fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
sgr_numqx : sgz (numq x) = sgz x.
Proof.
apply/eqP; case: (sgzP x); rewrite sgz_cp0 ?(numq_gt0, numq_lt0) //.
by move->.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | sgr_numq | |
denq_mulr_sign(b : bool) x : denq ((-1) ^+ b * x) = denq x.
Proof. by case: b; rewrite ?(mul1r, mulN1r) // denqN. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | denq_mulr_sign | |
denq_normx : denq `|x| = denq x.
Proof. by rewrite normrEsign denq_mulr_sign. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | denq_norm | |
floorx : int := (numq x %/ denq x)%Z. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | floor | |
ceilx : int := - (- numq x %/ denq x)%Z. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ceil | |
truncnx : nat :=
if 0 <= x then (`|numq x| %/ `|denq x|)%N else 0%N.
Let is_int x := denq x == 1.
Let is_nat x := (0 <= x) && (denq x == 1).
Fact floorP x :
if x \is Num.real then (floor x)%:~R <= x < (floor x + 1)%:~R
else floor x == 0.
Proof.
rewrite num_real /floor; case: (ratP x) => n d _ {x}; rewrite ler_pdi... | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | truncn | |
Definition_ :=
Num.NumDomain_hasFloorCeilTruncn.Build rat
ratArchimedean.floorP ratArchimedean.ceilP ratArchimedean.truncnP
ratArchimedean.intrP ratArchimedean.natrP. | HB.instance | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | Definition | |
floorErat(x : rat) : Num.floor x = (numq x %/ denq x)%Z.
Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | floorErat | |
ceilErat(x : rat) : Num.ceil x = - (- numq x %/ denq x)%Z.
Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ceilErat | |
Qint_def(x : rat) : (x \is a Num.int) = (denq x == 1). Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | Qint_def | |
numqK: {in Num.int, cancel (fun x => numq x) intr}.
Proof. by move=> _ /intrP [x ->]; rewrite numq_int. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | numqK | |
natq_divm n : (n %| m)%N -> (m %/ n)%:R = m%:R / n%:R :> rat.
Proof. exact/pchar0_natf_div/pchar_num. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | natq_div | |
ratrx : R := (numq x)%:~R / (denq x)%:~R. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ratr | |
ratr_intz : ratr z%:~R = z%:~R.
Proof. by rewrite /ratr numq_int denq_int divr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ratr_int | |
ratr_natn : ratr n%:R = n%:R.
Proof. exact: ratr_int n. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ratr_nat | |
rpred_rat(S : divringClosed R) a : ratr a \in S.
Proof. by rewrite rpred_div ?rpred_int. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | rpred_rat | |
fmorph_rat(aR : fieldType) rR (f : {rmorphism aR -> rR}) a :
f (ratr _ a) = ratr _ a.
Proof. by rewrite fmorph_div !rmorph_int. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | fmorph_rat | |
fmorph_eq_ratrR (f : {rmorphism rat -> rR}) : f =1 ratr _.
Proof. by move=> a; rewrite -{1}[a]divq_num_den fmorph_div !rmorph_int. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | fmorph_eq_rat | |
rat_linearU V (f : U -> V) : zmod_morphism f -> scalable f.
Proof.
move=> fB a u.
pose aM := GRing.isZmodMorphism.Build U V f fB.
pose phi : {additive U -> V} := HB.pack f aM.
rewrite -[f]/(phi : _ -> _) -{2}[a]divq_num_den mulrC -scalerA.
apply: canRL (scalerK _) _; first by rewrite intr_eq0 denq_neq0.
rewrite 2!scale... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | rat_linear | |
ratr_is_additive:= ratr_is_zmod_morphism.
Fact ratr_is_monoid_morphism : monoid_morphism (@ratr F).
Proof.
have injZtoQ: @injective rat int intr by apply: intr_inj.
have nz_den x: (denq x)%:~R != 0 :> F by rewrite intr_eq0 denq_eq0.
split=> [|x y]; first by rewrite /ratr divr1.
rewrite /ratr mulrC mulrAC; apply: canLR ... | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ratr_is_additive | |
ratr_is_multiplicative:=
(fun g => (g.2,g.1)) ratr_is_monoid_morphism.
HB.instance Definition _ := GRing.isZmodMorphism.Build rat F (@ratr F)
ratr_is_zmod_morphism.
HB.instance Definition _ := GRing.isMonoidMorphism.Build rat F (@ratr F)
ratr_is_monoid_morphism. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ratr_is_multiplicative | |
ler_rat: {mono (@ratr F) : x y / x <= y}.
