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 |
|---|---|---|---|---|---|---|---|---|---|---|
gcdp_eqp1 p q : (gcdp p q %= 1) = coprimep p q. | Proof. by rewrite coprimep_def size_poly_eq1. Qed. | Lemma | gcdp_eqp1 | 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... | [
"coprimep",
"coprimep_def",
"gcdp",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_sym p q : coprimep p q = coprimep q p. | Proof. by rewrite -!gcdp_eqp1; apply: eqp_ltrans; rewrite gcdpC. Qed. | Lemma | coprimep_sym | 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",
"coprimep",
"eqp_ltrans",
"gcdpC",
"gcdp_eqp1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime1p p : coprimep 1 p. | Proof. by rewrite /coprimep -[1%N](size_poly1 R); exact/eqP/eqp_size/gcd1p. Qed. | Lemma | coprime1p | 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... | [
"coprimep",
"eqp_size",
"gcd1p",
"size_poly1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep1 p : coprimep p 1. | Proof. by rewrite coprimep_sym; apply: coprime1p. Qed. | Lemma | coprimep1 | 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",
"coprime1p",
"coprimep",
"coprimep_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep0 p : coprimep p 0 = (p %= 1). | Proof. by rewrite /coprimep gcdp0 size_poly_eq1. Qed. | Lemma | coprimep0 | 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... | [
"coprimep",
"gcdp0",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime0p p : coprimep 0 p = (p %= 1). | Proof. by rewrite coprimep_sym coprimep0. Qed. | Lemma | coprime0p | 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... | [
"coprimep",
"coprimep0",
"coprimep_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimepP p q :
reflect (forall d, d %| p -> d %| q -> d %= 1) (coprimep p q). | Proof.
rewrite /coprimep; apply: (iffP idP) => [/eqP hs d dvddp dvddq | h].
have/dvdp_eqp1: d %| gcdp p q by rewrite dvdp_gcd dvddp dvddq.
by rewrite -size_poly_eq1 hs; exact.
by rewrite size_poly_eq1; case/andP: (dvdp_gcdlr p q); apply: h.
Qed. | Lemma | coprimepP | 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",
"coprimep",
"dvdp_eqp1",
"dvdp_gcd",
"dvdp_gcdlr",
"gcdp",
"size_poly_eq1"
] | This is different from coprimeP in div. shall we keep this? | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprimepPn p q : p != 0 ->
reflect (exists d, (d %| gcdp p q) && ~~ (d %= 1)) (~~ coprimep p q). | Proof.
move=> p0; apply: (iffP idP).
by rewrite -gcdp_eqp1=> ng1; exists (gcdp p q); rewrite dvdpp /=.
case=> d /andP [dg]; apply: contra; rewrite -gcdp_eqp1=> g1.
by move: dg; rewrite (eqp_dvdr _ g1) dvdp1 size_poly_eq1.
Qed. | Lemma | coprimepPn | 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",
"coprimep",
"dvdp1",
"dvdpp",
"eqp_dvdr",
"gcdp",
"gcdp_eqp1",
"p0",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_dvdl q p r : r %| q -> coprimep p q -> coprimep p r. | Proof.
move=> rp /coprimepP cpq'; apply/coprimepP => d dp dr.
exact/cpq'/(dvdp_trans dr).
Qed. | Lemma | coprimep_dvdl | 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",
"coprimep",
"coprimepP",
"dvdp_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_dvdr p q r : r %| p -> coprimep p q -> coprimep r q. | Proof.
by move=> rp; rewrite ![coprimep _ q]coprimep_sym; apply/coprimep_dvdl.
Qed. | Lemma | coprimep_dvdr | 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",
"coprimep",
"coprimep_dvdl",
"coprimep_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_modl p q : coprimep (p %% q) q = coprimep p q. | Proof.
rewrite !coprimep_def [in RHS]gcdpE.
by case: ltnP => // hpq; rewrite modp_small // gcdpE hpq.
Qed. | Lemma | coprimep_modl | 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... | [
"coprimep",
"coprimep_def",
"gcdpE",
"ltnP",
"modp_small"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_modr q p : coprimep q (p %% q) = coprimep q p. | Proof. by rewrite ![coprimep q _]coprimep_sym coprimep_modl. Qed. | Lemma | coprimep_modr | 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... | [
"coprimep",
"coprimep_modl",
"coprimep_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcoprimep_coprimep q p : rcoprimep q p = coprimep q p. | Proof. by rewrite /coprimep /rcoprimep (eqp_size (eqp_rgcd_gcd _ _)). Qed. | Lemma | rcoprimep_coprimep | 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... | [
"coprimep",
"eqp_rgcd_gcd",
"eqp_size",
"rcoprimep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_coprimepr p q r : q %= r -> coprimep p q = coprimep p r. | Proof. by rewrite -!gcdp_eqp1; move/(eqp_gcdr p)/eqp_ltrans. Qed. | Lemma | eqp_coprimepr | 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... | [
"coprimep",
"eqp_gcdr",
"eqp_ltrans",
"gcdp_eqp1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_coprimepl p q r : q %= r -> coprimep q p = coprimep r p. | Proof. by rewrite !(coprimep_sym _ p); apply: eqp_coprimepr. Qed. | Lemma | eqp_coprimepl | 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",
"coprimep",
"coprimep_sym",
"eqp_coprimepr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdp_rec p q k {struct k} : {poly R} * {poly R} | :=
if k is k'.+1 then
if q == 0 then (1, 0) else
let: (u, v) := egcdp_rec q (p %% q) k' in
(lead_coef q ^+ scalp p q *: v, (u - v * (p %/ q)))
else (1, 0). | Fixpoint | egcdp_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",
"poly",
"scalp"
] | This should be implemented with an extended remainder sequence | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
egcdp p q | :=
if size q <= size p then egcdp_rec p q (size q)
else let e := egcdp_rec q p (size p) in (e.2, e.1). | Definition | egcdp | 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... | [
"egcdp_rec",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdp0 p : egcdp p 0 = (1, 0). | Proof. by rewrite /egcdp size_poly0. Qed. | Lemma | egcdp0 | 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... | [
"egcdp",
"size_poly0"
] | No provable egcd0p | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
egcdp_recP : forall k p q, q != 0 -> size q <= k -> size q <= size p ->
let e := (egcdp_rec p q k) in
[/\ size e.1 <= size q, size e.2 <= size p & gcdp p q %= e.1 * p + e.2 * q]. | Proof.
