: p * ((lead_coef d)^+ k)%:P = q1 * d -> rdvdp d p.	Proof. move=> he. have Hnq0 := rreg_lead0 Rreg; set lq := lead_coef d. pose v := rscalp p d; pose m := maxn v k. rewrite /rdvdp -(rreg_polyMC_eq0 _ (@rregX _ _ (m - v) Rreg)). suff: ((rdivp p d) * (lq ^+ (m - v))%:P - q1 * (lq ^+ (m - k))%:P) * d + (rmodp p d) * (lq ^+ (m - v))%:P == 0. rewrite rreg_div0 //; last ...	Lemma	eq_rdvdp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "Cdl", "Rreg", "addnC", "addrAC", "commrX", "exprD", "last", "lead_coef", "leq_maxl", "leq_maxr", "ltn_rmodp", "maxn", "mulNr", "mulrA", "mulrDl", "polyCM", "polyC_exp", "rdivp", "rdivp_eq", "rdvdp", "rmodp", "rregX", "rreg_div0", "rreg_lead0", "rreg_polyMC_eq0", "r...	section variables impose an inconvenient order on parameters	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdp_spec p q : {poly R} -> bool -> Type	:= \| Rdvdp k q1 & p * ((lead_coef q)^+ k)%:P = q1 * q : rdvdp_spec p q 0 true \| RdvdpN & rmodp p q != 0 : rdvdp_spec p q (rmodp p q) false.	Variant	rdvdp_spec	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "lead_coef", "poly", "rmodp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdp_eqP p : rdvdp_spec p d (rmodp p d) (rdvdp d p).	Proof. case hdvd: (rdvdp d p); last by apply: RdvdpN; move/rmodp_eq0P/eqP: hdvd. move/rmodp_eq0P: (hdvd)->; apply: (@Rdvdp _ _ (rscalp p d) (rdivp p d)). by rewrite rdivp_eq //; move/rmodp_eq0P: (hdvd)->; rewrite addr0. Qed.	Lemma	rdvdp_eqP	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "addr0", "apply", "last", "rdivp", "rdivp_eq", "rdvdp", "rdvdp_spec", "rmodp", "rmodp_eq0P", "rscalp" ]	Is that version useable ?	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdp_mull p : rdvdp d (p * d).	Proof. by apply: (@eq_rdvdp 0 p); rewrite expr0 mulr1. Qed.	Lemma	rdvdp_mull	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "apply", "eq_rdvdp", "expr0", "mulr1", "rdvdp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodp_mull p : rmodp (p * d) d = 0.	Proof. exact/eqP/rdvdp_mull. Qed.	Lemma	rmodp_mull	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdvdp_mull", "rmodp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmodpp : rmodp d d = 0.	Proof. by rewrite -[d in rmodp d _]mul1r rmodp_mull. Qed.	Lemma	rmodpp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "mul1r", "rmodp", "rmodp_mull" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivpp : rdivp d d = (lead_coef d ^+ rscalp d d)%:P.	Proof. have dn0 : d != 0 by rewrite -lead_coef_eq0 rreg_neq0. move: (rdivp_eq d); rewrite rmodpp addr0. suff ->: GRing.comm d (lead_coef d ^+ rscalp d d)%:P by move/(rreg_lead Rreg)->. by rewrite polyC_exp; apply: commrX. Qed.	Lemma	rdivpp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "Rreg", "addr0", "apply", "comm", "commrX", "lead_coef", "lead_coef_eq0", "polyC_exp", "rdivp", "rdivp_eq", "rmodpp", "rreg_lead", "rreg_neq0", "rscalp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdvdpp : rdvdp d d.	Proof. exact/eqP/rmodpp. Qed.	Lemma	rdvdpp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "rdvdp", "rmodpp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivpK p : rdvdp d p -> rdivp p d * d = p * (lead_coef d ^+ rscalp p d)%:P.	Proof. by rewrite rdivp_eq /rdvdp; move/eqP->; rewrite addr0. Qed.	Lemma	rdivpK	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "addr0", "lead_coef", "rdivp", "rdivp_eq", "rdvdp", "rscalp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mond : d \is monic.		Hypothesis	mond	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "monic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
redivp_eq q r : size r < size d -> let k := (redivp (q * d + r) d).1.1 in redivp (q * d + r) d = (k, q, r).	Proof. case: (monic_comreg mond)=> Hc Hr /(redivp_eq Hc Hr q). by rewrite (eqP mond) => -> /=; rewrite expr1n !mulr1. Qed.	Lemma	redivp_eq	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "expr1n", "mond", "monic_comreg", "mulr1", "redivp", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivp_eq p : p = rdivp p d * d + rmodp p d.	Proof. rewrite -rdivp_eq (eqP mond); first exact: commr1. by rewrite expr1n mulr1. Qed.	Lemma	rdivp_eq	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "commr1", "expr1n", "mond", "mulr1", "rdivp", "rmodp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivpp : rdivp d d = 1.	Proof. by case: (monic_comreg mond) => hc hr; rewrite rdivpp // (eqP mond) expr1n. Qed.	Lemma	rdivpp	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "expr1n", "mond", "monic_comreg", "rdivp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivp_addl_mul_small q r : size r < size d -> rdivp (q * d + r) d = q.	Proof. by move=> Hd; case: (monic_comreg mond)=> Hc Hr; rewrite /rdivp redivp_eq. Qed.	Lemma	rdivp_addl_mul_small	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "mond", "monic_comreg", "rdivp", "redivp_eq", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivp_addl_mul q r : rdivp (q * d + r) d = q + rdivp r d.	Proof. case: (monic_comreg mond)=> Hc Hr; rewrite [r in _ * _ + r]rdivp_eq addrA. by rewrite -mulrDl rdivp_addl_mul_small // ltn_rmodp monic_neq0. Qed.	Lemma	rdivp_addl_mul	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "addrA", "ltn_rmodp", "mond", "monic_comreg", "monic_neq0", "mulrDl", "rdivp", "rdivp_addl_mul_small", "rdivp_eq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivpDl q r : rdvdp d q -> rdivp (q + r) d = rdivp q d + rdivp r d.	Proof. case: (monic_comreg mond)=> Hc Hr; rewrite [r in q + r]rdivp_eq addrA. rewrite [q in q + _ + _]rdivp_eq; move/rmodp_eq0P->. by rewrite addr0 -mulrDl rdivp_addl_mul_small // ltn_rmodp monic_neq0. Qed.	Lemma	rdivpDl	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "addr0", "addrA", "ltn_rmodp", "mond", "monic_comreg", "monic_neq0", "mulrDl", "rdivp", "rdivp_addl_mul_small", "rdivp_eq", "rdvdp", "rmodp_eq0P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rdivpDr q r : rdvdp d r -> rdivp (q + r) d = rdivp q d + rdivp r d.	Proof. by rewrite addrC; move/rdivpDl->; rewrite addrC. Qed.	Lemma	rdivpDr	algebra	algebra/polydiv.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...	[ "addrC", "rdivp", "rdivpDl", "rdvdp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

