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 |
|---|---|---|---|---|---|---|
oppr_ge0x : (0 <= - x) = (x <= 0).
Proof. by rewrite -sub0r 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 | oppr_ge0 | |
ler01: 0 <= 1 :> R.
Proof.
have n1_nz: `|1 : R| != 0 by apply: contraNneq (@oner_neq0 R) => /normr0_eq0->.
by rewrite ger0_def -(inj_eq (mulfI n1_nz)) -normrM !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 | ler01 | |
ltr01: 0 < 1 :> R. Proof. by rewrite lt_def oner_neq0 ler01. 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 | ltr01 | |
le0rx : (0 <= x) = (x == 0) || (0 < x).
Proof. by rewrite le_eqVlt eq_sym. 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 | le0r | |
addr_ge0x y : 0 <= x -> 0 <= y -> 0 <= x + y.
Proof.
rewrite le0r; case/predU1P=> [-> | x_pos]; rewrite ?add0r // le0r.
by case/predU1P=> [-> | y_pos]; rewrite ltW ?addr0 ?addr_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 | addr_ge0 | |
pmulr_rgt0x y : 0 < x -> (0 < x * y) = (0 < y).
Proof.
rewrite !lt_def !ger0_def normrM mulf_eq0 negb_or => /andP[x_neq0 /eqP->].
by rewrite x_neq0 (inj_eq (mulfI x_neq0)).
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 | pmulr_rgt0 | |
posrEx : (x \is pos) = (0 < x). 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 ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] | algebra/num_theory/numdomain.v | posrE | |
nnegrEx : (x \is nneg) = (0 <= x). 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 ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] | algebra/num_theory/numdomain.v | nnegrE | |
realEx : (x \is real) = (0 <= x) || (x <= 0). Proof. by []. Qed.
Fact pos_divr_closed : divr_closed (@pos R).
Proof.
split=> [|x y x_gt0 y_gt0]; rewrite posrE ?ltr01 //.
have [Uy|/invr_out->] := boolP (y \is a GRing.unit); last by rewrite pmulr_rgt0.
by rewrite -(pmulr_rgt0 _ y_gt0) mulrC divrK.
Qed.
#[export]
HB.instance Definition _ := GRing.isDivClosed.Build R pos_num_pred
pos_divr_closed.
Fact nneg_divr_closed : divr_closed (@nneg R).
Proof.
split=> [|x y]; rewrite !nnegrE ?ler01 ?le0r // -!posrE.
case/predU1P=> [-> _ | x_gt0]; first by rewrite mul0r eqxx.
by case/predU1P=> [-> | y_gt0]; rewrite ?invr0 ?mulr0 ?eqxx // orbC rpred_div.
Qed.
#[export]
HB.instance Definition _ := GRing.isDivClosed.Build R nneg_num_pred
nneg_divr_closed.
Fact nneg_addr_closed : addr_closed (@nneg R).
Proof. by split; [apply: lexx | apply: addr_ge0]. Qed.
#[export]
HB.instance Definition _ := GRing.isAddClosed.Build R nneg_num_pred
nneg_addr_closed.
Fact real_oppr_closed : oppr_closed (@real R).
Proof. by move=> x; rewrite /= !realE oppr_ge0 orbC -!oppr_ge0 opprK. Qed.
#[export]
HB.instance Definition _ := GRing.isOppClosed.Build R real_num_pred
real_oppr_closed.
Fact real_addr_closed : addr_closed (@real R).
Proof.
split=> [|x y Rx Ry]; first by rewrite realE lexx.
without loss{Rx} x_ge0: x y Ry / 0 <= x.
case/orP: Rx => [? | x_le0]; first exact.
by rewrite -rpredN opprD; apply; rewrite ?rpredN ?oppr_ge0. | 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 | realE | |
num_real(R : realDomainType) (x : R) : x \is real.