elim=> [|k ihk] p q /= qn0; first by rewrite size_poly_leq0 (negPf qn0).
move=> sqSn qsp; rewrite (negPf qn0).
have sp : size p > 0 by apply: leq_trans qsp; rewrite size_poly_gt0.
have [r0 | rn0] /= := eqVneq (p %%q) 0.
rewrite r0 /egcdp_rec; case: k ihk sqSn => [|n] ihn sqSn /=.
rewrite !scaler0 !mul0r su... | Lemma | egcdp_recP | 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... | [
"add0r",
"addKn",
"addSn",
"addnBA",
"addrC",
"addrCA",
"addrK",
"apply",
"divp_eq",
"dvdp",
"dvdp_gcd_idr",
"egcdp_rec",
"eqVneq",
"eqp_ltrans",
"eqp_trans",
"eqxx",
"gcdp",
"gcdpC",
"gcdpE",
"geq_max",
"last",
"lc_expn_scalp_neq0",
"leq_modp",
"leq_pred",
"leq_trans... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdpP p q : p != 0 -> q != 0 -> forall (e := egcdp p q),
[/\ size e.1 <= size q, size e.2 <= size p & gcdp p q %= e.1 * p + e.2 * q]. | Proof.
rewrite /egcdp => pn0 qn0; case: (leqP (size q) (size p)) => /= [|/ltnW] hp.
exact: egcdp_recP.
case: (egcdp_recP pn0 (leqnn (size p)) hp) => h1 h2 h3; split => //.
by rewrite (eqp_ltrans (gcdpC _ _)) addrC.
Qed. | Lemma | egcdpP | 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",
"egcdp",
"egcdp_recP",
"eqp_ltrans",
"gcdp",
"gcdpC",
"leqP",
"leqnn",
"ltnW",
"size",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
egcdpE p q (e := egcdp p q) : gcdp p q %= e.1 * p + e.2 * q. | Proof.
rewrite {}/e; have [-> /= | qn0] := eqVneq q 0.
by rewrite gcdp0 egcdp0 mul1r mulr0 addr0.
have [-> | pn0] := eqVneq p 0; last by case: (egcdpP pn0 qn0).
by rewrite gcd0p /egcdp size_poly0 size_poly_leq0 (negPf qn0) /= !simp.
Qed. | Lemma | egcdpE | 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",
"egcdp",
"egcdp0",
"egcdpP",
"eqVneq",
"gcd0p",
"gcdp",
"gcdp0",
"last",
"mul1r",
"mulr0",
"simp",
"size_poly0",
"size_poly_leq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bezoutp p q : exists u, u.1 * p + u.2 * q %= (gcdp p q). | Proof.
have [-> | pn0] := eqVneq p 0.
by rewrite gcd0p; exists (0, 1); rewrite mul0r mul1r add0r.
have [-> | qn0] := eqVneq q 0.
by rewrite gcdp0; exists (1, 0); rewrite mul0r mul1r addr0.
pose e := egcdp p q; exists e; rewrite eqp_sym.
by case: (egcdpP pn0 qn0).
Qed. | Lemma | Bezoutp | 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... | [
"add0r",
"addr0",
"egcdp",
"egcdpP",
"eqVneq",
"eqp_sym",
"gcd0p",
"gcdp",
"gcdp0",
"mul0r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bezout_coprimepP p q :
reflect (exists u, u.1 * p + u.2 * q %= 1) (coprimep p q). | Proof.
rewrite -gcdp_eqp1; apply: (iffP idP)=> [g1|].
by case: (Bezoutp p q) => [[u v] Puv]; exists (u, v); apply: eqp_trans g1.
case=> [[u v]]; rewrite eqp_sym=> Puv; rewrite /eqp (eqp_dvdr _ Puv).
by rewrite dvdp_addr dvdp_mull ?dvdp_gcdl ?dvdp_gcdr //= dvd1p.
Qed. | Lemma | Bezout_coprimepP | 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... | [
"Bezoutp",
"apply",
"coprimep",
"dvd1p",
"dvdp_addr",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mull",
"eqp",
"eqp_dvdr",
"eqp_sym",
"eqp_trans",
"gcdp_eqp1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_root p q x : coprimep p q -> root p x -> q.[x] != 0. | Proof.
case/Bezout_coprimepP=> [[u v] euv] px0.
move/eqpP: euv => [[c1 c2]] /andP /= [c1n0 c2n0 e].
suffices: c1 * (v.[x] * q.[x]) != 0.
by rewrite !mulf_eq0 !negb_or c1n0 /=; case/andP.
have := f_equal (horner^~ x) e; rewrite /= !hornerZ hornerD.
by rewrite !hornerM (eqP px0) mulr0 add0r hornerC mulr1; move->.
Qed. | Lemma | coprimep_root | 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... | [
"Bezout_coprimepP",
"add0r",
"c1",
"c2",
"coprimep",
"eqpP",
"horner",
"hornerC",
"hornerD",
"hornerM",
"hornerZ",
"mulf_eq0",
"mulr0",
"mulr1",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_dvdpl p q d: coprimep d q -> (d %| p * q) = (d %| p). | Proof.
move/Bezout_coprimepP=>[[u v] Puv]; apply/idP/idP; last exact: dvdp_mulr.
move/(eqp_mull p): Puv; rewrite mulr1 mulrDr eqp_sym=> peq dpq.
rewrite (eqp_dvdr _ peq) dvdp_addr; last by rewrite mulrA mulrAC dvdp_mulr.
by rewrite mulrA dvdp_mull ?dvdpp.
