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 |
|---|---|---|---|---|---|---|
normrEsgx : `|x| = sg x * x.
Proof. by case: sgrP; rewrite ?(mul0r, mul1r, mulN1r). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
normrEsg
| |
numEsgx : x = sg x * `|x|.
Proof. by case: sgrP; rewrite !(mul1r, mul0r, mulrNN). Qed.
#[deprecated(since="mathcomp 2.3.0", note="use `numEsg` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
numEsg
| |
mulr_sg_normx : sg x * `|x| = x. Proof. by rewrite -numEsg. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
mulr_sg_norm
| |
sgrMx y : sg (x * y) = sg x * sg y.
Proof.
rewrite !sgr_def mulr_lt0 andbA mulrnAr mulrnAl -mulrnA mulnb -negb_or mulf_eq0.
by case: (~~ _) => //; rewrite signr_addb.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sgrM
| |
sgrNx : sg (- x) = - sg x.
Proof. by rewrite -mulrN1 sgrM sgrN1 mulrN1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sgrN
| |
sgrXn x : sg (x ^+ n) = (sg x) ^+ n.
Proof. by elim: n => [|n IHn]; rewrite ?sgr1 // !exprS sgrM IHn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sgrX
| |
sgr_smulx y : sg (sg x * y) = sg x * sg y.
Proof. by rewrite sgrM sgr_id. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sgr_smul
| |
sgr_gt0x : (sg x > 0) = (x > 0).
Proof. by rewrite -[LHS]sgr_cp0 sgr_id sgr_cp0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sgr_gt0
| |
sgr_ge0x : (sgr x >= 0) = (x >= 0).
Proof. by rewrite !leNgt sgr_lt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sgr_ge0
| |
ler_normx : (x <= `|x|).
Proof. exact: real_ler_norm. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_norm
| |
ler_normlx y : (`|x| <= y) = (- y <= x <= y).
Proof. exact: real_ler_norml. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_norml
| |
ler_normlPx y : reflect ((- x <= y) * (x <= y)) (`|x| <= y).
Proof. exact: real_ler_normlP. Qed.
Arguments ler_normlP {x y}.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_normlP
| |
eqr_normlx y : (`|x| == y) = ((x == y) || (x == -y)) && (0 <= y).
Proof. exact: real_eqr_norml. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
eqr_norml
| |
eqr_norm2x y : (`|x| == `|y|) = (x == y) || (x == -y).
Proof. exact: real_eqr_norm2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
eqr_norm2
| |
ltr_normlx y : (`|x| < y) = (- y < x < y).
Proof. exact: real_ltr_norml. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_norml
| |
lter_norml:= (ler_norml, ltr_norml).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lter_norml
| |
ltr_normlPx y : reflect ((-x < y) * (x < y)) (`|x| < y).
Proof. exact: real_ltr_normlP. Qed.
Arguments ltr_normlP {x y}.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_normlP
| |
ltr_normlWx y : `|x| < y -> x < y. Proof. exact: real_ltr_normlW. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_normlW
| |
ltrNnormlWx y : `|x| < y -> - y < x. Proof. exact: real_ltrNnormlW. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrNnormlW
| |
ler_normlWx y : `|x| <= y -> x <= y. Proof. exact: real_ler_normlW. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_normlW
| |
lerNnormlWx y : `|x| <= y -> - y <= x. Proof. exact: real_lerNnormlW. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerNnormlW
| |
ler_normrx y : (x <= `|y|) = (x <= y) || (x <= - y).
Proof. exact: real_ler_normr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_normr
| |
ltr_normrx y : (x < `|y|) = (x < y) || (x < - y).
Proof. exact: real_ltr_normr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_normr
| |
lter_normr:= (ler_normr, ltr_normr).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lter_normr
| |
ler_distlx y e : (`|x - y| <= e) = (y - e <= x <= y + e).
Proof. exact: real_ler_distl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_distl
| |
ltr_distlx y e : (`|x - y| < e) = (y - e < x < y + e).
Proof. exact: real_ltr_distl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_distl
| |
lter_distl:= (ler_distl, ltr_distl).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lter_distl
| |
ltr_distlCx y e : (`|x - y| < e) = (x - e < y < x + e).
Proof. by rewrite distrC ltr_distl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_distlC
| |
ler_distlCx y e : (`|x - y| <= e) = (x - e <= y <= x + e).
Proof. by rewrite distrC ler_distl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_distlC
| |
lter_distlC:= (ler_distlC, ltr_distlC).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lter_distlC
| |
ltr_distlDrx y e : `|x - y| < e -> x < y + e.
