fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
oppr_lte0:= (oppr_le0, oppr_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
|
oppr_lte0
| |
oppr_cp0:= (oppr_gte0, oppr_lte0).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
oppr_cp0
| |
lterNE:= (oppr_cp0, lterN2).
|
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
|
lterNE
| |
ge0_cpx : 0 <= x -> (- x <= 0) * (- x <= x).
Proof. by move=> hx; rewrite oppr_cp0 hx (@le_trans _ _ 0) ?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
|
ge0_cp
| |
gt0_cpx : 0 < x ->
(0 <= x) * (- x <= 0) * (- x <= x) * (- x < 0) * (- x < x).
Proof.
move=> hx; move: (ltW hx) => hx'; rewrite !ge0_cp hx' //.
by rewrite oppr_cp0 hx // (@lt_trans _ _ 0) ?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
|
gt0_cp
| |
le0_cpx : x <= 0 -> (0 <= - x) * (x <= - x).
Proof. by move=> hx; rewrite oppr_cp0 hx (@le_trans _ _ 0) ?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
|
le0_cp
| |
lt0_cpx :
x < 0 -> (x <= 0) * (0 <= - x) * (x <= - x) * (0 < - x) * (x < - x).
Proof.
move=> hx; move: (ltW hx) => hx'; rewrite !le0_cp // hx'.
by rewrite oppr_cp0 hx // (@lt_trans _ _ 0) ?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
|
lt0_cp
| |
ger0_realx : 0 <= x -> x \is real.
Proof. by rewrite realE => ->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ger0_real
| |
ler0_realx : x <= 0 -> x \is real.
Proof. by rewrite realE 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
|
ler0_real
| |
gtr0_realx : 0 < x -> x \is real. Proof. by move=> /ltW/ger0_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
gtr0_real
| |
ltr0_realx : x < 0 -> x \is real. Proof. by move=> /ltW/ler0_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltr0_real
| |
real0: 0 \is @real R. Proof. exact: rpred0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real0
| |
real1: 1 \is @real R. Proof. exact: rpred1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real1
| |
realnn : n%:R \is @real R. Proof. exact: rpred_nat. Qed.
#[local] Hint Resolve real0 real1 : core.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
realn
| |
ler_leVgex y : x <= 0 -> y <= 0 -> (x <= y) || (y <= x).
Proof. by rewrite -!oppr_ge0 => /(ger_leVge _) /[apply]; rewrite !lerN2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_leVge
| |
real_leVgex y : x \is real -> y \is real -> (x <= y) || (y <= x).
Proof. by rewrite -comparabler0 -comparable0r => /comparabler_trans P/P. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_leVge
| |
real_comparablex y : x \is real -> y \is real -> x >=< y.
Proof. exact: real_leVge. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_comparable
| |
realB: {in real &, forall x y, x - y \is real}.
Proof. exact: rpredB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
realB
| |
realN: {mono (@GRing.opp R) : x / x \is real}.
Proof. exact: rpredN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
realN
| |
realBCx y : (x - y \is real) = (y - x \is real).
Proof. exact: rpredBC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
realBC
| |
realD: {in real &, forall x y, x + y \is real}.
Proof. exact: rpredD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
realD
| |
ler_xor_gt(x y : R) : R -> R -> R -> R -> R -> R ->
bool -> bool -> Set :=
| LerNotGt of x <= y : ler_xor_gt x y x x y y (y - x) (y - x) true false
| GtrNotLe of y < x : ler_xor_gt x y y y x x (x - y) (x - y) false true.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_xor_gt
| |
ltr_xor_ge(x y : R) : R -> R -> R -> R -> R -> R ->
bool -> bool -> Set :=
| LtrNotGe of x < y : ltr_xor_ge x y x x y y (y - x) (y - x) false true
| GerNotLt of y <= x : ltr_xor_ge x y y y x x (x - y) (x - y) true false.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_xor_ge
| |
comparerx y : R -> R -> R -> R -> R -> R ->
bool -> bool -> bool -> bool -> bool -> bool -> Set :=
| ComparerLt of x < y : comparer x y x x y y (y - x) (y - x)
false false false true false true
| ComparerGt of x > y : comparer x y y y x x (x - y) (x - y)
false false true false true false
| ComparerEq of x = y : comparer x y x x x x 0 0
true true true true false false.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
comparer
| |
real_lePx y : x \is real -> y \is real ->
ler_xor_gt x y (min y x) (min x y) (max y x) (max x y)
`|x - y| `|y - x| (x <= y) (y < x).
