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 |
|---|---|---|---|---|---|---|
nmulr_lge0x y : x < 0 -> (0 <= y * x) = (y <= 0).
Proof. by move=> x_lt0; rewrite mulrC nmulr_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
|
nmulr_lge0
| |
nmulr_llt0x y : x < 0 -> (y * x < 0) = (0 < y).
Proof. by move=> x_lt0; rewrite mulrC nmulr_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
|
nmulr_llt0
| |
nmulr_lle0x y : x < 0 -> (y * x <= 0) = (0 <= y).
Proof. by move=> x_lt0; rewrite mulrC nmulr_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
|
nmulr_lle0
| |
mulr_ge0x y : 0 <= x -> 0 <= y -> 0 <= x * y.
Proof. by move=> x_ge0 y_ge0; rewrite -(mulr0 x) 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
|
mulr_ge0
| |
mulr_le0x y : x <= 0 -> y <= 0 -> 0 <= x * y.
Proof. by move=> x_le0 y_le0; rewrite -(mulr0 x) ler_wnM2l. 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_le0
| |
mulr_ge0_le0x y : 0 <= x -> y <= 0 -> x * y <= 0.
Proof. by move=> x_le0 y_le0; rewrite -(mulr0 x) 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
|
mulr_ge0_le0
| |
mulr_le0_ge0x y : x <= 0 -> 0 <= y -> x * y <= 0.
Proof. by move=> x_le0 y_le0; rewrite -(mulr0 x) ler_wnM2l. 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_le0_ge0
| |
mulr_gt0x y : 0 < x -> 0 < y -> 0 < x * y.
Proof. by move=> x_gt0 y_gt0; rewrite 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
|
mulr_gt0
| |
mulr_ge0_gt0x y : 0 <= x -> 0 <= y -> (0 < x * y) = (0 < x) && (0 < y).
Proof.
rewrite le_eqVlt => /predU1P[<-|x0]; first by rewrite mul0r ltxx.
rewrite le_eqVlt => /predU1P[<-|y0]; first by rewrite mulr0 ltxx andbC.
by apply/idP/andP=> [|_]; rewrite 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
|
mulr_ge0_gt0
| |
prodr_ge0I r (P : pred I) (E : I -> R) :
(forall i, P i -> 0 <= E i) -> 0 <= \prod_(i <- r | P i) E i.
Proof. by move=> Ege0; rewrite -nnegrE rpred_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
|
prodr_ge0
| |
prodr_gt0I r (P : pred I) (E : I -> R) :
(forall i, P i -> 0 < E i) -> 0 < \prod_(i <- r | P i) E i.
Proof. by move=> Ege0; rewrite -posrE rpred_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
|
prodr_gt0
| |
ler_prodI r (P : pred I) (E1 E2 : I -> R) :
(forall i, P i -> 0 <= E1 i <= E2 i) ->
\prod_(i <- r | P i) E1 i <= \prod_(i <- r | P i) E2 i.
Proof.
move=> leE12; elim/(big_load (fun x => 0 <= x)): _.
elim/big_rec2: _ => // i x2 x1 /leE12/andP[le0Ei leEi12] [x1ge0 le_x12].
by rewrite mulr_ge0 // ler_pM.
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_prod
| |
ltr_prodI r (P : pred I) (E1 E2 : I -> R) :
has P r -> (forall i, P i -> 0 <= E1 i < E2 i) ->
\prod_(i <- r | P i) E1 i < \prod_(i <- r | P i) E2 i.
Proof.
elim: r => //= i r IHr; rewrite !big_cons; case: ifP => {IHr}// Pi _ ltE12.
have /andP[le0E1i ltE12i] := ltE12 i Pi; set E2r := \prod_(j <- r | P j) E2 j.
apply: le_lt_trans (_ : E1 i * E2r < E2 i * E2r).
by rewrite ler_wpM2l ?ler_prod // => j /ltE12/andP[-> /ltW].
by rewrite ltr_pM2r ?prodr_gt0 // => j /ltE12/andP[le0E1j /le_lt_trans->].