a

:= p`_2.

Let

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" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

b

:= p`_1.

Let

b

algebra

algebra/poly.v

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

c

:= p`_0.

Let

c

algebra

algebra/poly.v

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

pneq0 : p != 0.

Proof. by rewrite -size_poly_gt0 degp. Qed.

Let

pneq0

algebra

algebra/poly.v

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

[ "degp", "size_poly_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aneq0 : a != 0.

Proof. by move: pneq0; rewrite -lead_coef_eq0 lead_coefE degp. Qed.

Let

aneq0

algebra

algebra/poly.v

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

[ "degp", "lead_coefE", "lead_coef_eq0", "pneq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

a2neq0 : 2 * a != 0.

Proof. by rewrite mulf_neq0. Qed.

Let

a2neq0

algebra

algebra/poly.v

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

[ "mulf_neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sqa2neq0 : (2 * a) ^+ 2 != 0.

Proof. exact: expf_neq0. Qed.

Let

sqa2neq0

algebra

algebra/poly.v

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

[ "expf_neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

aa4 : 4 * a * a = (2 * a)^+2.

Proof. by rewrite expr2 mulrACA mulrA -natrM. Qed.

Let

aa4

algebra

algebra/poly.v

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

[ "expr2", "mulrA", "mulrACA", "natrM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

splitr (x : F) : x = x / 2 + x / 2.

Proof. by rewrite -mulr2n -[RHS]mulr_natr divfK. Qed.

Let

splitr

algebra

algebra/poly.v

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

[ "divfK", "mulr2n", "mulr_natr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pE : p = a *: 'X^2 + b *: 'X + c%:P.

Proof. apply/polyP => + /[!coefE] => -[|[|[|i]]] /=; rewrite !Monoid.simpm//. by rewrite nth_default// degp. Qed.

Let

pE

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "coefE", "degp", "nth_default", "polyP", "simpm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

delta

:= b ^+ 2 - 4 * a * c.

Let

delta

algebra

algebra/poly.v

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

deg2_poly_canonical : p = a *: (('X + (b / (2 * a))%:P)^+2 - (delta / (4 * a ^+ 2))%:P).

Proof. rewrite pE sqrrD -!addrA scalerDr; congr +%R; rewrite addrA scalerDr; congr +%R. - rewrite -mulrDr -polyCD -!mul_polyC mulrA mulrAC -polyCM. by rewrite [a * _]mulrC mulrDl invfM -!mulrA mulVf// mulr1 -splitr. - rewrite [a ^+ 2]expr2 mulrA aa4 -polyC_exp -polyCB expr_div_n -mulrBl subKr. by rewrite scale_poly...

Lemma

deg2_poly_canonical

algebra

algebra/poly.v

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

[ "aa4", "addrA", "delta", "expr2", "expr_div_n", "invfM", "mulVf", "mul_polyC", "mulfV", "mulr1", "mulrA", "mulrAC", "mulrACA", "mulrBl", "mulrC", "mulrCA", "mulrDl", "mulrDr", "pE", "polyCB", "polyCD", "polyCM", "polyC_exp", "scale_polyC", "scalerDr", "splitr", "s...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r_sqrt_delta : r ^+ 2 = delta.

Hypothesis

r_sqrt_delta

algebra

algebra/poly.v

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

[ "delta" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r1

:= (- b - r) / (2 * a).

Let

r1

algebra

algebra/poly.v

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

r2

:= (- b + r) / (2 * a).

Let

r2

algebra

algebra/poly.v

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

deg2_poly_factor : p = a *: ('X - r1%:P) * ('X - r2%:P).