Qed. | Lemma | Gauss_dvdpl | 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... | [
"Bezout_coprimepP",
"apply",
"coprimep",
"dvdp_addr",
"dvdp_mull",
"dvdp_mulr",
"dvdpp",
"eqp_dvdr",
"eqp_mull",
"eqp_sym",
"last",
"mulr1",
"mulrA",
"mulrAC",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_dvdpr p q d: coprimep d q -> (d %| q * p) = (d %| p). | Proof. by rewrite mulrC; apply: Gauss_dvdpl. Qed. | Lemma | Gauss_dvdpr | 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... | [
"Gauss_dvdpl",
"apply",
"coprimep",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_dvdp m n p : coprimep m n -> (m * n %| p) = (m %| p) && (n %| p). | Proof.
have [-> | mn0] := eqVneq m 0.
by rewrite coprime0p => /eqp_dvdl->; rewrite !mul0r dvd0p dvd1p andbT.
have [-> | nn0] := eqVneq n 0.
by rewrite coprimep0 => /eqp_dvdl->; rewrite !mulr0 dvd1p.
move=> hc; apply/idP/idP => [mnmp | /andP [dmp dnp]].
move/Gauss_dvdpl: hc => <-; move: (dvdp_mull m mnmp); rewrite... | Lemma | Gauss_dvdp | 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... | [
"Gauss_dvdpl",
"apply",
"c2",
"c3",
"coprime0p",
"coprimep",
"coprimep0",
"dvd0p",
"dvd1p",
"dvdp_eq",
"dvdp_mul2l",
"dvdp_mul2r",
"dvdp_mull",
"dvdp_mulr",
"eqVneq",
"eq_dvdp",
"eqp_dvdl",
"expf_neq0",
"lead_coef_eq0",
"mul0r",
"mul_polyC",
"mulf_neq0",
"mulr0",
"mulrA... | This could be simplified with the introduction of lcmp | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Gauss_gcdpr p m n : coprimep p m -> gcdp p (m * n) %= gcdp p n. | Proof.
move=> co_pm; apply/eqP; rewrite /eqp !dvdp_gcd !dvdp_gcdl /= andbC.
rewrite dvdp_mull ?dvdp_gcdr // -(@Gauss_dvdpl _ m); last first.
by rewrite mulrC dvdp_gcdr.
apply/coprimepP=> d; rewrite dvdp_gcd; case/andP=> hdp _ hdm.
by move/coprimepP: co_pm; apply.
Qed. | Lemma | Gauss_gcdpr | 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... | [
"Gauss_dvdpl",
"apply",
"coprimep",
"coprimepP",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mull",
"eqp",
"gcdp",
"last",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gauss_gcdpl p m n : coprimep p n -> gcdp p (m * n) %= gcdp p m. | Proof. by move=> co_pn; rewrite mulrC Gauss_gcdpr. Qed. | Lemma | Gauss_gcdpl | 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... | [
"Gauss_gcdpr",
"coprimep",
"gcdp",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimepMr p q r : coprimep p (q * r) = (coprimep p q && coprimep p r). | Proof.
apply/coprimepP/andP=> [hp | [/coprimepP-hq hr]].
by split; apply/coprimepP=> d dp dq; rewrite hp //;
[apply/dvdp_mulr | apply/dvdp_mull].
move=> d dp dqr; move/(_ _ dp) in hq.
rewrite Gauss_dvdpl in dqr; last exact: hq.
by move/coprimep_dvdr: hr; apply.
Qed. | Lemma | coprimepMr | 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... | [
"Gauss_dvdpl",
"apply",
"coprimep",
"coprimepP",
"coprimep_dvdr",
"dvdp_mull",
"dvdp_mulr",
"last",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimepMl p q r: coprimep (q * r) p = (coprimep q p && coprimep r p). | Proof. by rewrite ![coprimep _ p]coprimep_sym coprimepMr. Qed. | Lemma | coprimepMl | 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... | [
"coprimep",
"coprimepMr",
"coprimep_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modp_coprime k u n : k != 0 -> (k * u) %% n %= 1 -> coprimep k n. | Proof.
move=> kn0 hmod; apply/Bezout_coprimepP.
exists (((lead_coef n)^+(scalp (k * u) n) *: u), (- (k * u %/ n))).
by rewrite -scalerAl mulrC (divp_eq (u * k) n) mulNr addrC addKr mulrC.
Qed. | Lemma | modp_coprime | 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... | [
"Bezout_coprimepP",
"addKr",
"addrC",
"apply",
"coprimep",
"divp_eq",
"lead_coef",
"mulNr",
"mulrC",
"scalerAl",
"scalp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_pexpl k m n : 0 < k -> coprimep (m ^+ k) n = coprimep m n. | Proof.
case: k => // k _; elim: k => [|k IHk]; first by rewrite expr1.
by rewrite exprS coprimepMl -IHk andbb.
Qed. | Lemma | coprimep_pexpl | 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... | [
"coprimep",
"coprimepMl",
"expr1",
"exprS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_pexpr k m n : 0 < k -> coprimep m (n ^+ k) = coprimep m n. | Proof. by move=> k_gt0; rewrite !(coprimep_sym m) coprimep_pexpl. Qed. | Lemma | coprimep_pexpr | 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... | [
"coprimep",
"coprimep_pexpl",
"coprimep_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_expl k m n : coprimep m n -> coprimep (m ^+ k) n. | Proof. by case: k => [|k] co_pm; rewrite ?coprime1p // coprimep_pexpl. Qed. | Lemma | coprimep_expl | 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... | [
"coprime1p",
"coprimep",
"coprimep_pexpl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_expr k m n : coprimep m n -> coprimep m (n ^+ k). | Proof. by rewrite !(coprimep_sym m); apply: coprimep_expl. Qed. | Lemma | coprimep_expr | 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",
"coprimep",
"coprimep_expl",
"coprimep_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_mul2l p q r : gcdp (p * q) (p * r) %= (p * gcdp q r). | Proof.
have [->|hp] := eqVneq p 0; first by rewrite !mul0r gcdp0 eqpxx.
rewrite /eqp !dvdp_gcd !dvdp_mul2l // dvdp_gcdr dvdp_gcdl !andbT.
move: (Bezoutp q r) => [[u v]] huv.
rewrite eqp_sym in huv; rewrite (eqp_dvdr _ (eqp_mull _ huv)).
rewrite mulrDr ![p * (_ * _)]mulrCA.