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_prod
| |
ltr_prod_nat(E1 E2 : nat -> R) (n m : nat) :
(m < n)%N -> (forall i, (m <= i < n)%N -> 0 <= E1 i < E2 i) ->
\prod_(m <= i < n) E1 i < \prod_(m <= i < n) E2 i.
Proof.
move=> lt_mn ltE12; rewrite !big_nat ltr_prod {ltE12}//.
by apply/hasP; exists m; rewrite ?mem_index_iota leqnn.
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_prod_nat
| |
realMrx y : x != 0 -> x \is real -> (x * y \is real) = (y \is real).
Proof.
move=> x_neq0 xR; case: real_ltgtP x_neq0 => // hx _; rewrite !realE.
by rewrite nmulr_rge0 // nmulr_rle0 // orbC.
by rewrite pmulr_rge0 // pmulr_rle0 // orbC.
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
|
realMr
| |
realrMx y : y != 0 -> y \is real -> (x * y \is real) = (x \is real).
Proof. by move=> y_neq0 yR; rewrite mulrC realMr. 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
|
realrM
| |
realM: {in real &, forall x y, x * y \is real}.
Proof. exact: rpredM. 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
|
realM
| |
realrMnx n : (n != 0)%N -> (x *+ n \is real) = (x \is real).
Proof. by move=> n_neq0; rewrite -mulr_natl realMr ?realn ?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
|
realrMn
| |
ger_pMlx y : 0 < y -> (x * y <= y) = (x <= 1).
Proof. by move=> hy; rewrite -{2}[y]mul1r 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
|
ger_pMl
| |
gtr_pMlx y : 0 < y -> (x * y < y) = (x < 1).
Proof. by move=> hy; rewrite -{2}[y]mul1r ltr_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
|
gtr_pMl
| |
ger_pMrx y : 0 < y -> (y * x <= y) = (x <= 1).
Proof. by move=> hy; rewrite -{2}[y]mulr1 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
|
ger_pMr
| |
gtr_pMrx y : 0 < y -> (y * x < y) = (x < 1).
Proof. by move=> hy; rewrite -{2}[y]mulr1 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
|
gtr_pMr
| |
ler_pMlx y : 0 < y -> (y <= x * y) = (1 <= x).
Proof. by move=> hy; rewrite -{1}[y]mul1r 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
|
ler_pMl
| |
ltr_pMlx y : 0 < y -> (y < x * y) = (1 < x).
Proof. by move=> hy; rewrite -{1}[y]mul1r ltr_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_pMl
| |
ler_pMrx y : 0 < y -> (y <= y * x) = (1 <= x).
Proof. by move=> hy; rewrite -{1}[y]mulr1 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_pMr
| |
ltr_pMrx y : 0 < y -> (y < y * x) = (1 < x).
Proof. by move=> hy; rewrite -{1}[y]mulr1 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_pMr
| |
ger_nMlx y : y < 0 -> (x * y <= y) = (1 <= x).
Proof. by move=> hy; rewrite -{2}[y]mul1r 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
|
ger_nMl
| |
gtr_nMlx y : y < 0 -> (x * y < y) = (1 < x).
Proof. by move=> hy; rewrite -{2}[y]mul1r ltr_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
|
gtr_nMl
| |
ger_nMrx y : y < 0 -> (y * x <= y) = (1 <= x).
Proof. by move=> hy; rewrite -{2}[y]mulr1 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
|
ger_nMr
| |
gtr_nMrx y : y < 0 -> (y * x < y) = (1 < x).
Proof. by move=> hy; rewrite -{2}[y]mulr1 ltr_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
|
gtr_nMr
| |
ler_nMlx y : y < 0 -> (y <= x * y) = (x <= 1).
Proof. by move=> hy; rewrite -{1}[y]mul1r 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
|
ler_nMl
| |
ltr_nMlx y : y < 0 -> (y < x * y) = (x < 1).
Proof. by move=> hy; rewrite -{1}[y]mul1r ltr_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_nMl
| |
ler_nMrx y : y < 0 -> (y <= y * x) = (x <= 1).