Proof. rewrite [p]deg2_poly_canonical//= -/a -/b -/c -/delta /r1 /r2. rewrite ![(- b + _) * _]mulrDl 2!polyCD 2!opprD 2!addrA !mulNr !polyCN !opprK. rewrite -scalerAl [in RHS]mulrC -subr_sqr -polyC_exp -[4]/(2 * 2)%:R natrM. by rewrite -expr2 -exprMn [in RHS]exprMn exprVn r_sqrt_delta. Qed.

Lemma

deg2_poly_factor

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "deg2_poly_canonical", "delta", "expr2", "exprMn", "exprVn", "mulNr", "mulrC", "mulrDl", "natrM", "opprD", "opprK", "polyCD", "polyCN", "polyC_exp", "r1", "r2", "r_sqrt_delta", "scalerAl", "subr_sqr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deg2_poly_root1 : root p r1.

Proof. apply/factor_theorem. by exists (a *: ('X - r2%:P)); rewrite deg2_poly_factor -!scalerAl mulrC. Qed.

Lemma

deg2_poly_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" ]

[ "apply", "deg2_poly_factor", "factor_theorem", "mulrC", "r1", "r2", "root", "scalerAl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deg2_poly_root2 : root p r2.

Proof. apply/factor_theorem. by exists (a *: ('X - r1%:P)); rewrite deg2_poly_factor -!scalerAl. Qed.

Lemma

deg2_poly_root2

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "deg2_poly_factor", "factor_theorem", "r1", "r2", "root", "scalerAl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monicp : p \is monic.

Hypothesis

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" ]

[ "monic" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

a1 : a = 1.

Proof. by move: (monicP monicp); rewrite lead_coefE degp. Qed.

Let

a1

algebra

algebra/poly.v

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

[ "degp", "lead_coefE", "monicP", "monicp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

delta

:= b ^+ 2 - 4 * c.

Let

delta

algebra

algebra/poly.v

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

deg2_poly_canonical : p = (('X + (b / 2)%:P)^+2 - (delta / 4)%:P).

Proof. by rewrite [p]deg2_poly_canonical// -/a a1 scale1r expr1n !mulr1. Qed.

Lemma

deg2_poly_canonical

algebra

algebra/poly.v

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

[ "a1", "delta", "expr1n", "mulr1", "scale1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r1

:= (- b - r) / 2.

Let

r1

algebra

algebra/poly.v

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

r2

:= (- b + r) / 2.

Let

r2

algebra

algebra/poly.v

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

deg2_poly_factor : p = ('X - r1%:P) * ('X - r2%:P).

Proof. by rewrite [p](@deg2_poly_factor _ _ _ _ r)// -/a a1 !mulr1 ?scale1r. Qed.

Lemma

deg2_poly_factor

algebra

algebra/poly.v

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

[ "a1", "mulr1", "r1", "r2", "scale1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deg2_poly_root1 : root p r1.

Proof. rewrite /r1 -[2]mulr1 -[X in 2 * X]a1. by apply: deg2_poly_root1; rewrite // -/a a1 mulr1. Qed.

Lemma

deg2_poly_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" ]

[ "a1", "apply", "mulr1", "r1", "root" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deg2_poly_root2 : root p r2.

Proof. rewrite /r2 -[2]mulr1 -[X in 2 * X]a1. by apply: deg2_poly_root2; rewrite // -/a a1 mulr1. Qed.

Lemma

deg2_poly_root2

algebra

algebra/poly.v

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

[ "a1", "apply", "mulr1", "r2", "root" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dec_factor_theorem (p : {poly F}) : {s : seq F & {q : {poly F} | p = q * \prod_(x <- s) ('X - x%:P) /\ (q != 0 -> forall x, ~~ root q x)}}.

Proof. pose polyT (p : seq F) := (foldr (fun c f => f * 'X_0 + c%:T) (0%R)%:T p)%T. have eval_polyT (q : {poly F}) x : GRing.eval [:: x] (polyT q) = q.[x]. by rewrite /horner; elim: (val q) => //= ? ? ->. have [n] := ubnP (size p); elim: n => // n IHn in p *. have /decPcases /= := @satP F [::] ('exists 'X_0, polyT p ...

Lemma

dec_factor_theorem

algebra

algebra/poly.v

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

[ "addn2", "apply", "big_cat", "big_nil", "big_seq1", "cats1", "eqVneq", "eqxx", "eval", "factor_theorem", "foldr", "horner", "last", "monicXsubC", "mul0r", "mulr1", "mulrA", "poly", "rcons", "root", "rootP", "satP", "seq", "sig_eqW", "size", "size_Mmonic", "size_Xs...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

closedF : GRing.closed_field_axiom F.

Hypothesis

closedF

algebra

algebra/poly.v

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

[ "closed_field_axiom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

closed_rootP p : reflect (exists x, root p x) (size p != 1).

Proof. have [-> | nz_p] := eqVneq p 0. by rewrite size_poly0; left; exists 0; rewrite root0. rewrite neq_ltn [in _ < 1]polySpred //=. apply: (iffP idP) => [p_gt1 | [a]]; last exact: root_size_gt1. pose n := (size p).-1; have n_gt0: n > 0 by rewrite -ltnS -polySpred. have [a Dan] := closedF (fun i => - p`_i / lead_coe...

Lemma

closed_rootP

algebra

algebra/poly.v

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

[ "apply", "big1", "big_ord_recr", "big_split", "closedF", "eqVneq", "horner_coef", "last", "lead_coef", "lead_coef_eq0", "ltnS", "mulNr", "mulVKf", "mulrA", "mulrCA", "mulr_sumr", "n_gt0", "neq_ltn", "nz_p", "p_gt1", "polySpred", "root", "root0", "rootP", "root_size_gt...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

closed_nonrootP p : reflect (exists x, ~~ root p x) (p != 0).