Proof.
move=> x y /=; case: (ratP x) => nx dx cndx; case: (ratP y) => ny dy cndy.
rewrite !fmorph_div /= !ratr_int !ler_pdivlMr ?ltr0z //.
by rewrite ![_ / _ * _]mulrAC !ler_pdivrMr ?ltr0z // -!rmorphM /= !ler_int.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ler_rat | |
ltr_rat: {mono (@ratr F) : x y / x < y}.
Proof. exact: leW_mono ler_rat. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ltr_rat | |
ler0qx : (0 <= ratr F x) = (0 <= x).
Proof. by rewrite (_ : 0 = ratr F 0) ?ler_rat ?rmorph0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ler0q | |
lerq0x : (ratr F x <= 0) = (x <= 0).
Proof. by rewrite (_ : 0 = ratr F 0) ?ler_rat ?rmorph0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | lerq0 | |
ltr0qx : (0 < ratr F x) = (0 < x).
Proof. by rewrite (_ : 0 = ratr F 0) ?ltr_rat ?rmorph0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ltr0q | |
ltrq0x : (ratr F x < 0) = (x < 0).
Proof. by rewrite (_ : 0 = ratr F 0) ?ltr_rat ?rmorph0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ltrq0 | |
ratr_sgx : ratr F (sgr x) = sgr (ratr F x).
Proof. by rewrite !sgr_def fmorph_eq0 ltrq0 rmorphMn /= rmorph_sign. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ratr_sg | |
ratr_normx : ratr F `|x| = `|ratr F x|.
Proof.
by rewrite {2}[x]numEsign rmorphMsign normrMsign [`|ratr F _|]ger0_norm ?ler0q.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ratr_norm | |
minr_rat: {morph ratr F : x y / Num.min x y}.
Proof. by move=> x y; rewrite !minEle ler_rat; case: leP. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | minr_rat | |
maxr_rat: {morph ratr F : x y / Num.max x y}.
Proof. by move=> x y; rewrite !maxEle ler_rat; case: leP. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | maxr_rat | |
floor_rat: {mono (@ratr F) : x / Num.floor x}.
Proof.
move=> x; apply: floor_def; apply/andP; split.
- by rewrite -ratr_int ler_rat floor_le.
- by rewrite -ratr_int ltr_rat floorD1_gt.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | floor_rat | |
ceil_rat: {mono (@ratr F) : x / Num.ceil x}.
Proof. by move=> x; rewrite !ceilNfloor -rmorphN floor_rat. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ceil_rat | |
Qint_dvdz(m d : int) : (d %| m)%Z -> (m%:~R / d%:~R : rat) \is a Num.int.
Proof.
case/dvdzP=> z ->; rewrite rmorphM /=; have [->|dn0] := eqVneq d 0.
by rewrite mulr0 mul0r.
by rewrite mulfK ?intr_eq0.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | Qint_dvdz | |
Qnat_dvd(m d : nat) : (d %| m)%N -> (m%:R / d%:R : rat) \is a Num.nat.
Proof. by move=> h; rewrite natrEint divr_ge0 ?ler0n // !pmulrn Qint_dvdz. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | Qnat_dvd | |
size_rat_int_polyp : size (pZtoQ p) = size p.
Proof. by apply: size_map_inj_poly; first apply: intr_inj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | size_rat_int_poly | |
rat_poly_scale(p : {poly rat}) :
{q : {poly int} & {a | a != 0 & p = a%:~R^-1 *: pZtoQ q}}.
Proof.
pose a := \prod_(i < size p) denq p`_i.
have nz_a: a != 0 by apply/prodf_neq0=> i _; apply: denq_neq0.
exists (map_poly numq (a%:~R *: p)), a => //.
apply: canRL (scalerK _) _; rewrite ?intr_eq0 //.
apply/polyP=> i; rew... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | rat_poly_scale | |
dvdp_rat_intp q : (pZtoQ p %| pZtoQ q) = (p %| q).
Proof.
apply/dvdpP/Pdiv.Idomain.dvdpP=> [[/= r1 Dq] | [[/= a r] nz_a Dq]]; last first.
exists (a%:~R^-1 *: pZtoQ r).
by rewrite -scalerAl -rmorphM -Dq /= linearZ/= scalerK ?intr_eq0.
have [r [a nz_a Dr1]] := rat_poly_scale r1; exists (a, r) => //=.
apply: (map_inj_... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | dvdp_rat_int | |
dvdpP_rat_intp q :
p %| pZtoQ q ->
{p1 : {poly int} & {a | a != 0 & p = a *: pZtoQ p1} & {r | q = p1 * r}}.