by apply: dvdp_add; rewrite dvdp_mull// (dvdp_... | Lemma | gcdp_mul2l | 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... | [
"Bezoutp",
"apply",
"dvdp_add",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mul2l",
"dvdp_mull",
"eqVneq",
"eqp",
"eqp_dvdr",
"eqp_mull",
"eqp_sym",
"eqpxx",
"gcdp",
"gcdp0",
"mul0r",
"mulrCA",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_mul2r q r p : gcdp (q * p) (r * p) %= gcdp q r * p. | Proof. by rewrite ![_ * p]mulrC gcdp_mul2l. Qed. | Lemma | gcdp_mul2r | 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... | [
"gcdp",
"gcdp_mul2l",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulp_gcdr p q r : r * (gcdp p q) %= gcdp (r * p) (r * q). | Proof. by rewrite eqp_sym gcdp_mul2l. Qed. | Lemma | mulp_gcdr | 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... | [
"eqp_sym",
"gcdp",
"gcdp_mul2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulp_gcdl p q r : (gcdp p q) * r %= gcdp (p * r) (q * r). | Proof. by rewrite eqp_sym gcdp_mul2r. Qed. | Lemma | mulp_gcdl | 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... | [
"eqp_sym",
"gcdp",
"gcdp_mul2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_div_gcd p q : (p != 0) || (q != 0) ->
coprimep (p %/ (gcdp p q)) (q %/ gcdp p q). | Proof.
rewrite -negb_and -gcdp_eq0 -gcdp_eqp1 => gpq0.
rewrite -(@eqp_mul2r (gcdp p q)) // mul1r (eqp_ltrans (mulp_gcdl _ _ _)).
have: gcdp p q %| p by rewrite dvdp_gcdl.
have: gcdp p q %| q by rewrite dvdp_gcdr.
rewrite !dvdp_eq => /eqP <- /eqP <-.
have lcn0 k : (lead_coef (gcdp p q)) ^+ k != 0.
by rewrite expf_neq0... | Lemma | coprimep_div_gcd | 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",
"coprimep",
"dvdp_eq",
"dvdp_gcdl",
"dvdp_gcdr",
"eqp_gcd",
"eqp_ltrans",
"eqp_mul2r",
"eqp_scale",
"expf_neq0",
"gcdp",
"gcdp_eq0",
"gcdp_eqp1",
"lcn0",
"lead_coef",
"lead_coef_eq0",
"mul1r",
"mulp_gcdl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_eq0 p q : (p %/ q == 0) = [|| p == 0, q ==0 | size p < size q]. | Proof.
apply/eqP/idP=> [d0|]; last first.
case/or3P; [by move/eqP->; rewrite div0p| by move/eqP->; rewrite divp0|].
by move/divp_small.
case: eqVneq => // _; case: eqVneq => // qn0.
move: (divp_eq p q); rewrite d0 mul0r add0r.
move/(f_equal (fun x : {poly R} => size x)).
by rewrite size_scale ?lc_expn_scalp_neq0 //... | Lemma | divp_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... | [
"add0r",
"apply",
"div0p",
"divp0",
"divp_eq",
"divp_small",
"eqVneq",
"last",
"lc_expn_scalp_neq0",
"ltn_modp",
"mul0r",
"poly",
"size",
"size_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_div_eq0 p q : q %| p -> (p %/ q == 0) = (p == 0). | Proof.
move=> dvdp_qp; have [->|p_neq0] := eqVneq p 0; first by rewrite div0p eqxx.
rewrite divp_eq0 ltnNge dvdp_leq // (negPf p_neq0) orbF /=.
by apply: contraTF dvdp_qp=> /eqP ->; rewrite dvd0p.
Qed. | Lemma | dvdp_div_eq0 | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"nmodule",
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"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"div0p",
"divp_eq0",
"dvd0p",
"dvdp_leq",
"eqVneq",
"eqxx",
"ltnNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bezout_coprimepPn p q : p != 0 -> q != 0 ->
reflect (exists2 uv : {poly R} * {poly R},
(0 < size uv.1 < size q) && (0 < size uv.2 < size p) &
uv.1 * p = uv.2 * q)
(~~ (coprimep p q)). | Proof.
move=> pn0 qn0; apply: (iffP idP); last first.
case=> [[u v] /= /andP [/andP [ps1 s1] /andP [ps2 s2]] e].
have: ~~(size (q * p) <= size (u * p)).
rewrite -ltnNge !size_mul // -?size_poly_gt0 // (polySpred pn0) !addnS.
by rewrite ltn_add2r.
apply: contra => ?; apply: dvdp_leq; rewrite ?mulf_neq0 // ... | Lemma | Bezout_coprimepPn | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
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"addn1",
"addnS",
"apply",
"c1",
"c2",
"coprimep",
"coprimep_def",
"divp_eq0",
"dvdp_eq",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_leq",
"dvdp_mull",
"gcdp",
"gcdp_eq0",
"last",
"lc_expn_scalp_neq0",
"leqNgt",
"leq_add2l",
"leq_gcdpl",
"leq_gcdpr",
"ltnNge",
"ltnS"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_pexp2r m n k : k > 0 -> (m ^+ k %| n ^+ k) = (m %| n). | Proof.
move=> k_gt0; apply/idP/idP; last exact: dvdp_exp2r.
have [-> // | nn0] := eqVneq n 0; have [-> | mn0] := eqVneq m 0.
move/prednK: k_gt0=> {1}<-; rewrite exprS mul0r //= !dvd0p expf_eq0.
by case/andP=> _ ->.
set d := gcdp m n; have := dvdp_gcdr m n; rewrite -/d dvdp_eq.
set c1 := _ ^+ _; set n' := _ %/ _; mo... | Lemma | dvdp_pexp2r | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fintype",
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"divalg",
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"poly",
"GRing.Theory",
"CommonRing",
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"RingMonic",
"Ring",
"ComRing",
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"IdomainD... | [
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"def_n",
"dvd0p",
"dvdpZl",
"dvdpZr",
"dvdp_eq",
"dvdp_exp2r",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mul2r",
"dvdp_mull",
"dvdpp",
"eqSS",
"eqVneq",
"eqn0Ngt",
"expf_eq0",
"e... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_gcd p q x : root (gcdp p q) x = root p x && root q x. | Proof.
rewrite /= !root_factor_theorem; apply/idP/andP=> [dg| [dp dq]].
by split; apply: dvdp_trans dg _; rewrite ?(dvdp_gcdl, dvdp_gcdr).
have:= Bezoutp p q => [[[u v]]]; rewrite eqp_sym=> e.
by rewrite (eqp_dvdr _ e) dvdp_addl dvdp_mull.
