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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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_pdivlMr//.
by rewrite ltr_pdivrMr// -!intrM ler_int ltr_int lez_floor ?ltz_ceil.
Qed.
Fact ceilP x : ceil x = - floor (- x).
Proof. by rewrite /ceil /floor numqN denqN. Qed.
Fact truncnP x : truncn x = if floor x is Posz n then n else 0.
Proof.
rewrite /truncn /floor; case: (ratP x) => n d _ {x} /=.
by rewrite !ler_pdivlMr// mul0r; case: n => n; rewrite ler0z//= mul1n.
Qed.
Fact intrP x : reflect (exists n, x = n%:~R) (is_int x).
Proof.
apply: (iffP idP) => [/eqP d1 | [i ->]]; [|by rewrite /is_int denq_int].
by exists (numq x); case: (ratP x) d1 => n d _ ->; rewrite divr1.
Qed.
Fact natrP x : reflect (exists n, x = n%:R) (is_nat x).
Proof.
apply: (iffP idP) => [/andP[]/[swap]/intrP[i ->]|[n ->]].
by rewrite ler0z; case: i => [n _|//]; exists n.
by rewrite /is_nat pmulrn ler0z denq_int.
Qed.
|
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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!scaler_int -3!raddfMz /=.
by rewrite -scalerMzr scalerMzl -mulrzr -numqE scaler_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 polydiv intdiv matrix mxalgebra vector"
] |
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 (mulKf (nz_den _)) _; rewrite !mulrA.
do 2!apply: canRL (mulfK (nz_den _)) _; rewrite -!rmorphM; congr _%:~R.
apply: injZtoQ; rewrite !rmorphM [x * y]lock /= !numqE -lock.
by rewrite -!mulrA mulrA mulrCA -!mulrA (mulrCA y).
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `ratr_is_monoid_morphism` instead")]
|
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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; rewrite !(coefZ, coef_map_id0) // numqK // Qint_def mulrC.
have [ltip | /(nth_default 0)->] := ltnP i (size p); last by rewrite mul0r.
by rewrite [a](bigD1 (Ordinal ltip)) // rmorphM mulrA -numqE -rmorphM denq_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 polydiv intdiv matrix mxalgebra vector"
] |
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_poly _ _ : injective pZtoQ) => //; first exact: intr_inj.
by rewrite linearZ /= Dq Dr1 -scalerAl -rmorphM scalerKV ?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 polydiv intdiv matrix mxalgebra vector"
] |
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; rewrite ?oner_eq0 // zprimitive0 map_poly0 !scaler0.
exists ((zcontents p)%:~R / a%:~R).
by rewrite mulf_neq0 ?invr_eq0 ?intr_eq0 ?zcontents_eq0.
by rewrite mulrC -scalerA -map_polyZ -zpolyEprim.
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 polydiv intdiv matrix mxalgebra vector"
] |
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.
rewrite qE size_scale// size_rat_int_poly => r1.
apply/(eqp_trans (eqp_scale _ c0)).
rewrite /eqp !dvdp_rat_int; apply/p_irr => //.
by rewrite sE dvdp_mulIl.
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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.
have s_Zs a: \sum_(i < m) s`_i *~ a i \in <<s>>%VS.
by apply/rpred_sum => i _; apply/rpredMz.
case s_v: (v \in <<s>>%VS); last by right=> [[a Dv]]; rewrite Dv s_Zs in s_v.
move SE : (\matrix_(i < m, j < _) coord (vbasis <<s>>) j s`_i) => S.
move rE : (\rank S) => r; move kE : (m - r)%N => k.
have Dm: (m = k + r)%N by rewrite -kE -rE subnK ?rank_leq_row.
rewrite Dm in s s_s s_Zs s_v S SE rE kE *.
move=> {Dm m}; pose m := (k + r)%N.
have [K kerK]: {K : 'M_(k, m) | map_mx intr K == kermx S}%MS.
move: (mxrank_ker S); rewrite rE kE => krk.
pose B := row_base (kermx S); pose d := \prod_ij denq (B ij.1 ij.2).
exists (castmx (krk, erefl m) (map_mx numq (intr d *: B))).
rewrite map_castmx !eqmx_cast -map_mx_comp map_mx_id_in => [|i j]; last first.
rewrite mxE mulrC [d](bigD1 (i, j)) //= rmorphM mulrA.