Proof. exact: real_ltr_distlDr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_distlDr
| |
ler_distlDrx y e : `|x - y| <= e -> x <= y + e.
Proof. exact: real_ler_distlDr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_distlDr
| |
ltr_distlCDrx y e : `|x - y| < e -> y < x + e.
Proof. exact: real_ltr_distlCDr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_distlCDr
| |
ler_distlCDrx y e : `|x - y| <= e -> y <= x + e.
Proof. exact: real_ler_distlCDr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_distlCDr
| |
ltr_distlBlx y e : `|x - y| < e -> x - e < y.
Proof. exact: real_ltr_distlBl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_distlBl
| |
ler_distlBlx y e : `|x - y| <= e -> x - e <= y.
Proof. exact: real_ler_distlBl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_distlBl
| |
ltr_distlCBlx y e : `|x - y| < e -> y - e < x.
Proof. exact: real_ltr_distlCBl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr_distlCBl
| |
ler_distlCBlx y e : `|x - y| <= e -> y - e <= x.
Proof. exact: real_ler_distlCBl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_distlCBl
| |
exprn_even_ge0n x : ~~ odd n -> 0 <= x ^+ n.
Proof. by move=> even_n; rewrite real_exprn_even_ge0 ?num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
exprn_even_ge0
| |
exprn_even_gt0n x : ~~ odd n -> (0 < x ^+ n) = (n == 0)%N || (x != 0).
Proof. by move=> even_n; rewrite real_exprn_even_gt0 ?num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
exprn_even_gt0
| |
exprn_even_le0n x : ~~ odd n -> (x ^+ n <= 0) = (n != 0) && (x == 0).
Proof. by move=> even_n; rewrite real_exprn_even_le0 ?num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
exprn_even_le0
| |
exprn_even_lt0n x : ~~ odd n -> (x ^+ n < 0) = false.
Proof. by move=> even_n; rewrite real_exprn_even_lt0 ?num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
exprn_even_lt0
| |
exprn_odd_ge0n x : odd n -> (0 <= x ^+ n) = (0 <= x).
Proof. by move=> even_n; rewrite real_exprn_odd_ge0 ?num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
exprn_odd_ge0
| |
exprn_odd_gt0n x : odd n -> (0 < x ^+ n) = (0 < x).
Proof. by move=> even_n; rewrite real_exprn_odd_gt0 ?num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
exprn_odd_gt0
| |
exprn_odd_le0n x : odd n -> (x ^+ n <= 0) = (x <= 0).
Proof. by move=> even_n; rewrite real_exprn_odd_le0 ?num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
exprn_odd_le0
| |
exprn_odd_lt0n x : odd n -> (x ^+ n < 0) = (x < 0).
Proof. by move=> even_n; rewrite real_exprn_odd_lt0 ?num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
exprn_odd_lt0
| |
lteif_normlC x y :
(`|x| < y ?<= if C) = (- y < x ?<= if C) && (x < y ?<= if C).
Proof. by case: C; rewrite /= lter_norml. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lteif_norml
| |
lteif_normrC x y :
(x < `|y| ?<= if C) = (x < y ?<= if C) || (x < - y ?<= if C).
Proof. by case: C; rewrite /= lter_normr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lteif_normr
| |
lteif_distlC x y e :
(`|x - y| < e ?<= if C) = (y - e < x ?<= if C) && (x < y + e ?<= if C).
Proof. by case: C; rewrite /= lter_distl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lteif_distl
| |
sqr_ge0x : 0 <= x ^+ 2. Proof. by rewrite exprn_even_ge0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sqr_ge0
| |
sqr_norm_eq1x : (x ^+ 2 == 1) = (`|x| == 1).
Proof. by rewrite sqrf_eq1 eqr_norml ler01 andbT. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sqr_norm_eq1
| |
leif_mean_square_scaledx y :
x * y *+ 2 <= x ^+ 2 + y ^+ 2 ?= iff (x == y).
Proof. exact: real_leif_mean_square_scaled. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
leif_mean_square_scaled
| |
leif_AGM2_scaledx y : x * y *+ 4 <= (x + y) ^+ 2 ?= iff (x == y).
Proof. exact: real_leif_AGM2_scaled. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
leif_AGM2_scaled
| |
oppr_max: {morph -%R : x y / max x y >-> min x y : R}.
Proof. by move=> x y; apply: real_oppr_max. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
oppr_max
| |
oppr_min: {morph -%R : x y / min x y >-> max x y : R}.
Proof. by move=> x y; apply: real_oppr_min. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
oppr_min
| |
addr_minl: @left_distributive R R +%R min.