Proof. by move=> hy; rewrite -{1}[y]mulr1 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_nMr
| |
ltr_nMrx y : y < 0 -> (y < y * x) = (x < 1).
Proof. by move=> hy; rewrite -{1}[y]mulr1 ltr_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_nMr
| |
ler_peMlx y : 0 <= y -> 1 <= x -> y <= x * y.
Proof. by move=> hy hx; rewrite -{1}[y]mul1r ler_wpM2r. 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_peMl
| |
ler_neMlx y : y <= 0 -> 1 <= x -> x * y <= y.
Proof. by move=> hy hx; rewrite -{2}[y]mul1r ler_wnM2r. 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_neMl
| |
ler_peMrx y : 0 <= y -> 1 <= x -> y <= y * x.
Proof. by move=> hy hx; rewrite -{1}[y]mulr1 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_peMr
| |
ler_neMrx y : y <= 0 -> 1 <= x -> y * x <= y.
Proof. by move=> hy hx; rewrite -{2}[y]mulr1 ler_wnM2l. 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_neMr
| |
ler_piMlx y : 0 <= y -> x <= 1 -> x * y <= y.
Proof. by move=> hy hx; rewrite -{2}[y]mul1r ler_wpM2r. 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_piMl
| |
ler_niMlx y : y <= 0 -> x <= 1 -> y <= x * y.
Proof. by move=> hy hx; rewrite -{1}[y]mul1r ler_wnM2r. 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_niMl
| |
ler_piMrx y : 0 <= y -> x <= 1 -> y * x <= y.
Proof. by move=> hy hx; rewrite -{2}[y]mulr1 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_piMr
| |
ler_niMrx y : y <= 0 -> x <= 1 -> y <= y * x.
Proof. by move=> hx hy; rewrite -{1}[y]mulr1 ler_wnM2l. 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_niMr
| |
mulr_ile1x y : 0 <= x -> 0 <= y -> x <= 1 -> y <= 1 -> x * y <= 1.
Proof. by move=> *; rewrite (@le_trans _ _ y) ?ler_piMl. 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_ile1
| |
prodr_ile1{I : Type} (s : seq I) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i <= 1) -> \prod_(j <- s | P j) F j <= 1.
Proof.
elim: s => [_ | y s ih xs01]; rewrite ?big_nil// big_cons.
case: ifPn => Py; last by rewrite ih.
have /andP[y0 y1] : 0 <= F y <= 1 by rewrite xs01// mem_head.
rewrite mulr_ile1 ?andbT//; last first.
by rewrite ih// => e xs; rewrite xs01// in_cons xs orbT.
by rewrite prodr_ge0// => x /xs01 /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
|
prodr_ile1
| |
mulr_ilt1x y : 0 <= x -> 0 <= y -> x < 1 -> y < 1 -> x * y < 1.
Proof. by move=> *; rewrite (@le_lt_trans _ _ y) ?ler_piMl // ltW. 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_ilt1
| |
mulr_ilte1:= (mulr_ile1, mulr_ilt1).
|
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
|
mulr_ilte1
| |
mulr_ege1x y : 1 <= x -> 1 <= y -> 1 <= x * y.
Proof.
by move=> le1x le1y; rewrite (@le_trans _ _ y) ?ler_peMl // (le_trans 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
|
mulr_ege1
| |
mulr_egt1x y : 1 < x -> 1 < y -> 1 < x * y.
Proof.
by move=> le1x lt1y; rewrite (@lt_trans _ _ y) // ltr_pMl // (lt_trans 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
|
mulr_egt1
| |
mulr_egte1:= (mulr_ege1, mulr_egt1).
|
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
|
mulr_egte1
| |
mulr_cp1:= (mulr_ilte1, mulr_egte1).
|
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
|
mulr_cp1
| |
invr_gt0x : (0 < x^-1) = (0 < x).
Proof.
have [ux | nux] := boolP (x \is a GRing.unit); last by rewrite invr_out.
by apply/idP/idP=> /ltr_pM2r <-; rewrite mul0r (mulrV, mulVr) ?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
|
invr_gt0
| |
invr_ge0x : (0 <= x^-1) = (0 <= x).