Proof. apply: (iffP idP) => [nz_p | [x]]; last first. by apply: contraNneq => ->; apply: root0. have [[x /rootP p1x0]|] := altP (closed_rootP (p - 1)). by exists x; rewrite -[p](subrK 1) /root hornerD p1x0 add0r hornerC oner_eq0. rewrite negbK => /size_poly1P[c _ /(canRL (subrK 1)) Dp]. by exists 0; rewrite Dp -rad...

Lemma

closed_nonrootP

algebra

algebra/poly.v

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

[ "add0r", "apply", "closed_rootP", "contraNneq", "hornerC", "hornerD", "last", "nz_p", "oner_eq0", "polyC_eq0", "raddfD", "root", "root0", "rootC", "rootP", "size_poly1P", "subrK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

closedF

:= @solve_monicpoly F.

Let

closedF

algebra

algebra/poly.v

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

[ "solve_monicpoly" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

closed_rootP p : reflect (exists x, root p x) (size p != 1).

Proof. exact: PreClosedField.closed_rootP. Qed.

Lemma

closed_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", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

closed_nonrootP p : reflect (exists x, ~~ root p x) (p != 0).

Proof. exact: PreClosedField.closed_nonrootP. Qed.

Lemma

closed_nonrootP

algebra

algebra/poly.v

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

closed_field_poly_normal p : {r : seq F | p = lead_coef p *: \prod_(z <- r) ('X - z%:P)}.

Proof. apply: sig_eqW; have [r [q [->]]] /= := dec_factor_theorem p. have [->|] := eqVneq; first by exists [::]; rewrite mul0r lead_coef0 scale0r. have [[x rqx ? /(_ isT x) /negP /(_ rqx)] //|] := altP (closed_rootP q). rewrite negbK => /size_poly1P [c c_neq0-> _ _]; exists r. rewrite mul_polyC lead_coefZ (monicP _) ?m...

Lemma

closed_field_poly_normal

algebra

algebra/poly.v

[ "HB", "structures", "mathcomp", "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", "closed_rootP", "dec_factor_theorem", "eqVneq", "lead_coef", "lead_coef0", "lead_coefZ", "monicP", "monicXsubC", "monic_prod", "mul0r", "mul_polyC", "mulr1", "scale0r", "seq", "sig_eqW", "size_poly1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

redivp_rec (q : {poly R})

:= let sq := size q in let cq := lead_coef q in fix loop (k : nat) (qq r : {poly R})(n : nat) {struct n} := if size r < sq then (k, qq, r) else let m := (lead_coef r) *: 'X^(size r - sq) in let qq1 := qq * cq%:P + m in let r1 := r * cq%:P - m * q in if n is n1.+1 then loop k.+1 qq1 r1 n1 e...

Definition

redivp_rec

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "lead_coef", "nat", "poly", "r1", "size", "sq" ]

Pseudo division, defined on an arbitrary ring

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

redivp_expanded_def p q

:= if q == 0 then (0, 0, p) else redivp_rec q 0 0 p (size p).

Definition

redivp_expanded_def

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "redivp_rec", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

redivp_key : unit.

Proof. by []. Qed.

Fact

redivp_key

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

redivp : {poly R} -> {poly R} -> nat * {poly R} * {poly R}

:= locked_with redivp_key redivp_expanded_def.

Definition

redivp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "nat", "poly", "redivp_expanded_def", "redivp_key" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

redivp_unlockable

:= [unlockable fun redivp].

Canonical

redivp_unlockable

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "redivp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivp p q

:= ((redivp p q).1).2.

Definition

rdivp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "redivp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodp p q

:= (redivp p q).2.

Definition

rmodp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "redivp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rscalp p q

:= ((redivp p q).1).1.

Definition

rscalp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "redivp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdp p q

:= rmodp q p == 0.

Definition

rdvdp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rmodp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

redivp_def p q : redivp p q = (rscalp p q, rdivp p q, rmodp p q).

Proof. by rewrite /rscalp /rdivp /rmodp; case: (redivp p q) => [[]] /=. Qed.

Lemma

redivp_def

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdivp", "redivp", "rmodp", "rscalp" ]

Definition rmultp := [rel m d | rdvdp d m].

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdiv0p p : rdivp 0 p = 0.

Proof. rewrite /rdivp unlock; case: ifP => // Hp; rewrite /redivp_rec !size_poly0. by rewrite polySpred ?Hp. Qed.

Lemma

rdiv0p

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "polySpred", "rdivp", "redivp_rec", "size_poly0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivp0 p : rdivp p 0 = 0.

Proof. by rewrite /rdivp unlock eqxx. Qed.

Lemma

rdivp0

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "eqxx", "rdivp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivp_small p q : size p < size q -> rdivp p q = 0.

Proof. rewrite /rdivp unlock; have [-> | _ ltpq] := eqP; first by rewrite size_poly0. by case: (size p) => [|s]; rewrite /= ltpq. Qed.

Lemma

rdivp_small

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdivp", "size", "size_poly0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_rdivp p q : size (rdivp p q) <= size p.