Proof. by move=> x y z; apply: real_addr_minl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
addr_minl
| |
addr_minr: @right_distributive R R +%R min.
Proof. by move=> x y z; apply: real_addr_minr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
addr_minr
| |
addr_maxl: @left_distributive R R +%R max.
Proof. by move=> x y z; apply: real_addr_maxl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
addr_maxl
| |
addr_maxr: @right_distributive R R +%R max.
Proof. by move=> x y z; apply: real_addr_maxr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
addr_maxr
| |
minr_nMrx y z : x <= 0 -> x * min y z = max (x * y) (x * z).
Proof. by move=> x_le0; apply: real_minr_nMr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
minr_nMr
| |
maxr_nMrx y z : x <= 0 -> x * max y z = min (x * y) (x * z).
Proof. by move=> x_le0; apply: real_maxr_nMr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
maxr_nMr
| |
minr_nMlx y z : x <= 0 -> min y z * x = max (y * x) (z * x).
Proof. by move=> x_le0; apply: real_minr_nMl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
minr_nMl
| |
maxr_nMlx y z : x <= 0 -> max y z * x = min (y * x) (z * x).
Proof. by move=> x_le0; apply: real_maxr_nMl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
maxr_nMl
| |
maxrNx : max x (- x) = `|x|. Proof. exact: real_maxrN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
maxrN
| |
maxNrx : max (- x) x = `|x|. Proof. exact: real_maxNr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
maxNr
| |
minrNx : min x (- x) = - `|x|. Proof. exact: real_minrN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
minrN
| |
minNrx : min (- x) x = - `|x|. Proof. exact: real_minNr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
minNr
| |
poly_itv_bounda b : {ub | forall x, a <= x <= b -> `|p.[x]| <= ub}.
Proof.
have [ub le_p_ub] := poly_disk_bound p (Num.max `|a| `|b|).
exists ub => x /andP[le_a_x le_x_b]; rewrite le_p_ub // le_max !ler_normr.
by have [_|_] := ler0P x; rewrite ?lerN2 ?le_a_x ?le_x_b orbT.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
poly_itv_bound
| |
monic_Cauchy_bound: p \is monic -> {b | forall x, x >= b -> p.[x] > 0}.
Proof.
move/monicP=> mon_p; pose n := (size p - 2)%N.
have [p_le1 | p_gt1] := leqP (size p) 1.
exists 0 => x _; rewrite (size1_polyC p_le1) hornerC.
by rewrite -[p`_0]lead_coefC -size1_polyC // mon_p ltr01.
pose lb := \sum_(j < n.+1) `|p`_j|; exists (lb + 1) => x le_ub_x.
have x_ge1: 1 <= x; last have x_gt0 := lt_le_trans ltr01 x_ge1.
by rewrite -(lerD2l lb) ler_wpDl ?sumr_ge0 // => j _.
rewrite horner_coef -(subnK p_gt1) -/n addnS big_ord_recr /= addn1.
rewrite [in p`__]subnSK // subn1 -lead_coefE mon_p mul1r -ltrBlDl sub0r.
apply: le_lt_trans (_ : lb * x ^+ n < _); last first.
by rewrite exprS ltr_pM2r ?exprn_gt0// -(ltrD2r 1) ltr_pwDr.
rewrite -sumrN mulr_suml ler_sum // => j _; apply: le_trans (ler_norm _) _.
rewrite normrN normrM ler_wpM2l // normrX.
by rewrite ger0_norm ?(ltW x_gt0) // ler_weXn2l ?leq_ord.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
monic_Cauchy_bound
| |
RecordIntegralDomain_isNumRing R of GRing.IntegralDomain R := {
Rle : rel R;
Rlt : rel R;
norm : R -> R;
normD : forall x y, Rle (norm (x + y)) (norm x + norm y);
addr_gt0 : forall x y, Rlt 0 x -> Rlt 0 y -> Rlt 0 (x + y);
norm_eq0 : forall x, norm x = 0 -> x = 0;
ger_total : forall x y, Rle 0 x -> Rle 0 y -> Rle x y || Rle y x;
normM : {morph norm : x y / x * y};
le_def : forall x y, (Rle x y) = (norm (y - x) == y - x);
lt_def : forall x y, (Rlt x y) = (y != x) && (Rle x y)
}.
HB.builders Context R of IntegralDomain_isNumRing R.
Local Notation "x <= y" := (Rle x y) : ring_scope.
Local Notation "x < y" := (Rlt x y) : ring_scope.