Proof.
have{p} [p [a nz_a ->]] := rat_poly_scale p.
rewrite dvdpZl ?invr_eq0 ?intr_eq0 // dvdp_rat_int => dv_p_q.
exists (zprimitive p); last exact: dvdpP_int.
have [-> | nz_p] := eqVneq p 0.
by exists 1;... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | dvdpP_rat_int | |
irreducible_rat_intp :
irreducible_poly (pZtoQ p) <-> irreducible_poly p.
Proof.
rewrite /irreducible_poly size_rat_int_poly; split=> -[] p1 p_irr; split=> //.
move=> q q1; rewrite /eqp -!dvdp_rat_int => rq.
by apply/p_irr => //; rewrite size_rat_int_poly.
move=> q + /dvdpP_rat_int [] r [] c c0 qE [] s sE.
rewrit... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | irreducible_rat_int | |
inIntSpan(V : zmodType) m (s : m.-tuple V) v :=
exists a : int ^ m, v = \sum_(i < m) s`_i *~ a i. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | inIntSpan | |
solve_Qint_span(vT : vectType rat) m (s : m.-tuple vT) v :
{b : int ^ m &
{p : seq (int ^ m) &
forall a : int ^ m,
v = \sum_(i < m) s`_i *~ a i <->
exists c : seq int, a = b + \sum_(i < size p) p`_i *~ c`_i}} +
(~ inIntSpan s v).
Proof.
have s_s (i : 'I_m): s`_i \in <<s>>%VS by rewrite memv_span ?memt_nth.
... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | solve_Qint_span | |
dec_Qint_span(vT : vectType rat) m (s : m.-tuple vT) v :
decidable (inIntSpan s v).
Proof.
have [[b [p aP]]|] := solve_Qint_span s v; last by right.
left; exists b; apply/(aP b); exists [::]; rewrite big1 ?addr0 // => i _.
by rewrite nth_nil mulr0z.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | dec_Qint_span | |
eisenstein_crit(p : nat) (q : {poly int}) : prime p -> (size q != 1)%N ->
~~ (p %| lead_coef q)%Z -> ~~ (p ^+ 2 %| q`_0)%Z ->
(forall i, (i < (size q).-1)%N -> p %| q`_i)%Z ->
irreducible_poly q.
Proof.
move=> p_prime qN1 Ndvd_pql Ndvd_pq0 dvd_pq.
apply/irreducible_rat_int.
have qN0 : q != 0 by rewrite -lead_coef... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | eisenstein_crit | |
rat_to_ring:=
rewrite -?[0%Q]/(0 : rat)%R -?[1%Q]/(1 : rat)%R
-?[(_ - _)%Q]/(_ - _ : rat)%R -?[(_ / _)%Q]/(_ / _ : rat)%R
-?[(_ + _)%Q]/(_ + _ : rat)%R -?[(_ * _)%Q]/(_ * _ : rat)%R
-?[(- _)%Q]/(- _ : rat)%R -?[(_ ^-1)%Q]/(_ ^-1 : rat)%R /=. | Ltac | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | rat_to_ring | |
ring_to_rat:=
rewrite -?[0%R]/0%Q -?[1%R]/1%Q
-?[(_ - _)%R]/(_ - _)%Q -?[(_ / _)%R]/(_ / _)%Q
-?[(_ + _)%R]/(_ + _)%Q -?[(_ * _)%R]/(_ * _)%Q
-?[(- _)%R]/(- _)%Q -?[(_ ^-1)%R]/(_ ^-1)%Q /=. | Ltac | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | ring_to_rat | |
rat_vm_computen (x : rat) : vm_compute_eq n%:Q x -> n%:Q = x.
Proof. exact. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import prime fintype finfun bigop order tuple ssralg",
"From mathcomp Require Import countalg div ssrnum ssrint archimedean poly zmodp",
"From mathcomp Require Import... | algebra/rat.v | rat_vm_compute | |
RecordisZmodQuotient T eqT (zeroT : T) (oppT : T -> T) (addT : T -> T -> T)
(Q : Type) of GRing.Zmodule Q & EqQuotient T eqT Q := {
pi_zeror : \pi_Q zeroT = 0;
pi_oppr : {morph \pi_Q : x / oppT x >-> - x};
pi_addr : {morph \pi_Q : x y / addT x y >-> x + y}
}.
