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 |
|---|---|---|---|---|---|---|
ler_wpDry x z : 0 <= x -> y <= z -> y <= z + x.
Proof. by move=> *; rewrite addrC ler_wpDl. 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_wpDr | |
ltr_wpDry x z : 0 <= x -> y < z -> y < z + x.
Proof. by move=> *; rewrite addrC ltr_wpDl. 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_wpDr | |
ltr_pwDry x z : 0 < x -> y <= z -> y < z + x.
Proof. by move=> *; rewrite addrC ltr_pwDl. 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_pwDr | |
ltr_pDry x z : 0 < x -> y < z -> y < z + x.
Proof. by move=> *; rewrite addrC ltr_pDl. 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_pDr | |
ler_wnDry x z : x <= 0 -> y <= z -> y + x <= z.
Proof. by move=> *; rewrite addrC ler_wnDl. 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_wnDr | |
ltr_wnDry x z : x <= 0 -> y < z -> y + x < z.
Proof. by move=> *; rewrite addrC ltr_wnDl. 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_wnDr | |
ltr_nwDry x z : x < 0 -> y <= z -> y + x < z.
Proof. by move=> *; rewrite addrC ltr_nwDl. 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_nwDr | |
ltr_nDry x z : x < 0 -> y < z -> y + x < z.
Proof. by move=> *; rewrite addrC ltr_nDl. 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_nDr | |
paddr_eq0(x y : R) :
0 <= x -> 0 <= y -> (x + y == 0) = (x == 0) && (y == 0).
Proof.
rewrite le0r; case/orP=> [/eqP->|hx]; first by rewrite add0r eqxx.
by rewrite (gt_eqF hx) /= => hy; rewrite gt_eqF // ltr_pwDl.
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 | paddr_eq0 | |
naddr_eq0(x y : R) :
x <= 0 -> y <= 0 -> (x + y == 0) = (x == 0) && (y == 0).
Proof.
by move=> lex0 ley0; rewrite -oppr_eq0 opprD paddr_eq0 ?oppr_cp0 // !oppr_eq0.
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 | naddr_eq0 | |
addr_ss_eq0(x y : R) :
(0 <= x) && (0 <= y) || (x <= 0) && (y <= 0) ->
(x + y == 0) = (x == 0) && (y == 0).
Proof. by case/orP=> /andP []; [apply: paddr_eq0 | apply: naddr_eq0]. 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_ss_eq0 | |
sumr_ge0I (r : seq I) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) -> 0 <= \sum_(i <- r | P i) (F i).
Proof. exact: (big_ind _ _ (@ler_wpDl 0)). 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 | sumr_ge0 | |
sumr_le0I (r : seq I) (P : pred I) (F : I -> R) :
(forall i, P i -> F i <= 0) -> \sum_(i <- r | P i) F i <= 0.
Proof. by move=> F0; elim/big_ind : _ => // i x Pi; exact/ler_wnDl. 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 | sumr_le0 | |
ler_sumI (r : seq I) (P : pred I) (F G : I -> R) :
(forall i, P i -> F i <= G i) ->
\sum_(i <- r | P i) F i <= \sum_(i <- r | P i) G i.
Proof. exact: (big_ind2 _ (lexx _) lerD). 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_sum | |
ler_sum_nat(m n : nat) (F G : nat -> R) :
(forall i, (m <= i < n)%N -> F i <= G i) ->
\sum_(m <= i < n) F i <= \sum_(m <= i < n) G i.
Proof. by move=> le_FG; rewrite !big_nat ler_sum. 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_sum_nat | |
ltr_sumI (r : seq I) (P : pred I) (F G : I -> R) :
has P r -> (forall i, P i -> F i < G i) ->
\sum_(i <- r | P i) F i < \sum_(i <- r | P i) G i.
Proof.
rewrite -big_filter -[ltRHS]big_filter -size_filter_gt0.
case: filter (filter_all P r) => //= x {}r /andP[Px Pr] _ ltFG.
rewrite !big_cons ltr_leD// ?ltFG// -(all_filterP Pr) !big_filter.
by rewrite ler_sum => // i Pi; rewrite ltW ?ltFG.
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_sum | |
ltr_sum_nat(m n : nat) (F G : nat -> R) :
(m < n)%N -> (forall i, (m <= i < n)%N -> F i < G i) ->
\sum_(m <= i < n) F i < \sum_(m <= i < n) G i.
Proof.
move=> lt_mn i; rewrite big_nat [ltRHS]big_nat ltr_sum//.
by apply/hasP; exists m; rewrite ?mem_index_iota leqnn lt_mn.