Qed. | Lemma | root_gcd | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"Bezoutp",
"apply",
"dvdp_addl",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mull",
"dvdp_trans",
"eqp_dvdr",
"eqp_sym",
"gcdp",
"root",
"root_factor_theorem",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_biggcd x (ps : seq {poly R}) :
root (\big[gcdp/0]_(p <- ps) p) x = all (fun p => root p x) ps. | Proof.
elim: ps => [|p ps ihp]; first by rewrite big_nil root0.
by rewrite big_cons /= root_gcd ihp.
Qed. | Lemma | root_biggcd | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"all",
"big_cons",
"big_nil",
"gcdp",
"poly",
"root",
"root0",
"root_gcd",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gdcop_rec q p k | :=
if k is m.+1 then
if coprimep p q then p
else gdcop_rec q (divp p (gcdp p q)) m
else (q == 0)%:R. | Fixpoint | gdcop_rec | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
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"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"coprimep",
"divp",
"gcdp"
] | if P null, we pose that gdcop returns 1 if Q null, 0 otherwise | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gdcop q p | := gdcop_rec q p (size p). | Definition | gdcop | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
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"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"gdcop_rec",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gdcop_spec q p : {poly R} -> Type | :=
GdcopSpec r of (dvdp r p) & ((coprimep r q) || (p == 0))
& (forall d, dvdp d p -> coprimep d q -> dvdp d r)
: gdcop_spec q p r. | Variant | gdcop_spec | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"coprimep",
"dvdp",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gdcop0 q : gdcop q 0 = (q == 0)%:R. | Proof. by rewrite /gdcop size_poly0. Qed. | Lemma | gdcop0 | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"nmodule",
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"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"gdcop",
"size_poly0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gdcop_recP q p k : size p <= k -> gdcop_spec q p (gdcop_rec q p k). | Proof.
elim: k p => [p | k ihk p] /=.
move/size_poly_leq0P->.
have [->|q0] := eqVneq; split; rewrite ?coprime1p // ?eqxx ?orbT //.
by move=> d _; rewrite coprimep0 dvdp1 size_poly_eq1.
move=> hs; case cop : (coprimep _ _); first by split; rewrite ?dvdpp ?cop.
have [-> | p0] := eqVneq p 0.
by rewrite div0p; appl... | Lemma | gdcop_recP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
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"RingMonic",
"Ring",
"ComRing",
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"IdomainD... | [
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"add1n",
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"coprimep0",
"coprimepPn",
"coprimep_dvdl",
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"divp_dvd",
"divpp",
"dvd0p",
"dvd1p",
"dvdUp",
"dvdp1",
"dvdpN0",
"dvdp_eq",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_leq",
"dvdp_mulIl",
"dv... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gdcopP q p : gdcop_spec q p (gdcop q p). | Proof. by rewrite /gdcop; apply: gdcop_recP. Qed. | Lemma | gdcopP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
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"divalg",
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"poly",
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"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"gdcop",
"gdcop_recP",
"gdcop_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_gdco p q : (q != 0)%B -> coprimep (gdcop p q) p. | Proof. by move=> q_neq0; case: gdcopP=> d; rewrite (negPf q_neq0) orbF. Qed. | Lemma | coprimep_gdco | 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... | [
"coprimep",
"gdcop",
"gdcopP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size2_dvdp_gdco p q d : p != 0 -> size d = 2 ->
(d %| (gdcop q p)) = (d %| p) && ~~(d %| q). | Proof.
have [-> | dn0] := eqVneq d 0; first by rewrite size_poly0.
move=> p0 sd; apply/idP/idP.
case: gdcopP=> r rp crq maxr dr; move/negPf: (p0)=> p0f.
rewrite (dvdp_trans dr) //=.
apply: contraL crq => dq; rewrite p0f orbF; apply/coprimepPn.
by apply: contraNneq p0 => r0; move: rp; rewrite r0 dvd0p.
by ex... | Lemma | size2_dvdp_gdco | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"RingMonic",
"Ring",
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"coprimepP",
"coprimepPn",
"dvd0p",
"dvdp_gcd",
"dvdp_leq",
"dvdp_size_eqp",
"dvdp_trans",
"eqVneq",
"eqp_dvdl",
"gdcop",
"gdcopP",
"last",
"leq_eqVlt",
"ltnS",
"maxr",
"p0",
"predU1P",
"size",
"size_poly0",
"size_poly_eq1",
"size_poly_leq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_gdco p q : (gdcop p q) %| q. | Proof. by case: gdcopP. Qed. | Lemma | dvdp_gdco | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"gdcop",
"gdcopP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_gdco p q x : p != 0 -> root (gdcop q p) x = root p x && ~~(root q x). | Proof.
move=> p0 /=; rewrite !root_factor_theorem.
apply: size2_dvdp_gdco; rewrite ?p0 //.
by rewrite size_polyDl size_polyX // size_polyN size_polyC ltnS; case: (x != 0).
Qed. | Lemma | root_gdco | 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",
"gdcop",
"ltnS",
"p0",
"root",
"root_factor_theorem",
"size2_dvdp_gdco",
"size_polyC",
"size_polyDl",
"size_polyN",
"size_polyX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_comp_poly r p q : (p %| q) -> (p \Po r) %| (q \Po r). | Proof.
have [-> | pn0] := eqVneq p 0.
by rewrite comp_poly0 !dvd0p; move/eqP->; rewrite comp_poly0.
rewrite dvdp_eq; set c := _ ^+ _; set s := _ %/ _; move/eqP=> Hq.
apply: (@eq_dvdp c (s \Po r)); first by rewrite expf_neq0 // lead_coef_eq0.
by rewrite -comp_polyZ Hq comp_polyM.