Proof. by rewrite !le0r invr_gt0 invr_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
|
invr_ge0
| |
invr_lt0x : (x^-1 < 0) = (x < 0).
Proof. by rewrite -oppr_cp0 -invrN invr_gt0 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
|
invr_lt0
| |
invr_le0x : (x^-1 <= 0) = (x <= 0).
Proof. by rewrite -oppr_cp0 -invrN invr_ge0 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
|
invr_le0
| |
invr_gte0:= (invr_ge0, invr_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
|
invr_gte0
| |
invr_lte0:= (invr_le0, invr_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
|
invr_lte0
| |
divr_ge0x y : 0 <= x -> 0 <= y -> 0 <= x / y.
Proof. by move=> x_ge0 y_ge0; rewrite mulr_ge0 ?invr_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
|
divr_ge0
| |
divr_gt0x y : 0 < x -> 0 < y -> 0 < x / y.
Proof. by move=> x_gt0 y_gt0; rewrite pmulr_rgt0 ?invr_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
|
divr_gt0
| |
realV: {mono (@GRing.inv R) : x / x \is real}.
Proof. exact: rpredV. 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
|
realV
| |
exprn_ge0n x : 0 <= x -> 0 <= x ^+ n.
Proof. by move=> xge0; rewrite -nnegrE rpredX. 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_ge0
| |
realXn : {in real, forall x, x ^+ n \is real}.
Proof. exact: rpredX. 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
|
realX
| |
exprn_gt0n x : 0 < x -> 0 < x ^+ n.
Proof.
by rewrite !lt0r expf_eq0 => /andP[/negPf-> /exprn_ge0->]; rewrite 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
|
exprn_gt0
| |
exprn_gte0:= (exprn_ge0, exprn_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
|
exprn_gte0
| |
exprn_ile1n x : 0 <= x -> x <= 1 -> x ^+ n <= 1.
Proof.
move=> xge0 xle1; elim: n=> [|*]; rewrite ?expr0 // exprS.
by rewrite mulr_ile1 ?exprn_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
|
exprn_ile1
| |
exprn_ilt1n x : 0 <= x -> x < 1 -> x ^+ n < 1 = (n != 0).
Proof.
move=> xge0 xlt1.
case: n; [by rewrite eqxx ltxx | elim=> [|n ihn]; first by rewrite expr1].
by rewrite exprS mulr_ilt1 // exprn_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
|
exprn_ilt1
| |
exprn_ilte1:= (exprn_ile1, exprn_ilt1).
|
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
|
exprn_ilte1
| |
exprn_ege1n x : 1 <= x -> 1 <= x ^+ n.
Proof.
by move=> x_ge1; elim: n=> [|n ihn]; rewrite ?expr0 // exprS mulr_ege1.
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_ege1
| |
exprn_egt1n x : 1 < x -> 1 < x ^+ n = (n != 0).
Proof.
move=> xgt1; case: n; first by rewrite eqxx ltxx.
by elim=> [|n ihn]; rewrite ?expr1// exprS mulr_egt1 // exprn_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
|
exprn_egt1
| |
exprn_egte1:= (exprn_ege1, exprn_egt1).
|
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
|
exprn_egte1
| |
exprn_cp1:= (exprn_ilte1, exprn_egte1).
|
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
|
exprn_cp1
| |
ler_iXnrx n : (0 < n)%N -> 0 <= x -> x <= 1 -> x ^+ n <= x.
Proof. by case: n => n // *; rewrite exprS ler_piMr // exprn_ile1. 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_iXnr
| |
ltr_iXnrx n : 0 < x -> x < 1 -> (x ^+ n < x) = (1 < n)%N.
Proof.
case: n=> [|[|n]] //; first by rewrite expr0 => _ /lt_gtF ->.
by move=> x0 x1; rewrite exprS gtr_pMr // ?exprn_ilt1 // ltW.
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_iXnr
| |
lter_iXnr:= (ler_iXnr, ltr_iXnr).
|
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_iXnr
| |
ler_eXnrx n : (0 < n)%N -> 1 <= x -> x <= x ^+ n.