Proof. exact: le_total. 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 | num_real | |
ler_normDV (x y : V) : `|x + y| <= `|x| + `|y| :=
ler_normD x y. | 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 | ler_normD | |
addr_gt0x y : 0 < x -> 0 < y -> 0 < x + y := @addr_gt0 R x y. | 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 | addr_gt0 | |
normr0_eq0W (x : W) : `|x| = 0 -> x = 0 := @normr0_eq0 R W 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"
] | algebra/num_theory/numdomain.v | normr0_eq0 | |
ger_leVgex y : 0 <= x -> 0 <= y -> (x <= y) || (y <= x) :=
@ger_leVge R x y. | 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 | ger_leVge | |
normrM: {morph norm : x y / (x : R) * y} := @normrM R. | 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 | normrM | |
ler_defx y : (x <= y) = (`|y - x| == y - x) := ler_def x y. | 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 | ler_def | |
normrMnV (x : V) n : `|x *+ n| = `|x| *+ n := normrMn x n. | 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 | normrMn | |
normrNV (x : V) : `|- x| = `|x| := normrN 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"
] | algebra/num_theory/numdomain.v | normrN | |
posrEx : (x \is pos) = (0 < x). 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 ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] | algebra/num_theory/numdomain.v | posrE | |
negrEx : (x \is neg) = (x < 0). 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 ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] | algebra/num_theory/numdomain.v | negrE | |
nnegrEx : (x \is nneg) = (0 <= x). 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 ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] | algebra/num_theory/numdomain.v | nnegrE | |
nposrEx : (x \is npos) = (x <= 0). 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 ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] | algebra/num_theory/numdomain.v | nposrE | |
realEx : (x \is real) = (0 <= x) || (x <= 0). 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 ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] | algebra/num_theory/numdomain.v | realE | |
lt0rx : (0 < x) = (x != 0) && (0 <= x). Proof. exact: lt_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 | lt0r | |
le0rx : (0 <= x) = (x == 0) || (0 < x). Proof. exact: le0r. 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 | le0r | |
lt0r_neq0(x : R) : 0 < x -> x != 0. Proof. by move=> /gt_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"
] | algebra/num_theory/numdomain.v | lt0r_neq0 | |
ltr0_neq0(x : R) : x < 0 -> x != 0. Proof. by move=> /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"
] | algebra/num_theory/numdomain.v | ltr0_neq0 | |
pmulr_rgt0x y : 0 < x -> (0 < x * y) = (0 < y).
Proof. exact: pmulr_rgt0. 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 | pmulr_rgt0 | |
pmulr_rge0x y : 0 < x -> (0 <= x * y) = (0 <= y).
Proof. by move=> x_gt0; rewrite !le0r mulf_eq0 pmulr_rgt0 // gt_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"
] | algebra/num_theory/numdomain.v | pmulr_rge0 | |
ler01: 0 <= 1 :> R. Proof. exact: ler01. 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 | ler01 | |
ltr01: 0 < 1 :> R. Proof. exact: ltr01. 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 | ltr01 | |
ler0nn : 0 <= n%:R :> R. Proof. by rewrite -nnegrE rpred_nat. Qed.
Hint Extern 0 (is_true (@Order.le ring_display _ _ _)) =>
(apply: ler01) : core.
Hint Extern 0 (is_true (@Order.lt ring_display _ _ _)) =>
(apply: ltr01) : core.
Hint Extern 0 (is_true (@Order.le ring_display _ _ _)) =>
(apply: ler0n) : core. | 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 | ler0n | |
ltr0Snn : 0 < n.+1%:R :> R.
Proof. by elim: n => // n; apply: addr_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 | ltr0Sn | |
ltr0nn : (0 < n%:R :> R) = (0 < n)%N.
Proof. by case: n => //= n; apply: ltr0Sn. Qed.
Hint Extern 0 (is_true (@Order.lt ring_display _ _ _)) =>
(apply: ltr0Sn) : core. | 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 | ltr0n | |
pnatr_eq0n : (n%:R == 0 :> R) = (n == 0)%N.