Proof. have [/rdivp_small->|] := ltnP (size p) (size q); first by rewrite size_poly0. rewrite /rdivp /rmodp /rscalp unlock. have [->|q0] //= := eqVneq q 0. have: size (0 : {poly R}) <= size p by rewrite size_poly0. move: {2 3 4 6}(size p) (leqnn (size p)) => A. elim: (size p) 0%N (0 : {poly R}) {1 3 4}p (leqnn (size p)...

Lemma

leq_rdivp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "add1n", "addn0", "addn1", "addnA", "addnS", "apply", "coefB", "coefMC", "coefXnM", "coefZ", "eqVneq", "geq_max", "lead_coefE", "lead_coef_eq0", "leqLHS", "leq_add", "leq_add2r", "leq_eqVlt", "leq_ltn_trans", "leq_psubRL", "leq_sizeP", "leq_subLR", "leq_trans", "leqnn",...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmod0p p : rmodp 0 p = 0.

Proof. rewrite /rmodp unlock; case: ifP => // Hp; rewrite /redivp_rec !size_poly0. by rewrite polySpred ?Hp. Qed.

Lemma

rmod0p

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "polySpred", "redivp_rec", "rmodp", "size_poly0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodp0 p : rmodp p 0 = p.

Proof. by rewrite /rmodp unlock eqxx. Qed.

Lemma

rmodp0

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "eqxx", "rmodp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rscalp_small p q : size p < size q -> rscalp p q = 0.

Proof. rewrite /rscalp unlock; case: eqP => _ // spq. by case sp: (size p) => [| s] /=; rewrite spq. Qed.

Lemma

rscalp_small

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rscalp", "size", "sp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltn_rmodp p q : (size (rmodp p q) < size q) = (q != 0).

Proof. rewrite /rdivp /rmodp /rscalp unlock; have [->|q0] := eqVneq q 0. by rewrite /= size_poly0 ltn0. elim: (size p) 0%N 0 {1 3}p (leqnn (size p)) => [|n ihn] k q1 r. move/size_poly_leq0P->. by rewrite /= size_poly0 size_poly_gt0 q0 size_poly0 size_poly_gt0. move=> hr /=; case: (ltnP (size r)) => // hsrq; apply...

Lemma

ltn_rmodp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "add1n", "apply", "coefB", "coefMC", "coefXnM", "coefZ", "eqVneq", "lead_coefE", "leq_add", "leq_add2r", "leq_addr", "leq_eqVlt", "leq_sizeP", "leq_subRL", "leq_trans", "leqnn", "ltn0", "ltnNge", "ltnP", "ltnS", "ltn_subRL", "mul0r", "mulr0", "predU1P", "prednK", "p...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltn_rmodpN0 p q : q != 0 -> size (rmodp p q) < size q.

Proof. by rewrite ltn_rmodp. Qed.

Lemma

ltn_rmodpN0

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "ltn_rmodp", "rmodp", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodp1 p : rmodp p 1 = 0.

Proof. apply/eqP; have := ltn_rmodp p 1. by rewrite !oner_neq0 -size_poly_eq0 size_poly1 ltnS leqn0. Qed.

Lemma

rmodp1

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "leqn0", "ltnS", "ltn_rmodp", "oner_neq0", "rmodp", "size_poly1", "size_poly_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodp_small p q : size p < size q -> rmodp p q = p.

Proof. rewrite /rmodp unlock; have [->|_] := eqP; first by rewrite size_poly0. by case sp: (size p) => [| s] Hs /=; rewrite sp Hs /=. Qed.

Lemma

rmodp_small

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rmodp", "size", "size_poly0", "sp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_rmodp m d : size (rmodp m d) <= size m.

Proof. have [/rmodp_small -> //|h] := ltnP (size m) (size d). have [->|d0] := eqVneq d 0; first by rewrite rmodp0. by apply: leq_trans h; apply: ltnW; rewrite ltn_rmodp. Qed.

Lemma

leq_rmodp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "eqVneq", "leq_trans", "ltnP", "ltnW", "ltn_rmodp", "rmodp", "rmodp0", "rmodp_small", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodpC p c : c != 0 -> rmodp p c%:P = 0.

Proof. move=> Hc; apply/eqP; rewrite -size_poly_leq0 -ltnS. have -> : 1%N = nat_of_bool (c != 0) by rewrite Hc. by rewrite -size_polyC ltn_rmodp polyC_eq0. Qed.

Lemma

rmodpC

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "ltnS", "ltn_rmodp", "nat_of_bool", "polyC_eq0", "rmodp", "size_polyC", "size_poly_leq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdp0 d : rdvdp d 0.

Proof. by rewrite /rdvdp rmod0p. Qed.

Lemma

rdvdp0

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdvdp", "rmod0p" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvd0p n : rdvdp 0 n = (n == 0).

Proof. by rewrite /rdvdp rmodp0. Qed.

Lemma

rdvd0p

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdvdp", "rmodp0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvd0pP n : reflect (n = 0) (rdvdp 0 n).

Proof. by apply: (iffP idP); rewrite rdvd0p; move/eqP. Qed.

Lemma

rdvd0pP

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "rdvd0p", "rdvdp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdpN0 p q : rdvdp p q -> q != 0 -> p != 0.

Proof. by move=> pq hq; apply: contraTneq pq => ->; rewrite rdvd0p. Qed.

Lemma

rdvdpN0

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "contraTneq", "rdvd0p", "rdvdp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdp1 d : rdvdp d 1 = (size d == 1).