Proof.
case: n => // n _ x_ge1.
by rewrite exprS ler_peMr ?(le_trans _ x_ge1) // exprn_ege1.
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_eXnr
| |
ltr_eXnrx n : 1 < x -> (x < x ^+ n) = (1 < n)%N.
Proof.
move=> x_ge1; case: n=> [|[|n]] //; first by rewrite expr0 lt_gtF.
by rewrite exprS ltr_pMr ?(lt_trans _ x_ge1) ?exprn_egt1.
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_eXnr
| |
lter_eXnr:= (ler_eXnr, ltr_eXnr).
|
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_eXnr
| |
lter_Xnr:= (lter_iXnr, lter_eXnr).
|
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_Xnr
| |
ler_wiXn2lx :
0 <= x -> x <= 1 -> {homo GRing.exp x : m n / (n <= m)%N >-> m <= n}.
Proof.
move=> xge0 xle1 m n /= hmn.
by rewrite -(subnK hmn) exprD ler_piMl ?(exprn_ge0, exprn_ile1).
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_wiXn2l
| |
ler_weXn2lx : 1 <= x -> {homo GRing.exp x : m n / (m <= n)%N >-> m <= n}.
Proof.
move=> xge1 m n /= hmn; rewrite -(subnK hmn) exprD.
by rewrite ler_peMl ?(exprn_ge0, exprn_ege1) // (le_trans _ xge1) ?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
|
ler_weXn2l
| |
ieexprn_weq1x n : 0 <= x -> (x ^+ n == 1) = ((n == 0) || (x == 1)).
Proof.
move=> xle0; case: n => [|n]; first by rewrite expr0 eqxx.
case: (@real_ltgtP x 1); do ?by rewrite ?ger0_real.
+ by move=> x_lt1; rewrite 1?lt_eqF // exprn_ilt1.
+ by move=> x_lt1; rewrite 1?gt_eqF // exprn_egt1.
by move->; rewrite expr1n 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
|
ieexprn_weq1
| |
ieexprInx : 0 < x -> x != 1 -> injective (GRing.exp x).
Proof.
move=> x_gt0 x_neq1 m n; without loss /subnK <-: m n / (n <= m)%N.
by move=> IH eq_xmn; case/orP: (leq_total m n) => /IH->.
case: {m}(m - n)%N => // m /eqP/idPn[]; rewrite -[x ^+ n]mul1r exprD.
by rewrite (inj_eq (mulIf _)) ?ieexprn_weq1 ?ltW // expf_neq0 ?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
|
ieexprIn
| |
ler_iXn2lx :
0 < x -> x < 1 -> {mono GRing.exp x : m n / (n <= m)%N >-> m <= n}.
Proof.
move=> xgt0 xlt1; apply: (le_nmono (inj_nhomo_lt _ _)); last first.
by apply/ler_wiXn2l; exact/ltW.
by apply: ieexprIn; rewrite ?lt_eqF ?ltr_cpable.
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_iXn2l
| |
ltr_iXn2lx :
0 < x -> x < 1 -> {mono GRing.exp x : m n / (n < m)%N >-> m < n}.
Proof. by move=> xgt0 xlt1; apply: (leW_nmono (ler_iXn2l _ _)). 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_iXn2l
| |
lter_iXn2l:= (ler_iXn2l, ltr_iXn2l).
|
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_iXn2l
| |
ler_eXn2lx :
1 < x -> {mono GRing.exp x : m n / (m <= n)%N >-> m <= n}.
Proof.
move=> xgt1; apply: (le_mono (inj_homo_lt _ _)); last first.
by apply: ler_weXn2l; rewrite ltW.
by apply: ieexprIn; rewrite ?gt_eqF ?gtr_cpable //; apply: lt_trans xgt1.
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_eXn2l
| |
ltr_eXn2lx :
1 < x -> {mono (GRing.exp x) : m n / (m < n)%N >-> m < n}.