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_sum_nat | |
psumr_eq0(I : eqType) (r : seq I) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) ->
(\sum_(i <- r | P i) (F i) == 0) = (all (fun i => (P i) ==> (F i == 0)) r).
Proof.
elim: r=> [|a r ihr hr] /=; rewrite (big_nil, big_cons); first by rewrite eqxx.
by case: ifP=> pa /=; rewrite ?paddr_eq0 ?ihr ?hr // sumr_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 | psumr_eq0 | |
psumr_eq0P(I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) -> \sum_(i | P i) F i = 0 ->
(forall i, P i -> F i = 0).
Proof.
move=> F_ge0 /eqP; rewrite psumr_eq0 // -big_all big_andE => /forallP hF i Pi.
by move: (hF i); rewrite implyTb Pi /= => /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 | psumr_eq0P | |
psumr_neq0(I : eqType) (r : seq I) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) ->
(\sum_(i <- r | P i) (F i) != 0) = (has (fun i => P i && (0 < F i)) r).
Proof.
move=> F_ge0; rewrite psumr_eq0// -has_predC; apply: eq_has => x /=.
by case Px: (P x); rewrite //= lt_def F_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 | psumr_neq0 | |
psumr_neq0P(I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i) -> \sum_(i | P i) F i <> 0 ->
(exists i, P i && (0 < F i)).
Proof. by move=> ? /eqP; rewrite psumr_neq0// => /hasP[x _ ?]; exists x. 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 | psumr_neq0P | |
ler_pM2lx : 0 < x -> {mono *%R x : x y / x <= y}.
Proof.
by move=> x_gt0 y z /=; rewrite -subr_ge0 -mulrBr pmulr_rge0 // 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 | ler_pM2l | |
ltr_pM2lx : 0 < x -> {mono *%R x : x y / x < y}.
Proof. by move=> x_gt0; apply: leW_mono (ler_pM2l _). 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_pM2l | |
lter_pM2l:= (ler_pM2l, ltr_pM2l). | 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_pM2l | |
ler_pM2rx : 0 < x -> {mono *%R^~ x : x y / x <= y}.
Proof. by move=> x_gt0 y z /=; rewrite ![_ * x]mulrC ler_pM2l. 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_pM2r | |
ltr_pM2rx : 0 < x -> {mono *%R^~ x : x y / x < y}.
Proof. by move=> x_gt0; apply: leW_mono (ler_pM2r _). 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_pM2r | |
lter_pM2r:= (ler_pM2r, ltr_pM2r). | 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_pM2r | |
ler_nM2lx : x < 0 -> {mono *%R x : x y /~ x <= y}.
Proof. by move=> x_lt0 y z /=; rewrite -lerN2 -!mulNr ler_pM2l ?oppr_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 | ler_nM2l | |
ltr_nM2lx : x < 0 -> {mono *%R x : x y /~ x < y}.
Proof. by move=> x_lt0; apply: leW_nmono (ler_nM2l _). 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_nM2l | |
lter_nM2l:= (ler_nM2l, ltr_nM2l). | 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_nM2l | |
ler_nM2rx : x < 0 -> {mono *%R^~ x : x y /~ x <= y}.
Proof. by move=> x_lt0 y z /=; rewrite ![_ * x]mulrC ler_nM2l. 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_nM2r | |
ltr_nM2rx : x < 0 -> {mono *%R^~ x : x y /~ x < y}.
Proof. by move=> x_lt0; apply: leW_nmono (ler_nM2r _). 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_nM2r | |
lter_nM2r:= (ler_nM2r, ltr_nM2r). | 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_nM2r | |
ler_wpM2lx : 0 <= x -> {homo *%R x : y z / y <= z}.
Proof.
by rewrite le0r => /orP[/eqP-> y z | /ler_pM2l/mono2W//]; rewrite !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 | ler_wpM2l | |
ler_wpM2rx : 0 <= x -> {homo *%R^~ x : y z / y <= z}.
Proof. by move=> x_ge0 y z leyz; rewrite ![_ * x]mulrC ler_wpM2l. 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_wpM2r | |
ler_wnM2lx : x <= 0 -> {homo *%R x : y z /~ y <= z}.
by move=> x_le0 y z leyz; rewrite -![x * _]mulrNN ler_wpM2l ?lterNE. 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_wnM2l | |
ler_wnM2rx : x <= 0 -> {homo *%R^~ x : y z /~ y <= z}.
Proof. by move=> x_le0 y z leyz; rewrite -![_ * x]mulrNN ler_wpM2r ?lterNE. 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_wnM2r | |
ler_pMx1 y1 x2 y2 :
0 <= x1 -> 0 <= x2 -> x1 <= y1 -> x2 <= y2 -> x1 * x2 <= y1 * y2.