by rewrite -numqE -rmorphM numq_int.
suff nz_d: d%:Q != 0 by rewrite !eqmx_scale // !eq_row_base andbb.
by rewrite intr_eq0; apply/prodf_neq0 => i _; apply: denq_neq0.
have [L _ [G uG [D _ defK]]] := int_Smith_normal_form K.
have {K L D defK kerK} [kerGu kerS_sub_Gu]: map_mx intr (usubmx G) *m S = 0 /\
(kermx S <= map_mx intr (usubmx G))%MS.
pose Kl : 'M[rat]_k
...
|
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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_eq0; apply: contraNneq Ndvd_pql => ->.
split.
rewrite size_map_poly_id0 ?intr_eq0 ?lead_coef_eq0//.
by rewrite ltn_neqAle eq_sym qN1 size_poly_gt0.
move=> f' +/dvdpP_rat_int[f [d dN0 feq]]; rewrite {f'}feq size_scale// => fN1.
move=> /= [g q_eq]; rewrite q_eq (eqp_trans (eqp_scale _ _))//.
have fN0 : f != 0 by apply: contra_neq qN0; rewrite q_eq => ->; rewrite mul0r.
have gN0 : g != 0 by apply: contra_neq qN0; rewrite q_eq => ->; rewrite mulr0.
rewrite size_map_poly_id0 ?intr_eq0 ?lead_coef_eq0// in fN1.
have [/eqP/size_poly1P[c cN0 ->]|gN1] := eqVneq (size g) 1%N.
by rewrite mulrC mul_polyC map_polyZ/= eqp_sym eqp_scale// intr_eq0.
have c_neq0 : (lead_coef q)%:~R != 0 :> 'F_p
by rewrite -(dvdz_pcharf (pchar_Fp _)).
have : map_poly (intr : int -> 'F_p) q = (lead_coef q)%:~R *: 'X^((size q).-1).
apply/val_inj/(@eq_from_nth _ 0) => [|i]; rewrite size_map_poly_id0//.
by rewrite size_scale// size_polyXn -polySpred.
move=> i_small; rewrite coef_poly i_small coefZ coefXn lead_coefE.
move: i_small; rewrite polySpred// ltnS/=.
case: ltngtP => // [i_lt|->]; rewrite (mulr1, mulr0)//= => _.
by apply/eqP; rewrite -(dvdz_pcharf (pchar_Fp _))// dvd_pq.
rewrite [in LH
...
|
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 polydiv intdiv matrix mxalgebra vector"
] |
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 ZmodQuotient T eqT zeroT oppT addT :=
{Q of isZmodQuotient T eqT zeroT oppT addT Q &
GRing.Zmodule Q & EqQuotient T eqT Q}.
|
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 oppT addT Q & GRing.NzRing Q }.
#[deprecated(since="mathcomp 2.4.0",
note="Use nzRingQuotType instead.")]
|
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="unitRingQuotType")]
HB.structure Definition UnitRingQuotient T eqT zeroT oppT addT oneT mulT unitT invT :=
{Q of isUnitRingQuotient T eqT zeroT oppT addT oneT mulT unitT invT Q & GRing.UnitRing Q & isQuotient T Q & isEqQuotient T eqT Q & isZmodQuotient T eqT zeroT oppT addT Q & isNzRingQuotient T eqT zeroT oppT addT oneT mulT Q}.
|
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 := {S of isProperIdeal R S}.
#[short(type="idealr")]
HB.structure Definition Idealr (R : nzRingType) :=
{S of GRing.ZmodClosed R S & ProperIdeal R S}.
HB.mixin Record isPrimeIdealrClosed (R : nzRingType) (S : R -> bool) := {
prime_idealr_closed_subproof : prime_idealr_closed S
}.
#[short(type="prime_idealr")]
HB.structure Definition PrimeIdealr (R : nzRingType) :=
{S of Idealr R S & isPrimeIdealrClosed R S}.
HB.factory Record isIdealr (R : nzRingType) (S : R -> bool) := {
idealr_closed_subproof : idealr_closed S
}.
HB.builders Context R S of isIdealr R S.
HB.instance Definition _ := GRing.isZmodClosed.Build R S
(idealr_closedB idealr_closed_subproof).
HB.instance Definition _ := isProperIdeal.Build R S
(idealr_closed_nontrivial idealr_closed_subproof).
HB.end.
|
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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