#[short(type="zmodQuotType")]
HB.structure Definition ... | HB.mixin | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | Record | |
pi_zero_quot_morphzqT := PiMorph (@pi_zeror _ _ _ _ _ zqT). | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_zero_quot_morph | |
pi_opp_quot_morphzqT := PiMorph1 (@pi_oppr _ _ _ _ _ zqT). | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_opp_quot_morph | |
pi_add_quot_morphzqT := PiMorph2 (@pi_addr _ _ _ _ _ zqT). | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_add_quot_morph | |
pi_is_zmod_morphism: zmod_morphism \pi_Q.
Proof. by move=> x y /=; rewrite !piE. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `pi_is_monoid_morphism` instead")] | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_is_zmod_morphism | |
pi_is_additive:= pi_is_zmod_morphism.
HB.instance Definition _ := GRing.isZmodMorphism.Build V Q \pi_Q pi_is_zmod_morphism. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_is_additive | |
RecordisNzRingQuotient T eqT zeroT oppT
addT (oneT : T) (mulT : T -> T -> T) (Q : Type)
of ZmodQuotient T eqT zeroT oppT addT Q & GRing.NzRing Q:=
{
pi_oner : \pi_Q oneT = 1;
pi_mulr : {morph \pi_Q : x y / mulT x y >-> x * y}
}. | HB.mixin | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | Record | |
BuildT eqT zeroT oppT addT oneT mulT Q :=
(isNzRingQuotient.Build T eqT zeroT oppT addT oneT mulT Q) (only parsing). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | Build | |
isRingQuotientT eqT zeroT oppT addT oneT mulT Q :=
(isNzRingQuotient T eqT zeroT oppT addT oneT mulT Q) (only parsing).
#[short(type="nzRingQuotType")]
HB.structure Definition NzRingQuotient T eqT zeroT oppT addT oneT mulT :=
{Q of isNzRingQuotient T eqT zeroT oppT addT oneT mulT Q &
ZmodQuotient T eqT zeroT opp... | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | isRingQuotient | |
ringQuotType:= (nzRingQuotType) (only parsing). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | ringQuotType | |
pi_one_quot_morphrqT := PiMorph (@pi_oner _ _ _ _ _ _ _ rqT). | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_one_quot_morph | |
pi_mul_quot_morphrqT := PiMorph2 (@pi_mulr _ _ _ _ _ _ _ rqT). | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_mul_quot_morph | |
pi_is_monoid_morphism: monoid_morphism \pi_Q.
Proof. by split; do ?move=> x y /=; rewrite !piE. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `pi_is_monoid_morphism` instead")] | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_is_monoid_morphism | |
pi_is_multiplicative:=
(fun g => (g.2,g.1)) pi_is_monoid_morphism.
HB.instance Definition _ := GRing.isMonoidMorphism.Build R Q \pi_Q
pi_is_monoid_morphism. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_is_multiplicative | |
RecordisUnitRingQuotient T eqT zeroT oppT addT oneT mulT (unitT : pred T) (invT : T -> T)
(Q : Type) of NzRingQuotient T eqT zeroT oppT addT oneT mulT Q & GRing.UnitRing Q :=
{
pi_unitr : {mono \pi_Q : x / unitT x >-> x \in GRing.unit};
pi_invr : {morph \pi_Q : x / invT x >-> x^-1}
}.
#[short(type="unitRi... | HB.mixin | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | Record | |
pi_unit_quot_morphurqT := PiMono1 (@pi_unitr _ _ _ _ _ _ _ _ _ urqT). | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_unit_quot_morph | |
pi_inv_quot_morphurqT := PiMorph1 (@pi_invr _ _ _ _ _ _ _ _ _ urqT). | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_inv_quot_morph | |
proper_ideal(R : nzRingType) (S : {pred R}) : Prop :=
1 \notin S /\ forall a, {in S, forall u, a * u \in S}. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | proper_ideal | |
prime_idealr_closed(R : nzRingType) (S : {pred R}) : Prop :=
forall u v, u * v \in S -> (u \in S) || (v \in S). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | prime_idealr_closed | |
idealr_closed(R : nzRingType) (S : {pred R}) :=
[/\ 0 \in S, 1 \notin S & forall a, {in S &, forall u v, a * u + v \in S}]. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | idealr_closed | |
idealr_closed_nontrivialR S : @idealr_closed R S -> proper_ideal S.
Proof. by case=> S0 S1 hS; split => // a x xS; rewrite -[_ * _]addr0 hS. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | idealr_closed_nontrivial | |
idealr_closedBR S : @idealr_closed R S -> zmod_closed S.
Proof. by case=> S0 _ hS; split=> // x y xS yS; rewrite -mulN1r addrC hS. Qed.
HB.mixin Record isProperIdeal (R : nzRingType) (S : R -> bool) := {
proper_ideal_subproof : proper_ideal S
}.