Proof.
move=> x1ge0 x2ge0 le_xy1 le_xy2; have y1ge0 := le_trans x1ge0 le_xy1.
exact: le_trans (ler_wpM2r x2ge0 le_xy1) (ler_wpM2l y1ge0 le_xy2).
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_pM | |
ltr_pMx1 y1 x2 y2 :
0 <= x1 -> 0 <= x2 -> x1 < y1 -> x2 < y2 -> x1 * x2 < y1 * y2.
Proof.
move=> x1ge0 x2ge0 lt_xy1 lt_xy2; have y1gt0 := le_lt_trans x1ge0 lt_xy1.
by rewrite (le_lt_trans (ler_wpM2r x2ge0 (ltW lt_xy1))) ?ltr_pM2l.
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_pM | |
ler_pMn2rn : (0 < n)%N -> {mono (@GRing.natmul R)^~ n : x y / x <= y}.
Proof.
by case: n => // n _ x y /=; rewrite -mulr_natl -[y *+ _]mulr_natl ler_pM2l.
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_pMn2r | |
ltr_pMn2rn : (0 < n)%N -> {mono (@GRing.natmul R)^~ n : x y / x < y}.
Proof. by move/ler_pMn2r/leW_mono. 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_pMn2r | |
pmulrnIn : (0 < n)%N -> injective ((@GRing.natmul R)^~ n).
Proof. by move/ler_pMn2r/inc_inj. 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 | pmulrnI | |
eqr_pMn2rn : (0 < n)%N -> {mono (@GRing.natmul R)^~ n : x y / x == y}.
Proof. by move/pmulrnI/inj_eq. 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_pMn2r | |
pmulrn_lgt0x n : (0 < n)%N -> (0 < x *+ n) = (0 < x).
Proof. by move=> n_gt0; rewrite -(mul0rn _ n) ltr_pMn2r // mul0rn. 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 | pmulrn_lgt0 | |
pmulrn_llt0x n : (0 < n)%N -> (x *+ n < 0) = (x < 0).
Proof. by move=> n_gt0; rewrite -(mul0rn _ n) ltr_pMn2r // mul0rn. 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 | pmulrn_llt0 | |
pmulrn_lge0x n : (0 < n)%N -> (0 <= x *+ n) = (0 <= x).
Proof. by move=> n_gt0; rewrite -(mul0rn _ n) ler_pMn2r // mul0rn. 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 | pmulrn_lge0 | |
pmulrn_lle0x n : (0 < n)%N -> (x *+ n <= 0) = (x <= 0).
Proof. by move=> n_gt0; rewrite -(mul0rn _ n) ler_pMn2r // mul0rn. 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 | pmulrn_lle0 | |
ltr_wMn2rx y n : x < y -> (x *+ n < y *+ n) = (0 < n)%N.
Proof. by move=> ltxy; case: n=> // n; rewrite ltr_pMn2r. 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_wMn2r | |
ltr_wpMn2rn : (0 < n)%N -> {homo (@GRing.natmul R)^~ n : x y / x < y}.
Proof. by move=> n_gt0 x y /= / ltr_wMn2r ->. 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_wpMn2r | |
ler_wMn2rn : {homo (@GRing.natmul R)^~ n : x y / x <= y}.
Proof. by move=> x y hxy /=; case: n=> // n; rewrite ler_pMn2r. 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_wMn2r | |
mulrn_wge0x n : 0 <= x -> 0 <= x *+ n.
Proof. by move=> /(ler_wMn2r n); rewrite mul0rn. 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 | mulrn_wge0 | |
mulrn_wle0x n : x <= 0 -> x *+ n <= 0.
Proof. by move=> /(ler_wMn2r n); rewrite mul0rn. 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 | mulrn_wle0 | |
lerMn2rn x y : (x *+ n <= y *+ n) = ((n == 0) || (x <= y)).
Proof. by case: n => [|n]; rewrite ?lexx ?eqxx // ler_pMn2r. 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 | lerMn2r | |
ltrMn2rn x y : (x *+ n < y *+ n) = ((0 < n)%N && (x < y)).
Proof. by case: n => [|n]; rewrite ?lexx ?eqxx // ltr_pMn2r. 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 | ltrMn2r | |
eqrMn2rn x y : (x *+ n == y *+ n) = (n == 0)%N || (x == y).
Proof. by rewrite !(@eq_le _ R) !lerMn2r -orb_andr. 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 | eqrMn2r | |
mulrn_eq0x n : (x *+ n == 0) = ((n == 0)%N || (x == 0)).