Proof. by case: n => [|n]; rewrite ?mulr0n ?eqxx // gt_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"
] | algebra/num_theory/numdomain.v | pnatr_eq0 | |
pchar_num: [pchar R] =i pred0.
Proof. by case=> // p /=; rewrite !inE pnatr_eq0 andbF. 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 | pchar_num | |
ger0_defx : (0 <= x) = (`|x| == x). Proof. exact: ger0_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 | ger0_def | |
normr_idP{x} : reflect (`|x| = x) (0 <= x).
Proof. by rewrite ger0_def; apply: eqP. 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 | normr_idP | |
ger0_normx : 0 <= x -> `|x| = x. Proof. exact: normr_idP. 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 | ger0_norm | |
normr1: `|1 : R| = 1. Proof. exact: ger0_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 | normr1 | |
normr_natn : `|n%:R : R| = n%:R. Proof. exact: ger0_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 | normr_nat | |
normr_prodI r (P : pred I) (F : I -> R) :
`|\prod_(i <- r | P i) F i| = \prod_(i <- r | P i) `|F i|.
Proof. exact: (big_morph norm normrM normr1). 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 | normr_prod | |
normrXn x : `|x ^+ n| = `|x| ^+ n.
Proof. by rewrite -(card_ord n) -!prodr_const normr_prod. 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 | normrX | |
normr_unit: {homo (@norm _ R) : x / x \is a GRing.unit}.
Proof.
move=> x /= /unitrP [y [yx xy]]; apply/unitrP; exists `|y|.
by rewrite -!normrM xy yx normr1.
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 | normr_unit | |
normrV: {in GRing.unit, {morph (@norm _ R) : x / x ^-1}}.
Proof.
move=> x ux; apply: (mulrI (normr_unit ux)).
by rewrite -normrM !divrr ?normr1 ?normr_unit.
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 | normrV | |
normrN1: `|-1 : R| = 1.
Proof.
have: `|-1 : R| ^+ 2 == 1 by rewrite -normrX -signr_odd normr1.
rewrite sqrf_eq1 => /orP[/eqP //|]; rewrite -ger0_def le0r oppr_eq0 oner_eq0.
by move/(addr_gt0 ltr01); rewrite subrr ltxx.
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 | |
big_realx0 op I (P : pred I) F (s : seq I) :
{in real &, forall x y, op x y \is real} -> x0 \is real ->
{in P, forall i, F i \is real} -> \big[op/x0]_(i <- s | P i) F i \is real.
Proof. exact: comparable_bigr. 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 | big_real | |
sum_realI (P : pred I) (F : I -> R) (s : seq I) :
{in P, forall i, F i \is real} -> \sum_(i <- s | P i) F i \is real.
Proof. by apply/big_real; [apply: rpredD | apply: rpred0]. 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 | sum_real | |
prod_realI (P : pred I) (F : I -> R) (s : seq I) :
{in P, forall i, F i \is real} -> \prod_(i <- s | P i) F i \is real.
Proof. by apply/big_real; [apply: rpredM | apply: rpred1]. 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 | prod_real | |
normr0: `|0 : V| = 0.
Proof. by rewrite -(mulr0n 0) normrMn mulr0n. 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 | normr0 | |
distrCv w : `|v - w| = `|w - v|.
Proof. by rewrite -opprB normrN. 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 | distrC | |
normr_idv : `| `|v| | = `|v|.
Proof.
have nz2: 2 != 0 :> R by rewrite pnatr_eq0.
apply: (mulfI nz2); rewrite -{1}normr_nat -normrM mulr_natl mulr2n ger0_norm //.
by rewrite -{2}normrN -normr0 -(subrr v) ler_normD.
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 | normr_id | |
normr_ge0v : 0 <= `|v|. Proof. by rewrite ger0_def normr_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 | normr_ge0 | |
normr_lt0v : `|v| < 0 = false.