Proof. rewrite /rdvdp; have [->|] := eqVneq d 0. by rewrite rmodp0 size_poly0 (negPf (oner_neq0 _)). rewrite -size_poly_leq0 -ltnS; case: ltngtP => // [|/eqP] hd _. by rewrite rmodp_small ?size_poly1 // oner_eq0. have [c cn0 ->] := size_poly1P _ hd. rewrite /rmodp unlock -size_poly_eq0 size_poly1 /= size_poly1 size...

Lemma

rdvdp1

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "eqVneq", "lead_coefC", "ltnS", "ltngtP", "oner_eq0", "oner_neq0", "polyC_eq0", "rdvdp", "rmodp", "rmodp0", "rmodp_small", "scale1r", "size", "size_poly0", "size_poly1", "size_poly1P", "size_polyC", "size_poly_eq0", "size_poly_leq0", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvd1p m : rdvdp 1 m.

Proof. by rewrite /rdvdp rmodp1. Qed.

Lemma

rdvd1p

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdvdp", "rmodp1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Nrdvdp_small (n d : {poly R}) : n != 0 -> size n < size d -> rdvdp d n = false.

Proof. by move=> nn0 hs; rewrite /rdvdp (rmodp_small hs); apply: negPf. Qed.

Lemma

Nrdvdp_small

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "poly", "rdvdp", "rmodp_small", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodp_eq0P p q : reflect (rmodp p q = 0) (rdvdp q p).

Proof. exact: (iffP eqP). Qed.

Lemma

rmodp_eq0P

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdvdp", "rmodp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodp_eq0 p q : rdvdp q p -> rmodp p q = 0.

Proof. exact: rmodp_eq0P. Qed.

Lemma

rmodp_eq0

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdvdp", "rmodp", "rmodp_eq0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdp_leq p q : rdvdp p q -> q != 0 -> size p <= size q.

Proof. by move=> dvd_pq; rewrite leqNgt; apply: contra => /rmodp_small <-. Qed.

Lemma

rdvdp_leq

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "leqNgt", "rdvdp", "rmodp_small", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rgcdp p q

:= let: (p1, q1) := if size p < size q then (q, p) else (p, q) in if p1 == 0 then q1 else let fix loop (n : nat) (pp qq : {poly R}) {struct n} := let rr := rmodp pp qq in if rr == 0 then qq else if n is n1.+1 then loop n1 qq rr else rr in loop (size p1) p1 q1.

Definition

rgcdp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "nat", "poly", "rmodp", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rgcd0p : left_id 0 rgcdp.

Proof. move=> p; rewrite /rgcdp size_poly0 size_poly_gt0 if_neg. case: ifP => /= [_ | nzp]; first by rewrite eqxx. by rewrite polySpred !(rmodp0, nzp) //; case: _.-1 => [|m]; rewrite rmod0p eqxx. Qed.

Lemma

rgcd0p

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "eqxx", "polySpred", "rgcdp", "rmod0p", "rmodp0", "size_poly0", "size_poly_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rgcdp0 : right_id 0 rgcdp.

Proof. move=> p; have:= rgcd0p p; rewrite /rgcdp size_poly0 size_poly_gt0. by case: eqVneq => p0; rewrite ?(eqxx, p0) //= eqxx. Qed.

Lemma

rgcdp0

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "eqVneq", "eqxx", "p0", "rgcd0p", "rgcdp", "size_poly0", "size_poly_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rgcdpE p q : rgcdp p q = if size p < size q then rgcdp (rmodp q p) p else rgcdp (rmodp p q) q.

Proof. pose rgcdp_rec := fix rgcdp_rec (n : nat) (pp qq : {poly R}) {struct n} := let rr := rmodp pp qq in if rr == 0 then qq else if n is n1.+1 then rgcdp_rec n1 qq rr else rr. have Irec: forall m n p q, size q <= m -> size q <= n -> size q < size p -> rgcdp_rec m p q = rgcdp_rec n p q. + elim=> [|m H...

Lemma

rgcdpE

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "eqVneq", "eqxx", "last", "leqW", "leq_trans", "ltnP", "ltnS", "ltnW", "ltn_rmodp", "nat", "poly", "polySpred", "rgcd0p", "rgcdp", "rgcdp0", "rmod0p", "rmodp", "rmodp0", "size", "size_poly0", "size_poly_gt0", "size_poly_leq0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_redivp_spec m d : nat * {poly R} * {poly R} -> Type

:= ComEdivnSpec k (q r : {poly R}) of (GRing.comm d (lead_coef d)%:P -> m * (lead_coef d ^+ k)%:P = q * d + r) & (d != 0 -> size r < size d) : comm_redivp_spec m d (k, q, r).

Variant

comm_redivp_spec

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "comm", "lead_coef", "nat", "poly", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_redivpP m d : comm_redivp_spec m d (redivp m d).

Proof. rewrite unlock; have [->|Hd] := eqVneq d 0. by constructor; rewrite !(simp, eqxx). have: GRing.comm d (lead_coef d)%:P -> m * (lead_coef d ^+ 0)%:P = 0 * d + m. by rewrite !simp. elim: (size m) 0%N 0 {1 4 6}m (leqnn (size m)) => [|n IHn] k q r Hr /=. move/size_poly_leq0P: Hr ->. suff hsd: size (0: {poly ...