Proof. by rewrite -mulr_natl mulf_eq0 pnatr_eq0. 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 | mulrn_eq0 | |
eqNrx : (- x == x) = (x == 0).
Proof. by rewrite eq_sym -addr_eq0 -mulr2n mulrn_eq0. 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 | eqNr | |
mulrInx : x != 0 -> injective (GRing.natmul x).
Proof.
move=> x_neq0 m n; without loss /subnK <-: m n / (n <= m)%N.
by move=> IH eq_xmn; case/orP: (leq_total m n) => /IH->.
by move/eqP; rewrite mulrnDr -subr_eq0 addrK mulrn_eq0 => /predU1P[-> | /idPn].
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 | mulrIn | |
ler_wpMn2lx :
0 <= x -> {homo (@GRing.natmul R x) : m n / (m <= n)%N >-> m <= n}.
Proof. by move=> xge0 m n /subnK <-; rewrite mulrnDr ler_wpDl ?mulrn_wge0. 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_wpMn2l | |
ler_wnMn2lx :
x <= 0 -> {homo (@GRing.natmul R x) : m n / (n <= m)%N >-> m <= n}.
Proof.
by move=> xle0 m n hmn /=; rewrite -lerN2 -!mulNrn ler_wpMn2l // oppr_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 | ler_wnMn2l | |
mulrn_wgt0x n : 0 < x -> 0 < x *+ n = (0 < n)%N.
Proof. by case: n => // n hx; rewrite pmulrn_lgt0. 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 | mulrn_wgt0 | |
mulrn_wlt0x n : x < 0 -> x *+ n < 0 = (0 < n)%N.
Proof. by case: n => // n hx; rewrite pmulrn_llt0. 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 | mulrn_wlt0 | |
ler_pMn2lx :
0 < x -> {mono (@GRing.natmul R x) : m n / (m <= n)%N >-> m <= n}.
Proof.
move=> x_gt0 m n /=; case: leqP => hmn; first by rewrite ler_wpMn2l // ltW.
by rewrite -(subnK (ltnW hmn)) mulrnDr gerDr lt_geF // mulrn_wgt0 // subn_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 | ler_pMn2l | |
ltr_pMn2lx :
0 < x -> {mono (@GRing.natmul R x) : m n / (m < n)%N >-> m < n}.
Proof. by move=> x_gt0; apply: leW_mono (ler_pMn2l _). 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_pMn2l | |
ler_nMn2lx :
x < 0 -> {mono (@GRing.natmul R x) : m n / (n <= m)%N >-> m <= n}.
Proof. by move=> xlt0 m n /=; rewrite -lerN2 -!mulNrn ler_pMn2l// oppr_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 | ler_nMn2l | |
ltr_nMn2lx :
x < 0 -> {mono (@GRing.natmul R x) : m n / (n < m)%N >-> m < n}.
Proof. by move=> x_lt0; apply: leW_nmono (ler_nMn2l _). 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_nMn2l | |
ler_natm n : (m%:R <= n%:R :> R) = (m <= n)%N.
Proof. by rewrite ler_pMn2l. 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_nat | |
ltr_natm n : (m%:R < n%:R :> R) = (m < n)%N.
Proof. by rewrite ltr_pMn2l. 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_nat | |
eqr_natm n : (m%:R == n%:R :> R) = (m == n)%N.
Proof. by rewrite (inj_eq (mulrIn _)) ?oner_eq0. 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_nat | |
pnatr_eq1n : (n%:R == 1 :> R) = (n == 1)%N.
Proof. exact: eqr_nat 1. 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_eq1 | |
lern0n : (n%:R <= 0 :> R) = (n == 0).
Proof. by rewrite -[0]/0%:R ler_nat leqn0. 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 | lern0 | |
ltrn0n : (n%:R < 0 :> R) = false.
Proof. by rewrite -[0]/0%:R ltr_nat ltn0. 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 | ltrn0 | |
ler1nn : 1 <= n%:R :> R = (1 <= n)%N. Proof. by rewrite -ler_nat. 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 | ler1n | |
ltr1nn : 1 < n%:R :> R = (1 < n)%N. Proof. by rewrite -ltr_nat. 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 | ltr1n | |
lern1n : n%:R <= 1 :> R = (n <= 1)%N. Proof. by rewrite -ler_nat. 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 | lern1 | |
ltrn1n : n%:R < 1 :> R = (n < 1)%N. Proof. by rewrite -ltr_nat. 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 | ltrn1 | |
ltrN10: -1 < 0 :> R. Proof. by rewrite oppr_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 | ltrN10 | |
lerN10: -1 <= 0 :> R. Proof. by rewrite oppr_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 | lerN10 | |
ltr10: 1 < 0 :> R = false. Proof. by rewrite le_gtF. 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 | ltr10 | |
ler10: 1 <= 0 :> R = false. Proof. by rewrite lt_geF. 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 | ler10 | |
ltr0N1: 0 < -1 :> R = false. Proof. by rewrite le_gtF // lerN10. 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 | ltr0N1 | |
ler0N1: 0 <= -1 :> R = false. Proof. by rewrite lt_geF // ltrN10. Qed.