Proof. by rewrite le_gtF// normr_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 | normr_lt0 | |
gtr0_norm_neq0v : `|v| > 0 -> (v != 0).
Proof. by apply: contra_ltN => /eqP->; rewrite normr0. 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 | gtr0_norm_neq0 | |
gtr0_norm_eq0Fv : `|v| > 0 -> (v == 0) = false.
Proof. by move=> /gtr0_norm_neq0/negPf->. 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 | gtr0_norm_eq0F | |
normr0Pv : reflect (`|v| = 0) (v == 0).
Proof. by apply: (iffP eqP)=> [->|/normr0_eq0 //]; apply: normr0. 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 | normr0P | |
normr_eq0v := sameP (`|v| =P 0) (normr0P v). | 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 | normr_eq0 | |
normr_le0v : `|v| <= 0 = (v == 0).
Proof. by rewrite -normr_eq0 eq_le normr_ge0 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 | normr_le0 | |
normr_gt0v : `|v| > 0 = (v != 0).
Proof. by rewrite lt_def normr_eq0 normr_ge0 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 | normr_gt0 | |
normrE:= (normr_id, normr0, normr1, normrN1, normr_ge0, normr_eq0,
normr_lt0, normr_le0, normr_gt0, normrN). | 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 | normrE | |
ler0_defx : (x <= 0) = (`|x| == - x).
Proof. by rewrite ler_def sub0r normrN. 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 | ler0_def | |
ler0_normx : x <= 0 -> `|x| = - x.
Proof. by move=> x_le0; rewrite -[r in _ = r]ger0_norm ?normrN ?oppr_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 | ler0_norm | |
gtr0_normx (hx : 0 < x) := ger0_norm (ltW hx). | 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 | gtr0_norm | |
ltr0_normx (hx : x < 0) := ler0_norm (ltW hx). | 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 | ltr0_norm | |
ger0_le_norm:
{in nneg &, {mono (@normr _ R) : x y / x <= y}}.
Proof. by move=> x y; rewrite !nnegrE => x0 y0; rewrite !ger0_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 | ger0_le_norm | |
gtr0_le_norm:
{in pos &, {mono (@normr _ R) : x y / x <= y}}.
Proof. by move=> x y; rewrite !posrE => /ltW x0 /ltW y0; exact: ger0_le_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 | gtr0_le_norm | |
ler0_ge_norm:
{in npos &, {mono (@normr _ R) : x y / x <= y >-> x >= y}}.
Proof.
move=> x y; rewrite !nposrE => x0 y0.
by rewrite !ler0_norm// -[LHS]subr_ge0 opprK addrC 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 | ler0_ge_norm | |
ltr0_ge_norm:
{in neg &, {mono (@normr _ R) : x y / x <= y >-> x >= y}}.
Proof. by move=> x y; rewrite !negrE => /ltW x0 /ltW y0; exact: ler0_ge_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 | ltr0_ge_norm | |
subr_ge0x y : (0 <= y - x) = (x <= y). Proof. exact: 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_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 | |
subr_le0x y : (y - x <= 0) = (y <= x).
Proof. by rewrite -[LHS]subr_ge0 opprB add0r 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_le0 | |
subr_lt0x y : (y - x < 0) = (y < x).
Proof. by rewrite -[LHS]subr_gt0 opprB add0r 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 | subr_lt0 | |
subr_lte0:= (subr_le0, subr_lt0). | 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 | subr_lte0 | |
subr_gte0:= (subr_ge0, subr_gt0). | 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 | subr_gte0 | |
subr_cp0:= (subr_lte0, subr_gte0). | 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 | subr_cp0 | |
comparable0rx : (0 >=< x)%R = (x \is Num.real). 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 ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] | algebra/num_theory/numdomain.v | comparable0r | |
comparabler0x : (x >=< 0)%R = (x \is Num.real).
Proof. by rewrite comparable_sym. 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 | comparabler0 | |
subr_comparable0x y : (x - y >=< 0)%R = (x >=< y)%R.