Lemma

comm_redivpP

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "add1n", "addrA", "addrN", "apply", "coefB", "coefMC", "coefXnM", "coefZ", "comm", "comm_redivp_spec", "eqVneq", "eqxx", "exprSr", "last", "lead_coef", "leq_add2r", "leq_eqVlt", "leq_ltn_trans", "leq_sizeP", "leq_sub2r", "leq_subLR", "leq_trans", "leqnn", "ltnNge", "l...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodpp p : GRing.comm p (lead_coef p)%:P -> rmodp p p = 0.

Proof. move=> hC; rewrite /rmodp unlock; have [-> //|] := eqVneq. rewrite -size_poly_eq0 /redivp_rec; case sp: (size p)=> [|n] // _. rewrite sp ltnn subnn expr0 hC alg_polyC !simp subrr. by case: n sp => [|n] sp; rewrite size_polyC /= eqxx. Qed.

Lemma

rmodpp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "alg_polyC", "comm", "eqVneq", "eqxx", "expr0", "lead_coef", "ltnn", "redivp_rec", "rmodp", "simp", "size", "size_polyC", "size_poly_eq0", "sp", "subnn", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoprimep (p q : {poly R})

:= size (rgcdp p q) == 1.

Definition

rcoprimep

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "poly", "rgcdp", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rgdcop_rec q p n

:= if n is m.+1 then if rcoprimep p q then p else rgdcop_rec q (rdivp p (rgcdp p q)) m else (q == 0)%:R.

Fixpoint

rgdcop_rec

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rcoprimep", "rdivp", "rgcdp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rgdcop q p

:= rgdcop_rec q p (size p).

Definition

rgdcop

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rgdcop_rec", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rgdcop0 q : rgdcop q 0 = (q == 0)%:R.

Proof. by rewrite /rgdcop size_poly0. Qed.

Lemma

rgdcop0

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rgdcop", "size_poly0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Cdl : GRing.comm d (lead_coef d)%:P.

Hypothesis

Cdl

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "comm", "lead_coef" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Rreg : GRing.rreg (lead_coef d).

Hypothesis

Rreg

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "lead_coef", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

redivp_eq q r : size r < size d -> let k := (redivp (q * d + r) d).1.1 in let c := (lead_coef d ^+ k)%:P in redivp (q * d + r) d = (k, q * c, r * c).

Proof. move=> lt_rd; case: comm_redivpP=> k q1 r1 /(_ Cdl) Heq. have dn0: d != 0 by case: (size d) lt_rd (size_poly_eq0 d) => // n _ <-. move=> /(_ dn0) Hs. have eC : q * d * (lead_coef d ^+ k)%:P = q * (lead_coef d ^+ k)%:P * d. by rewrite -mulrA polyC_exp (commrX k Cdl) mulrA. suff e1 : q1 = q * (lead_coef d ^+ k)%...

Lemma

redivp_eq

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "Cdl", "addn0", "addnS", "addrACA", "addrI", "apply", "comm_redivpP", "commrX", "eC", "gtn_max", "last", "lead_coef", "leq_ltn_trans", "mulNr", "mulrA", "mulrDl", "opprB", "opprD", "polyC_exp", "r1", "redivp", "rreg_div0", "size", "size_polyC", "size_polyD", "size_p...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivp_eq p : p * (lead_coef d ^+ (rscalp p d))%:P = (rdivp p d) * d + (rmodp p d).

Proof. by rewrite /rdivp /rmodp /rscalp; case: comm_redivpP=> k q1 r1 Hc _; apply: Hc. Qed.

Lemma

rdivp_eq

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "comm_redivpP", "lead_coef", "r1", "rdivp", "rmodp", "rscalp" ]

this is a bad name

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_rdvdp k q1 p: p * ((lead_coef d)^+ k)%:P = q1 * d -> rdvdp d p.

Proof. move=> he. have Hnq0 := rreg_lead0 Rreg; set lq := lead_coef d. pose v := rscalp p d; pose m := maxn v k. rewrite /rdvdp -(rreg_polyMC_eq0 _ (@rregX _ _ (m - v) Rreg)). suff: ((rdivp p d) * (lq ^+ (m - v))%:P - q1 * (lq ^+ (m - k))%:P) * d + (rmodp p d) * (lq ^+ (m - v))%:P == 0. rewrite rreg_div0 //; last ...

Lemma

eq_rdvdp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "Cdl", "Rreg", "addnC", "addrAC", "commrX", "exprD", "last", "lead_coef", "leq_maxl", "leq_maxr", "ltn_rmodp", "maxn", "mulNr", "mulrA", "mulrDl", "polyCM", "polyC_exp", "rdivp", "rdivp_eq", "rdvdp", "rmodp", "rregX", "rreg_div0", "rreg_lead0", "rreg_polyMC_eq0", "r...

section variables impose an inconvenient order on parameters

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdp_spec p q : {poly R} -> bool -> Type

:= | Rdvdp k q1 & p * ((lead_coef q)^+ k)%:P = q1 * q : rdvdp_spec p q 0 true | RdvdpN & rmodp p q != 0 : rdvdp_spec p q (rmodp p q) false.

Variant

rdvdp_spec

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "lead_coef", "poly", "rmodp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdp_eqP p : rdvdp_spec p d (rmodp p d) (rdvdp d p).

Proof. case hdvd: (rdvdp d p); last by apply: RdvdpN; move/rmodp_eq0P/eqP: hdvd. move/rmodp_eq0P: (hdvd)->; apply: (@Rdvdp _ _ (rscalp p d) (rdivp p d)). by rewrite rdivp_eq //; move/rmodp_eq0P: (hdvd)->; rewrite addr0. Qed.