#[short(type="proper_ideal")]
HB.structure Definition ProperIdeal R := {... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | idealr_closedB | |
idealr1: 1 \in I = false.
Proof. apply: negPf; exact: proper_ideal_subproof.1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | idealr1 | |
idealMra u : u \in I -> a * u \in I.
Proof. exact: proper_ideal_subproof.2. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | idealMr | |
idealr0: 0 \in I. Proof. exact: rpred0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | idealr0 | |
prime_idealrMu v : (u * v \in I) = (u \in I) || (v \in I).
Proof.
apply/idP/idP; last by case/orP => /idealMr hI; rewrite // mulrC.
exact: prime_idealr_closed_subproof.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | prime_idealrM | |
equiv(x y : R) := (x - y) \in I. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | equiv | |
equivEx y : (equiv x y) = (x - y \in I). Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | equivE | |
equiv_is_equiv: equiv_class_of equiv.
Proof.
split=> [x|x y|y x z]; rewrite !equivE ?subrr ?rpred0 //.
by rewrite -opprB rpredN.
by move=> *; rewrite -[x](addrNK y) -addrA rpredD.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | equiv_is_equiv | |
equiv_equiv:= EquivRelPack equiv_is_equiv. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | equiv_equiv | |
equiv_encModRel:= defaultEncModRel equiv. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | equiv_encModRel | |
quot:= {eq_quot equiv}.
#[export]
HB.instance Definition _ : EqQuotient R equiv quot := EqQuotient.on quot.
#[export]
HB.instance Definition _ := Choice.on quot. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | quot | |
idealrBEx y : (x - y) \in I = (x == y %[mod quot]).
Proof. by rewrite piE equivE. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | idealrBE | |
idealrDEx y : (x + y) \in I = (x == - y %[mod quot]).
Proof. by rewrite -idealrBE opprK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | idealrDE | |
zero: quot := lift_cst quot 0. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | zero | |
add:= lift_op2 quot +%R. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | add | |
opp:= lift_op1 quot -%R. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | opp | |
pi_zero_morph:= PiConst zero. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_zero_morph | |
pi_opp: {morph \pi : x / - x >-> opp x}.
Proof.
move=> x; unlock opp; apply/eqP; rewrite piE equivE.
by rewrite -opprD rpredN idealrDE opprK reprK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_opp | |
pi_opp_morph:= PiMorph1 pi_opp. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_opp_morph | |
pi_add: {morph \pi : x y / x + y >-> add x y}.
Proof.
move=> x y /=; unlock add; apply/eqP; rewrite piE equivE.
rewrite opprD addrAC addrA -addrA.
by rewrite rpredD // (idealrBE, idealrDE) ?pi_opp ?reprK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_add | |
pi_add_morph:= PiMorph2 pi_add. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_add_morph | |
addqA: associative add.
Proof. by move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK !piE addrA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | addqA | |
addqC: commutative add.
Proof. by move=> x y; rewrite -[x]reprK -[y]reprK !piE addrC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | addqC | |
add0q: left_id zero add.
Proof. by move=> x; rewrite -[x]reprK !piE add0r. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | add0q | |
addNq: left_inverse zero opp add.
Proof. by move=> x; rewrite -[x]reprK !piE addNr. Qed.
#[export]
HB.instance Definition _ := GRing.isZmodule.Build quot addqA addqC add0q addNq.
#[export]
HB.instance Definition _ := @isZmodQuotient.Build R equiv 0 -%R +%R quot
(lock _) pi_opp pi_add. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | addNq | |
one: {quot idealI} := lift_cst {quot idealI} 1. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | one | |
mul:= lift_op2 {quot idealI} *%R. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | mul | |
pi_one_morph:= PiConst one. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_one_morph | |
pi_mul: {morph \pi : x y / x * y >-> mul x y}.
Proof.
move=> x y; unlock mul; apply/eqP; rewrite piE equivE.
rewrite -[_ * _](addrNK (x * repr (\pi_{quot idealI} y))) -mulrBr.
rewrite -addrA -mulrBl rpredD //.
by rewrite idealMr // idealrDE opprK reprK.
by rewrite mulrC idealMr // idealrDE opprK reprK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_mul | |
pi_mul_morph:= PiMorph2 pi_mul. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | pi_mul_morph | |
mulqA: associative mul.
Proof. by move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK !piE mulrA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | mulqA | |
mulqC: commutative mul.
Proof. by move=> x y; rewrite -[x]reprK -[y]reprK !piE mulrC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq ssralg generic_quotient"
] | algebra/ring_quotient.v | mulqC |
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