Proof. by rewrite /comparable subr_ge0 subr_le0. 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_comparable0 | |
comparablerEx y : (x >=< y)%R = (x - y \is Num.real).
Proof. by rewrite -comparabler0 subr_comparable0. 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 | comparablerE | |
comparabler_trans: transitive (comparable : rel R).
Proof.
move=> y x z; rewrite !comparablerE => xBy_real yBz_real.
by have := rpredD xBy_real yBz_real; rewrite addrA addrNK.
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 | comparabler_trans | |
lter01:= (ler01, ltr01). | 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 | lter01 | |
addr_ge0x y : 0 <= x -> 0 <= y -> 0 <= x + y.
Proof. exact: addr_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 | addr_ge0 | |
char_num:= pchar_num (only parsing).
Arguments ler01 {R}.
Arguments ltr01 {R}.
Arguments normr_idP {R x}.
Arguments normr0P {R V v}.
#[global] Hint Extern 0 (is_true (@Order.le ring_display _ _ _)) =>
(apply: ler01) : core.
#[global] Hint Extern 0 (is_true (@Order.lt ring_display _ _ _)) =>
(apply: ltr01) : core.
#[global] Hint Extern 0 (is_true (@Order.le ring_display _ _ _)) =>
(apply: ler0n) : core.
#[global] Hint Extern 0 (is_true (@Order.lt ring_display _ _ _)) =>
(apply: ltr0Sn) : core.
#[global] Hint Extern 0 (is_true (0 <= norm _)) => apply: normr_ge0 : core. | 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"
] | algebra/num_theory/numdomain.v | char_num | |
normr_nneg(R : numDomainType) (x : R) : `|x| \is Num.nneg.
Proof. by rewrite qualifE /=. Qed.
#[global] Hint Resolve normr_nneg : core. | 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 | normr_nneg | |
lerN2: {mono -%R : x y /~ x <= y :> R}.
Proof. by move=> x y /=; rewrite -subr_ge0 opprK addrC subr_ge0. Qed.
Hint Resolve lerN2 : core. | 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 | lerN2 | |
ltrN2: {mono -%R : x y /~ x < y :> R}.
Proof. by move=> x y /=; rewrite leW_nmono. Qed.
Hint Resolve ltrN2 : core. | 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 | ltrN2 | |
lterN2:= (lerN2, ltrN2). | 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 | lterN2 | |
lerNrx y : (x <= - y) = (y <= - x).
Proof. by rewrite (monoRL opprK lerN2). 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 | lerNr | |
ltrNrx y : (x < - y) = (y < - x).
Proof. by rewrite (monoRL opprK (leW_nmono _)). 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 | ltrNr | |
lterNr:= (lerNr, ltrNr). | 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 | lterNr | |
lerNlx y : (- x <= y) = (- y <= x).
Proof. by rewrite (monoLR opprK lerN2). 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 | lerNl | |
ltrNlx y : (- x < y) = (- y < x).
Proof. by rewrite (monoLR opprK (leW_nmono _)). 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 | ltrNl | |
lterNl:= (lerNl, ltrNl). | 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 | lterNl | |
oppr_ge0x : (0 <= - x) = (x <= 0). Proof. by rewrite lerNr oppr0. 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_ge0 | |
oppr_gt0x : (0 < - x) = (x < 0). Proof. by rewrite ltrNr oppr0. 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_gt0 | |
oppr_gte0:= (oppr_ge0, oppr_gt0). | 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 | oppr_gte0 | |
oppr_le0x : (- x <= 0) = (0 <= x). Proof. by rewrite lerNl oppr0. 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_le0 | |
oppr_lt0x : (- x < 0) = (0 < x). Proof. by rewrite ltrNl oppr0. 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_lt0 | |
gtrNx : 0 < x -> - x < x.
Proof. by move=> n0; rewrite -subr_lt0 -opprD oppr_lt0 addr_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 | gtrN |
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