Local Notation "`| x |" := (norm x) : ring_scope.
|
HB.factory
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
Record
| |
ltrrx : x < x = false. Proof. by rewrite lt_def eqxx. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrr
| |
ge0_defx : (0 <= x) = (`|x| == x).
Proof. by rewrite le_def subr0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ge0_def
| |
subr_ge0x y : (0 <= x - y) = (y <= x).
Proof. by rewrite ge0_def -le_def. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
subr_ge0
| |
subr_gt0x y : (0 < y - x) = (x < y).
Proof. by rewrite !lt_def subr_eq0 subr_ge0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
subr_gt0
| |
lt_trans: transitive Rlt.
Proof.
move=> y x z le_xy le_yz.
by rewrite -subr_gt0 -(subrK y z) -addrA addr_gt0 // subr_gt0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lt_trans
| |
le01: 0 <= 1.
Proof.
have n1_nz: `|1| != 0 :> R by apply: contraNneq (@oner_neq0 R) => /norm_eq0->.
by rewrite ge0_def -(inj_eq (mulfI n1_nz)) -normM !mulr1.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
le01
| |
lt01: 0 < 1.
Proof. by rewrite lt_def oner_neq0 le01. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lt01
| |
ltWx y : x < y -> x <= y. Proof. by rewrite lt_def => /andP[]. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltW
| |
lerrx : x <= x.
Proof.
have n2: `|2| == 2 :> R by rewrite -ge0_def ltW ?addr_gt0 ?lt01.
rewrite le_def subrr -(inj_eq (addrI `|0|)) addr0 -mulr2n -mulr_natr.
by rewrite -(eqP n2) -normM mul0r.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerr
| |
le_def'x y : (x <= y) = (x == y) || (x < y).
Proof. by rewrite lt_def; case: eqVneq => //= ->; rewrite lerr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
le_def'
| |
le_trans: transitive Rle.
by move=> y x z; rewrite !le_def' => /predU1P [->|hxy] // /predU1P [<-|hyz];
rewrite ?hxy ?(lt_trans hxy hyz) orbT.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
le_trans
| |
normrMnx n : `|x *+ n| = `|x| *+ n.
Proof.
rewrite -mulr_natr -[RHS]mulr_natr normM.
congr (_ * _); apply/eqP; rewrite -ge0_def.
elim: n => [|n ih]; [exact: lerr | apply: (le_trans ih)].
by rewrite le_def -natrB // subSnn -[_%:R]subr0 -le_def mulr1n le01.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
normrMn
| |
normrN1: `|-1| = 1 :> R.
Proof.
have: `|-1| ^+ 2 == 1 :> R
by rewrite expr2 /= -normM mulrNN mul1r -[1]subr0 -le_def le01.
rewrite sqrf_eq1 => /predU1P [] //; rewrite -[-1]subr0 -le_def.
have ->: 0 <= -1 = (-1 == 0 :> R) || (0 < -1)
by rewrite lt_def; case: eqP => // ->; rewrite lerr.
by rewrite oppr_eq0 oner_eq0 => /(addr_gt0 lt01); rewrite subrr ltrr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
normrN1
| |
normrNx : `|- x| = `|x|.
Proof. by rewrite -mulN1r normM -[RHS]mul1r normrN1. Qed.
HB.instance Definition _ :=
Order.LtLe_isPOrder.Build ring_display R le_def' ltrr lt_trans.
HB.instance Definition _ :=
Zmodule_isNormed.Build _ R normD norm_eq0 normrMn normrN.
HB.instance Definition _ :=
isNumRing.Build R addr_gt0 ger_total normM le_def.
HB.end.
HB.factory Record NumDomain_isReal R of NumDomain R := {
real : real_axiom R
}.
HB.builders Context R of NumDomain_isReal R.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
normrN
| |
le_total: Order.POrder_isTotal ring_display R.
Proof.
constructor=> x y; move: (real (x - y)).
by rewrite unfold_in /= !ler_def subr0 add0r opprB orbC.
Qed.
HB.instance Definition _ := le_total.
HB.end.
HB.factory Record IntegralDomain_isLeReal R of GRing.IntegralDomain R := {
Rle : rel R;
Rlt : rel R;
norm : R -> R;
le0_add : forall x y, Rle 0 x -> Rle 0 y -> Rle 0 (x + y);
le0_mul : forall x y, Rle 0 x -> Rle 0 y -> Rle 0 (x * y);
le0_anti : forall x, Rle 0 x -> Rle x 0 -> x = 0;
sub_ge0 : forall x y, Rle 0 (y - x) = Rle x y;
le0_total : forall x, Rle 0 x || Rle x 0;
normN : forall x, norm (- x) = norm x;
ge0_norm : forall x, Rle 0 x -> norm x = x;
lt_def : forall x y, Rlt x y = (y != x) && Rle x y;
}.