Qed. | Lemma | dvdp_comp_poly | 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",
"comp_poly0",
"comp_polyM",
"comp_polyZ",
"dvd0p",
"dvdp_eq",
"eqVneq",
"eq_dvdp",
"expf_neq0",
"lead_coef_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_comp_poly r p q : gcdp p q \Po r %= gcdp (p \Po r) (q \Po r). | Proof.
apply/andP; split.
by rewrite dvdp_gcd !dvdp_comp_poly ?dvdp_gcdl ?dvdp_gcdr.
case: (Bezoutp p q) => [[u v]] /andP [].
move/(dvdp_comp_poly r) => Huv _.
rewrite (dvdp_trans _ Huv) // comp_polyD !comp_polyM.
by rewrite dvdp_add // dvdp_mull //; [ exact: dvdp_gcdl | exact: dvdp_gcdr].
Qed. | Lemma | gcdp_comp_poly | 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... | [
"Bezoutp",
"apply",
"comp_polyD",
"comp_polyM",
"dvdp_add",
"dvdp_comp_poly",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mull",
"dvdp_trans",
"gcdp",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_comp_poly r p q : coprimep p q -> coprimep (p \Po r) (q \Po r). | Proof.
rewrite -!gcdp_eqp1 -!size_poly_eq1 -!dvdp1; move/(dvdp_comp_poly r).
rewrite comp_polyC => Hgcd.
by apply: dvdp_trans Hgcd; case/andP: (gcdp_comp_poly r p q).
Qed. | Lemma | coprimep_comp_poly | 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",
"comp_polyC",
"coprimep",
"dvdp1",
"dvdp_comp_poly",
"dvdp_trans",
"gcdp_comp_poly",
"gcdp_eqp1",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_addl_mul p q r : coprimep r (p * r + q) = coprimep r q. | Proof. by rewrite !coprimep_def (eqp_size (gcdp_addl_mul _ _ _)). Qed. | Lemma | coprimep_addl_mul | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
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"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"coprimep",
"coprimep_def",
"eqp_size",
"gcdp_addl_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irreducible_poly p | :=
(size p > 1) * (forall q, size q != 1 -> q %| p -> q %= p) : Prop. | Definition | irreducible_poly | 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... | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irredp_neq0 p : irreducible_poly p -> p != 0. | Proof. by rewrite -size_poly_gt0 => [[/ltnW]]. Qed. | Lemma | irredp_neq0 | 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... | [
"irreducible_poly",
"ltnW",
"size_poly_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
apply_irredp p (irr_p : irreducible_poly p) | := irr_p.2. | Definition | apply_irredp | 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... | [
"irr_p",
"irreducible_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
apply_irredp : irreducible_poly >-> Funclass. | Coercion | apply_irredp | 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... | [
"irreducible_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
modp_XsubC p c : p %% ('X - c%:P) = p.[c]%:P. | Proof.
have/factor_theorem [q /(canRL (subrK _)) Dp]: root (p - p.[c]%:P) c.
by rewrite /root !hornerE subrr.
rewrite modpE /= lead_coefXsubC unitr1 expr1n invr1 scale1r [in LHS]Dp.
rewrite RingMonic.rmodp_addl_mul_small // ?monicXsubC// size_XsubC size_polyC.
by case: (p.[c] == 0).
Qed. | Lemma | modp_XsubC | 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",
"factor_theorem",
"hornerE",
"invr1",
"lead_coefXsubC",
"modpE",
"monicXsubC",
"rmodp_addl_mul_small",
"root",
"scale1r",
"size_XsubC",
"size_polyC",
"subrK",
"subrr",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_XsubC p c : coprimep p ('X - c%:P) = ~~ root p c. | Proof.
rewrite -coprimep_modl modp_XsubC /root -alg_polyC.
have [-> | /coprimepZl->] := eqVneq; last exact: coprime1p.
by rewrite scale0r /coprimep gcd0p size_XsubC.
Qed. | Lemma | coprimep_XsubC | 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",
"coprime1p",
"coprimep",
"coprimepZl",
"coprimep_modl",
"eqVneq",
"gcd0p",
"last",
"modp_XsubC",
"root",
"scale0r",
"size_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_XsubC2 (a b : R) : b - a != 0 ->
coprimep ('X - a%:P) ('X - b%:P). | Proof. by move=> bBa_neq0; rewrite coprimep_XsubC rootE hornerXsubC. Qed. | Lemma | coprimep_XsubC2 | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
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"divalg",
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"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"coprimep",
"coprimep_XsubC",
"hornerXsubC",
"rootE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimepX p : coprimep p 'X = ~~ root p 0. | Proof. by rewrite -['X]subr0 coprimep_XsubC. Qed. | Lemma | coprimepX | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
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"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"coprimep",
"coprimep_XsubC",
"root",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_monic : {in monic &, forall p q, (p %= q) = (p == q)}. | Proof.
move=> p q monic_p monic_q; apply/idP/eqP=> [|-> //].
case/eqpP=> [[a b] /= /andP[a_neq0 _] eq_pq].
apply: (@mulfI _ a%:P); first by rewrite polyC_eq0.
rewrite !mul_polyC eq_pq; congr (_ *: q); apply: (mulIf (oner_neq0 _)).
by rewrite -[in LHS](monicP monic_q) -(monicP monic_p) -!lead_coefZ eq_pq.
Qed. | Lemma | eqp_monic | 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",
"eqpP",
"lead_coefZ",
"monic",
"monicP",
"mulIf",
"mul_polyC",
"mulfI",
"oner_neq0",
"polyC_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_mul_XsubC p q c :
(p %| ('X - c%:P) * q) = ((if root p c then p %/ ('X - c%:P) else p) %| q). | Proof.
case: ifPn => [| not_pc0]; last by rewrite Gauss_dvdpr ?coprimep_XsubC.
rewrite root_factor_theorem -eqp_div_XsubC mulrC => /eqP{1}->.
by rewrite dvdp_mul2l ?polyXsubC_eq0.