#[deprecated(since="mathcomp 2.4.0", note="use `mulrn_wgt0` 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 | ler0N1 | |
pmulrn_rgt0x n : 0 < x -> 0 < x *+ n = (0 < n)%N.
Proof. exact: mulrn_wgt0. 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 | pmulrn_rgt0 | |
pmulrn_rlt0x n : 0 < x -> x *+ n < 0 = false.
Proof. by move=> x_gt0; rewrite -(mulr0n x) ltr_pMn2l. 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 | pmulrn_rlt0 | |
pmulrn_rge0x n : 0 < x -> 0 <= x *+ n.
Proof. by move=> x_gt0; rewrite -(mulr0n x) ler_pMn2l. 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 | pmulrn_rge0 | |
pmulrn_rle0x n : 0 < x -> x *+ n <= 0 = (n == 0)%N.
Proof. by move=> x_gt0; rewrite -(mulr0n x) ler_pMn2l ?leqn0. 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 | pmulrn_rle0 | |
nmulrn_rgt0x n : x < 0 -> 0 < x *+ n = false.
Proof. by move=> x_lt0; rewrite -(mulr0n x) ltr_nMn2l. 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 | nmulrn_rgt0 | |
nmulrn_rge0x n : x < 0 -> 0 <= x *+ n = (n == 0)%N.
Proof. by move=> x_lt0; rewrite -(mulr0n x) ler_nMn2l ?leqn0. 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 | nmulrn_rge0 | |
nmulrn_rle0x n : x < 0 -> x *+ n <= 0.
Proof. by move=> x_lt0; rewrite -(mulr0n x) ler_nMn2l. 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 | nmulrn_rle0 | |
pmulr_rlt0x y : 0 < x -> (x * y < 0) = (y < 0).
Proof.
by move=> x_gt0; rewrite -[LHS]oppr_gt0 -mulrN pmulr_rgt0 // oppr_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 | pmulr_rlt0 | |
pmulr_rle0x y : 0 < x -> (x * y <= 0) = (y <= 0).
Proof.
by move=> x_gt0; rewrite -[LHS]oppr_ge0 -mulrN pmulr_rge0 // 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 | pmulr_rle0 | |
pmulr_lgt0x y : 0 < x -> (0 < y * x) = (0 < y).
Proof. by move=> x_gt0; rewrite mulrC 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_lgt0 | |
pmulr_lge0x y : 0 < x -> (0 <= y * x) = (0 <= y).
Proof. by move=> x_gt0; rewrite mulrC pmulr_rge0. 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_lge0 | |
pmulr_llt0x y : 0 < x -> (y * x < 0) = (y < 0).
Proof. by move=> x_gt0; rewrite mulrC pmulr_rlt0. 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_llt0 | |
pmulr_lle0x y : 0 < x -> (y * x <= 0) = (y <= 0).
Proof. by move=> x_gt0; rewrite mulrC pmulr_rle0. 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_lle0 | |
nmulr_rgt0x y : x < 0 -> (0 < x * y) = (y < 0).
Proof. by move=> x_lt0; rewrite -mulrNN pmulr_rgt0 lterNE. 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 | nmulr_rgt0 | |
nmulr_rge0x y : x < 0 -> (0 <= x * y) = (y <= 0).
Proof. by move=> x_lt0; rewrite -mulrNN pmulr_rge0 lterNE. 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 | nmulr_rge0 | |
nmulr_rlt0x y : x < 0 -> (x * y < 0) = (0 < y).
Proof. by move=> x_lt0; rewrite -mulrNN pmulr_rlt0 lterNE. 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 | nmulr_rlt0 | |
nmulr_rle0x y : x < 0 -> (x * y <= 0) = (0 <= y).
Proof. by move=> x_lt0; rewrite -mulrNN pmulr_rle0 lterNE. 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 | nmulr_rle0 | |
nmulr_lgt0x y : x < 0 -> (0 < y * x) = (y < 0).
Proof. by move=> x_lt0; rewrite mulrC nmulr_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 | nmulr_lgt0 |
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