Proof. by move=> xgt1; apply: (leW_mono (ler_eXn2l _)). 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_eXn2l
| |
lter_eXn2l:= (ler_eXn2l, ltr_eXn2l).
|
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_eXn2l
| |
ltrXn2rn x y : 0 <= x -> x < y -> x ^+ n < y ^+ n = (n != 0).
Proof.
move=> xge0 xlty; case: n; first by rewrite ltxx.
elim=> [|n IHn]; rewrite ?[_ ^+ _.+2]exprS //.
rewrite (@le_lt_trans _ _ (x * y ^+ n.+1)) ?ler_wpM2l ?ltr_pM2r ?IHn //.
by rewrite ltW.
by rewrite exprn_gt0 // (le_lt_trans xge0).
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
|
ltrXn2r
| |
lerXn2rn : {in nneg & , {homo (@GRing.exp R)^~ n : x y / x <= y}}.
Proof.
move=> x y /= x0 y0 xy; elim: n => [|n IHn]; rewrite !(expr0, exprS) //.
by rewrite (@le_trans _ _ (x * y ^+ n)) ?ler_wpM2l ?ler_wpM2r ?exprn_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
|
lerXn2r
| |
lterXn2r:= (lerXn2r, ltrXn2r).
|
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
|
lterXn2r
| |
ltr_wpXn2rn :
(0 < n)%N -> {in nneg & , {homo (@GRing.exp R)^~ n : x y / x < y}}.
Proof. by move=> ngt0 x y /= x0 y0 hxy; rewrite ltrXn2r // -lt0n. 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_wpXn2r
| |
ler_pXn2rn :
(0 < n)%N -> {in nneg & , {mono (@GRing.exp R)^~ n : x y / x <= y}}.
Proof.
case: n => // n _ x y; rewrite !qualifE /= => x_ge0 y_ge0.
have [-> | nzx] := eqVneq x 0; first by rewrite exprS mul0r exprn_ge0.
rewrite -subr_ge0 subrXX pmulr_lge0 ?subr_ge0 //= big_ord_recr /=.
rewrite subnn expr0 mul1r /= ltr_pwDr // ?exprn_gt0 ?lt0r ?nzx //.
by rewrite sumr_ge0 // => i _; rewrite mulr_ge0 ?exprn_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_pXn2r
| |
ltr_pXn2rn :
(0 < n)%N -> {in nneg & , {mono (@GRing.exp R)^~ n : x y / x < y}}.
Proof.
by move=> n_gt0 x y x_ge0 y_ge0; rewrite !lt_neqAle !eq_le !ler_pXn2r.
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_pXn2r
| |
lter_pXn2r:= (ler_pXn2r, ltr_pXn2r).
|
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_pXn2r
| |
pexpIrnn : (0 < n)%N -> {in nneg &, injective ((@GRing.exp R)^~ n)}.
Proof. by move=> n_gt0; apply: inc_inj_in (ler_pXn2r _). 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
|
pexpIrn
| |
expr_le1n x : (0 < n)%N -> 0 <= x -> (x ^+ n <= 1) = (x <= 1).
Proof.
by move=> ngt0 xge0; rewrite -{1}[1](expr1n _ n) ler_pXn2r // [_ \in _]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
|
expr_le1
| |
expr_lt1n x : (0 < n)%N -> 0 <= x -> (x ^+ n < 1) = (x < 1).
Proof.
by move=> ngt0 xge0; rewrite -{1}[1](expr1n _ n) ltr_pXn2r // [_ \in _]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
|
expr_lt1
| |
expr_lte1:= (expr_le1, expr_lt1).
|
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
|
expr_lte1
| |
expr_ge1n x : (0 < n)%N -> 0 <= x -> (1 <= x ^+ n) = (1 <= x).
Proof.
by move=> ngt0 xge0; rewrite -{1}[1](expr1n _ n) ler_pXn2r // [_ \in _]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
|
expr_ge1
| |
expr_gt1n x : (0 < n)%N -> 0 <= x -> (1 < x ^+ n) = (1 < x).
Proof.
by move=> ngt0 xge0; rewrite -{1}[1](expr1n _ n) ltr_pXn2r // [_ \in _]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
|
expr_gt1
|
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