Qed. | Lemma | dvdp_mul_XsubC | 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... | [
"Gauss_dvdpr",
"coprimep_XsubC",
"dvdp_mul2l",
"eqp_div_XsubC",
"last",
"mulrC",
"polyXsubC_eq0",
"root",
"root_factor_theorem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_prod_XsubC (I : Type) (r : seq I) (F : I -> R) p :
p %| \prod_(i <- r) ('X - (F i)%:P) ->
{m | p %= \prod_(i <- mask m r) ('X - (F i)%:P)}. | Proof.
elim: r => [|i r IHr] in p *.
by rewrite big_nil dvdp1; exists nil; rewrite // big_nil -size_poly_eq1.
rewrite big_cons dvdp_mul_XsubC root_factor_theorem -eqp_div_XsubC.
case: eqP => [{2}-> | _] /IHr[m Dp]; last by exists (false :: m).
by exists (true :: m); rewrite /= mulrC big_cons eqp_mul2l ?polyXsubC_eq0.... | Lemma | dvdp_prod_XsubC | 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... | [
"big_cons",
"big_nil",
"dvdp1",
"dvdp_mul_XsubC",
"eqp_div_XsubC",
"eqp_mul2l",
"last",
"mask",
"mulrC",
"polyXsubC_eq0",
"root_factor_theorem",
"seq",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irredp_XsubC (x : R) : irreducible_poly ('X - x%:P). | Proof.
split=> [|d size_d d_dv_Xx]; first by rewrite size_XsubC.
have: ~ d %= 1 by apply/negP; rewrite -size_poly_eq1.
have [|m /=] := @dvdp_prod_XsubC _ [:: x] id d; first by rewrite big_seq1.
by case: m => [|[] [|_ _] /=]; rewrite (big_nil, big_seq1).
Qed. | Lemma | irredp_XsubC | 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",
"big_nil",
"big_seq1",
"dvdp_prod_XsubC",
"id",
"irreducible_poly",
"size_XsubC",
"size_poly_eq1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irredp_XaddC (x : R) : irreducible_poly ('X + x%:P). | Proof. by rewrite -[x]opprK rmorphN; apply: irredp_XsubC. Qed. | Lemma | irredp_XaddC | 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",
"irredp_XsubC",
"irreducible_poly",
"opprK",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irredp_XsubCP d p :
irreducible_poly p -> d %| p -> {d %= 1} + {d %= p}. | Proof.
move=> irred_p dvd_dp; have [] := boolP (_ %= 1); first by left.
by rewrite -size_poly_eq1=> /irred_p /(_ dvd_dp); right.
Qed. | Lemma | irredp_XsubCP | 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... | [
"irreducible_poly",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_exp_XsubCP (p : {poly R}) (c : R) (n : nat) :
reflect (exists2 k, (k <= n)%N & p %= ('X - c%:P) ^+ k)
(p %| ('X - c%:P) ^+ n). | Proof.
apply: (iffP idP) => [|[k lkn /eqp_dvdl->]]; last by rewrite dvdp_exp2l.
move=> /Pdiv.WeakIdomain.dvdpP[[/= a q] a_neq0].
have [m [r]] := multiplicity_XsubC p c; have [->|pN0]/= := eqVneq p 0.
rewrite mulr0 => _ _ /eqP; rewrite scale_poly_eq0 (negPf a_neq0)/=.
by rewrite expf_eq0/= andbC polyXsubC_eq0.
move... | Lemma | dvdp_exp_XsubCP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
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"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
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"coprimepP",
"coprimep_XsubC",
"coprimep_expr",
"dvdpP",
"dvdp_Pexp2l",
"dvdp_exp2l",
"dvdp_mull",
"dvdpp",
"eqVneq",
"eqp_dvdl",
"eqp_dvdr",
"eqp_mulr",
"eqp_scale",
"eqp_trans",
"eqpxx",
"expf_eq0",
"last",
"mul1r",
"mulr0",
"mulrA",
"mulrAC",
"multiplicity_Xsu... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
monq : q \is monic. | Hypothesis | monq | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
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"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"monic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
divpE p : p %/ q = rdivp p q. | Proof. by rewrite divpE (eqP monq) unitr1 expr1n invr1 scale1r. Qed. | Lemma | divpE | 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",
"invr1",
"monq",
"rdivp",
"scale1r",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpE p : p %% q = rmodp p q. | Proof. by rewrite modpE (eqP monq) unitr1 expr1n invr1 scale1r. Qed. | Lemma | modpE | 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",
"invr1",
"monq",
"rmodp",
"scale1r",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalpE p : scalp p q = 0. | Proof. by rewrite scalpE (eqP monq) unitr1. Qed. | Lemma | scalpE | 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... | [
"monq",
"scalp",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_eq p : p = (p %/ q) * q + (p %% q). | Proof. by rewrite -divp_eq (eqP monq) expr1n scale1r. Qed. | Lemma | divp_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",
"monq",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpp p : q %/ q = 1. | Proof. by rewrite divpp ?monic_neq0 // (eqP monq) expr1n. Qed. | Lemma | divpp | 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",
"monic_neq0",
"monq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_eq p : (q %| p) = (p == (p %/ q) * q). | Proof. by rewrite dvdp_eq (eqP monq) expr1n scale1r. Qed. | Lemma | dvdp_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",
"monq",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpP p : reflect (exists qq, p = qq * q) (q %| p). | Proof.
apply: (iffP idP); first by rewrite dvdp_eq; move/eqP=> e; exists (p %/ q).
by case=> qq ->; rewrite dvdp_mull // dvdpp.
Qed. | Lemma | dvdpP | 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",
"dvdp_eq",
"dvdp_mull",
"dvdpp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulpK p : p * q %/ q = p. | Proof. by rewrite mulpK ?monic_neq0 // (eqP monq) expr1n scale1r. Qed. | Lemma | mulpK | 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",
"monic_neq0",
"monq",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulKp p : q * p %/ q = p. | Proof. by rewrite mulrC mulpK. Qed. | Lemma | mulKp | 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... | [
"mulpK",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_poly_divp n p : drop_poly n p = p %/ 'X^n. | Proof. by rewrite RingMonic.drop_poly_rdivp divpE // monicXn. Qed. | Lemma | drop_poly_divp | 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... | [
"divpE",
"drop_poly",
"drop_poly_rdivp",
"monicXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_poly_modp n p : take_poly n p = p %% 'X^n. | Proof. by rewrite RingMonic.take_poly_rmodp modpE // monicXn. Qed. | Lemma | take_poly_modp | 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... | [
"modpE",
"monicXn",
"take_poly",
"take_poly_rmodp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ulcd : lead_coef d \in GRing.unit. | Hypothesis | ulcd | 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",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
divp_eq p : p = (p %/ d) * d + (p %% d). | Proof. by have := divp_eq p d; rewrite scalpE ulcd expr0 scale1r. Qed. | Lemma | divp_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... | [
"expr0",
"scale1r",
"scalpE",
"ulcd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
edivpP p q r : p = q * d + r -> size r < size d ->
q = (p %/ d) /\ r = p %% d. | Proof.