Lemma

rdvdp_eqP

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "addr0", "apply", "last", "rdivp", "rdivp_eq", "rdvdp", "rdvdp_spec", "rmodp", "rmodp_eq0P", "rscalp" ]

Is that version useable ?

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdp_mull p : rdvdp d (p * d).

Proof. by apply: (@eq_rdvdp 0 p); rewrite expr0 mulr1. Qed.

Lemma

rdvdp_mull

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "apply", "eq_rdvdp", "expr0", "mulr1", "rdvdp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodp_mull p : rmodp (p * d) d = 0.

Proof. exact/eqP/rdvdp_mull. Qed.

Lemma

rmodp_mull

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdvdp_mull", "rmodp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmodpp : rmodp d d = 0.

Proof. by rewrite -[d in rmodp d _]mul1r rmodp_mull. Qed.

Lemma

rmodpp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "mul1r", "rmodp", "rmodp_mull" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivpp : rdivp d d = (lead_coef d ^+ rscalp d d)%:P.

Proof. have dn0 : d != 0 by rewrite -lead_coef_eq0 rreg_neq0. move: (rdivp_eq d); rewrite rmodpp addr0. suff ->: GRing.comm d (lead_coef d ^+ rscalp d d)%:P by move/(rreg_lead Rreg)->. by rewrite polyC_exp; apply: commrX. Qed.

Lemma

rdivpp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "Rreg", "addr0", "apply", "comm", "commrX", "lead_coef", "lead_coef_eq0", "polyC_exp", "rdivp", "rdivp_eq", "rmodpp", "rreg_lead", "rreg_neq0", "rscalp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdvdpp : rdvdp d d.

Proof. exact/eqP/rmodpp. Qed.

Lemma

rdvdpp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "rdvdp", "rmodpp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivpK p : rdvdp d p -> rdivp p d * d = p * (lead_coef d ^+ rscalp p d)%:P.

Proof. by rewrite rdivp_eq /rdvdp; move/eqP->; rewrite addr0. Qed.

Lemma

rdivpK

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "addr0", "lead_coef", "rdivp", "rdivp_eq", "rdvdp", "rscalp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mond : d \is monic.

Hypothesis

mond

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "monic" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

redivp_eq q r : size r < size d -> let k := (redivp (q * d + r) d).1.1 in redivp (q * d + r) d = (k, q, r).

Proof. case: (monic_comreg mond)=> Hc Hr /(redivp_eq Hc Hr q). by rewrite (eqP mond) => -> /=; rewrite expr1n !mulr1. Qed.

Lemma

redivp_eq

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "expr1n", "mond", "monic_comreg", "mulr1", "redivp", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivp_eq p : p = rdivp p d * d + rmodp p d.

Proof. rewrite -rdivp_eq (eqP mond); first exact: commr1. by rewrite expr1n mulr1. Qed.

Lemma

rdivp_eq

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "commr1", "expr1n", "mond", "mulr1", "rdivp", "rmodp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivpp : rdivp d d = 1.

Proof. by case: (monic_comreg mond) => hc hr; rewrite rdivpp // (eqP mond) expr1n. Qed.

Lemma

rdivpp

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "expr1n", "mond", "monic_comreg", "rdivp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivp_addl_mul_small q r : size r < size d -> rdivp (q * d + r) d = q.

Proof. by move=> Hd; case: (monic_comreg mond)=> Hc Hr; rewrite /rdivp redivp_eq. Qed.

Lemma

rdivp_addl_mul_small

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "mond", "monic_comreg", "rdivp", "redivp_eq", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivp_addl_mul q r : rdivp (q * d + r) d = q + rdivp r d.

Proof. case: (monic_comreg mond)=> Hc Hr; rewrite [r in _ * _ + r]rdivp_eq addrA. by rewrite -mulrDl rdivp_addl_mul_small // ltn_rmodp monic_neq0. Qed.

Lemma

rdivp_addl_mul

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "addrA", "ltn_rmodp", "mond", "monic_comreg", "monic_neq0", "mulrDl", "rdivp", "rdivp_addl_mul_small", "rdivp_eq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivpDl q r : rdvdp d q -> rdivp (q + r) d = rdivp q d + rdivp r d.

Proof. case: (monic_comreg mond)=> Hc Hr; rewrite [r in q + r]rdivp_eq addrA. rewrite [q in q + _ + _]rdivp_eq; move/rmodp_eq0P->. by rewrite addr0 -mulrDl rdivp_addl_mul_small // ltn_rmodp monic_neq0. Qed.

Lemma

rdivpDl

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "addr0", "addrA", "ltn_rmodp", "mond", "monic_comreg", "monic_neq0", "mulrDl", "rdivp", "rdivp_addl_mul_small", "rdivp_eq", "rdvdp", "rmodp_eq0P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rdivpDr q r : rdvdp d r -> rdivp (q + r) d = rdivp q d + rdivp r d.

Proof. by rewrite addrC; move/rdivpDl->; rewrite addrC. Qed.

Lemma

rdivpDr

algebra

algebra/polydiv.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "poly", "GRing.Theory", "CommonRing", "RingComRreg", "RingMonic", "Ring", "ComRing", "UnitRing", "IdomainD...

[ "addrC", "rdivp", "rdivpDl", "rdvdp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d