HB.builders Context R of IntegralDomain_isLeReal R.
Local Notation le := Rle.
Local Notation lt := Rlt.
Local Notation "x <= y" := (le x y) : ring_scope.
Local Notation "x < y" := (lt x y) : ring_scope.
Local Notation "`| x |" := (norm x) : ring_scope.
Let le0N x : (0 <= - x) = (x <= 0). Proof. by rewrite -sub0r sub_ge0. Qed.
Let leN_total x : 0 <= x \/ 0 <= - x.
Proof. by apply/orP; rewrite le0N le0_total. Qed.
Let le00 : 0 <= 0. Proof. by have:= le0_total 0; rewrite orbb. Qed.
Fact lt0_add x y : 0 < x -> 0 < y -> 0 < x + y.
Proof.
rewrite !lt_def => /andP [x_neq0 l0x] /andP [y_neq0 l0y]; rewrite le0_add //.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
le_total
| |
DefinitionNumField := { R of GRing.UnitRing_isField R &
GRing.IntegralDomain R &
POrderedZmodule R &
NormedZmodule (POrderedZmodule.clone R _) R &
isNumRing R }.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
Definition
| |
RecordNumField_isImaginary R of NumField R := {
imaginary : R;
conj_subdef : {rmorphism R -> R};
sqrCi : imaginary ^+ 2 = - 1;
normCK_subdef : forall x, `|x| ^+ 2 = x * conj_subdef x;
}.
#[short(type="numClosedFieldType")]
HB.structure Definition ClosedField :=
{ R of NumField_isImaginary R & GRing.ClosedField R & NumField R }.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
Record
| |
conj{C : numClosedFieldType} : C -> C := @conj_subdef C.
#[export] HB.instance Definition _ C := GRing.RMorphism.on (@conj C).
#[deprecated(since="mathcomp 2.5.0",note="Use conj instead.")]
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
conj
| |
conj_op:= conj (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
conj_op
| |
DefinitionRealField :=
{ R of Order.Total ring_display R & NumField R }.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
Definition
| |
RecordRealField_isClosed R of RealField R := {
poly_ivt_subproof : real_closed_axiom R
}.
#[short(type="rcfType")]
HB.structure Definition RealClosedField :=
{ R of RealField_isClosed R & RealField R }.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
Record
| |
poly_ivt: real_closed_axiom R. Proof. exact: poly_ivt_subproof. Qed.
Fact sqrtr_subproof (x : R) :
exists2 y, 0 <= y & (if 0 <= x then y ^+ 2 == x else y == 0) : bool.
Proof.
case x_ge0: (0 <= x); last by exists 0.
have le0x1: 0 <= x + 1 by rewrite -nnegrE rpredD ?rpred1.
have [|y /andP[y_ge0 _]] := @poly_ivt ('X^2 - x%:P) _ _ le0x1.
rewrite !hornerE -subr_ge0 add0r expr0n sub0r opprK x_ge0 sqrrD mulr1.
by rewrite addrAC !addrA addrK -nnegrE !rpredD ?rpredX ?rpred1.
by rewrite rootE !hornerE subr_eq0; exists y.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
poly_ivt
| |
conjC:= conj.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
conjC
| |
sqrtr{R} x := s2val (sig2W (@sqrtr_subproof R x)).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrtr
| |
sqrt:= sqrtr.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
sqrt
| |
unitf_gt0x : 0 < x -> x \is a GRing.unit.
Proof. by move=> hx; rewrite unitfE eq_sym lt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
unitf_gt0
| |
unitf_lt0x : x < 0 -> x \is a GRing.unit.
Proof. by move=> hx; rewrite unitfE lt_eqF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
unitf_lt0
| |
lef_pV2: {in pos &, {mono (@GRing.inv F) : x y /~ x <= y}}.
Proof. by move=> x y hx hy /=; rewrite ler_pV2 ?inE ?unitf_gt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lef_pV2
| |
lef_nV2: {in neg &, {mono (@GRing.inv F) : x y /~ x <= y}}.
Proof. by move=> x y hx hy /=; rewrite ler_nV2 ?inE ?unitf_lt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
lef_nV2
| |
ltf_pV2: {in pos &, {mono (@GRing.inv F) : x y /~ x < y}}.
Proof. exact: leW_nmono_in lef_pV2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod numdomain"
] |
algebra/num_theory/numfield.v
|
ltf_pV2
|
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