move=> ep srd; have := divp_eq p; rewrite [LHS]ep.
move/eqP; rewrite -subr_eq -addrA addrC eq_sym -subr_eq -mulrBl; move/eqP.
have lcdn0 : lead_coef d != 0 by apply: contraTneq ulcd => ->; rewrite unitr0.
have [-> /esym /eqP|abs] := eqVneq (p %/ d) q.
by rewrite subrr mul0r subr_eq0 => /eqP<-.
have hleq : size... | Lemma | edivpP | 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... | [
"addSn",
"addrA",
"addrC",
"apply",
"contraTneq",
"divp_eq",
"eqVneq",
"eq_sym",
"gtn_max",
"lead_coef",
"lead_coef_eq0",
"leq_addl",
"leq_ltn_trans",
"leq_trans",
"ltn_modp",
"ltnn",
"mul0r",
"mulf_eq0",
"mulrBl",
"polySpred",
"size",
"size_polyD",
"size_polyN",
"size_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpP p q r : p = q * d + r -> size r < size d -> q = (p %/ d). | Proof. by move/edivpP=> h; case/h. Qed. | Lemma | divpP | 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... | [
"edivpP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpP p q r : p = q * d + r -> size r < size d -> r = (p %% d). | Proof. by move/edivpP=> h; case/h. Qed. | Lemma | modpP | 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... | [
"edivpP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ulc_eqpP p q : lead_coef q \is a GRing.unit ->
reflect (exists2 c : R, c != 0 & p = c *: q) (p %= q). | Proof.
have [->|] := eqVneq (lead_coef q) 0; first by rewrite unitr0.
rewrite lead_coef_eq0 => nz_q ulcq; apply: (iffP idP).
have [->|nz_p] := eqVneq p 0; first by rewrite eqp_sym eqp0 (negPf nz_q).
move/eqp_eq=> eq; exists (lead_coef p / lead_coef q).
by rewrite mulf_neq0 // ?invr_eq0 lead_coef_eq0.
by apply... | Lemma | ulc_eqpP | 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",
"divrr",
"eqVneq",
"eqp0",
"eqpP",
"eqp_eq",
"eqp_sym",
"invr_eq0",
"lead_coef",
"lead_coef_eq0",
"mulf_neq0",
"mulr1",
"mulrCA",
"nz_p",
"oner_eq0",
"scale1r",
"scalerA",
"scaler_injl",
"unit",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_eq p : (d %| p) = (p == p %/ d * d). | Proof.
apply/eqP/eqP=> [modp0 | ->]; last exact: modp_mull.
by rewrite [p in LHS]divp_eq modp0 addr0.
Qed. | Lemma | dvdp_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... | [
"addr0",
"apply",
"divp_eq",
"last",
"modp0",
"modp_mull"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ucl_eqp_eq p q : lead_coef q \is a GRing.unit ->
p %= q -> p = (lead_coef p / lead_coef q) *: q. | Proof.
move=> ulcq /eqp_eq; move/(congr1 ( *:%R (lead_coef q)^-1 )).
by rewrite !scalerA mulrC divrr // scale1r mulrC.
Qed. | Lemma | ucl_eqp_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... | [
"divrr",
"eqp_eq",
"lead_coef",
"mulrC",
"scale1r",
"scalerA",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpZl c p : (c *: p) %% d = c *: (p %% d). | Proof.
have [-> | cn0] := eqVneq c 0; first by rewrite !scale0r mod0p.
have e : (c *: p) = (c *: (p %/ d)) * d + c *: (p %% d).
by rewrite -scalerAl -scalerDr -divp_eq.
suff s: size (c *: (p %% d)) < size d by case: (edivpP e s) => _ ->.
rewrite -mul_polyC; apply: leq_ltn_trans (size_polyMleq _ _) _.
rewrite size_pol... | Lemma | modpZl | 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... | [
"add0n",
"addSn",
"apply",
"contraTneq",
"divp_eq",
"edivpP",
"eqVneq",
"lead_coef_eq0",
"leq_ltn_trans",
"ltn_modp",
"mod0p",
"mul_polyC",
"scale0r",
"scalerAl",
"scalerDr",
"size",
"size_polyC",
"size_polyMleq",
"ulcd",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpZl c p : (c *: p) %/ d = c *: (p %/ d). | Proof.
have [-> | cn0] := eqVneq c 0; first by rewrite !scale0r div0p.
have e : (c *: p) = (c *: (p %/ d)) * d + c *: (p %% d).
by rewrite -scalerAl -scalerDr -divp_eq.
suff s: size (c *: (p %% d)) < size d by case: (edivpP e s) => ->.
rewrite -mul_polyC; apply: leq_ltn_trans (size_polyMleq _ _) _.
rewrite size_polyC... | Lemma | divpZl | 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... | [
"add0n",
"addSn",
"apply",
"contraTneq",
"div0p",
"divp_eq",
"edivpP",
"eqVneq",
"lead_coef_eq0",
"leq_ltn_trans",
"ltn_modp",
"mul_polyC",
"scale0r",
"scalerAl",
"scalerDr",
"size",
"size_polyC",
"size_polyMleq",
"ulcd",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_modpl p q : p %= q -> (p %% d) %= (q %% d). | Proof.
case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e].
by apply/eqpP; exists (c1, c2); rewrite ?c1n0 //= -!modpZl e.
Qed. | Lemma | eqp_modpl | 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",
"c1",
"c2",
"eqpP",
"modpZl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_divl p q : p %= q -> (p %/ d) %= (q %/ d). | Proof.
case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e].
by apply/eqpP; exists (c1, c2); rewrite ?c1n0 // -!divpZl e.
Qed. | Lemma | eqp_divl | 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",
"c1",
"c2",
"divpZl",
"eqpP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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