Proof.
move=> xR yR; case: (comparable_leP (real_leVge xR yR)) => xy.
- by rewrite [`|x - y|]distrC !ger0_norm ?subr_cp0 //; constructor.
- by rewrite [`|y - x|]distrC !gtr0_norm ?subr_cp0 //; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_leP
| |
real_ltPx y : x \is real -> y \is real ->
ltr_xor_ge x y (min y x) (min x y) (max y x) (max x y)
`|x - y| `|y - x| (y <= x) (x < y).
Proof. by move=> xR yR; case: real_leP=> //; constructor. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_ltP
| |
real_ltNge: {in real &, forall x y, (x < y) = ~~ (y <= x)}.
Proof. by move=> x y xR yR /=; case: real_leP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_ltNge
| |
real_leNgt: {in real &, forall x y, (x <= y) = ~~ (y < x)}.
Proof. by move=> x y xR yR /=; case: real_leP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_leNgt
| |
real_ltgtPx y : x \is real -> y \is real ->
comparer x y (min y x) (min x y) (max y x) (max x y) `|x - y| `|y - x|
(y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y).
Proof.
move=> xR yR; case: (comparable_ltgtP (real_leVge yR xR)) => [?|?|->].
- by rewrite [`|y - x|]distrC !gtr0_norm ?subr_gt0//; constructor.
- by rewrite [`|x - y|]distrC !gtr0_norm ?subr_gt0//; constructor.
- by rewrite subrr normr0; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_ltgtP
| |
ger0_xor_lt0(x : R) : R -> R -> R -> R -> R ->
bool -> bool -> Set :=
| Ger0NotLt0 of 0 <= x : ger0_xor_lt0 x 0 0 x x x false true
| Ltr0NotGe0 of x < 0 : ger0_xor_lt0 x x x 0 0 (- x) true false.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ger0_xor_lt0
| |
ler0_xor_gt0(x : R) : R -> R -> R -> R -> R ->
bool -> bool -> Set :=
| Ler0NotLe0 of x <= 0 : ler0_xor_gt0 x x x 0 0 (- x) false true
| Gtr0NotGt0 of 0 < x : ler0_xor_gt0 x 0 0 x x x true false.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler0_xor_gt0
| |
comparer0x : R -> R -> R -> R -> R ->
bool -> bool -> bool -> bool -> bool -> bool -> Set :=
| ComparerGt0 of 0 < x : comparer0 x 0 0 x x x false false false true false true
| ComparerLt0 of x < 0 : comparer0 x x x 0 0 (- x) false false true false true false
| ComparerEq0 of x = 0 : comparer0 x 0 0 0 0 0 true true true true false false.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
comparer0
| |
real_ge0Px : x \is real -> ger0_xor_lt0 x
(min 0 x) (min x 0) (max 0 x) (max x 0)
`|x| (x < 0) (0 <= x).
Proof.
move=> hx; rewrite -[X in `|X|]subr0; case: real_leP;
by rewrite ?subr0 ?sub0r //; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_ge0P
| |
real_le0Px : x \is real -> ler0_xor_gt0 x
(min 0 x) (min x 0) (max 0 x) (max x 0)
`|x| (0 < x) (x <= 0).
Proof.
move=> hx; rewrite -[X in `|X|]subr0; case: real_ltP;
by rewrite ?subr0 ?sub0r //; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_le0P
| |
real_ltgt0Px : x \is real ->
comparer0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
`|x| (0 == x) (x == 0) (x <= 0) (0 <= x) (x < 0) (x > 0).
Proof.
move=> hx; rewrite -[X in `|X|]subr0; case: (@real_ltgtP 0 x);
by rewrite ?subr0 ?sub0r //; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_ltgt0P
| |
max_real: {in real &, forall x y, max x y \is real}.
Proof. exact: comparable_maxr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
max_real
| |
min_real: {in real &, forall x y, min x y \is real}.
Proof. exact: comparable_minr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
min_real
| |
bigmax_realI x0 (r : seq I) (P : pred I) (f : I -> R):
x0 \is real -> {in P, forall i : I, f i \is real} ->
\big[max/x0]_(i <- r | P i) f i \is real.
Proof. exact/big_real/max_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
bigmax_real
| |
bigmin_realI x0 (r : seq I) (P : pred I) (f : I -> R):
x0 \is real -> {in P, forall i : I, f i \is real} ->
\big[min/x0]_(i <- r | P i) f i \is real.
Proof. exact/big_real/min_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
bigmin_real
| |
real_neqr_lt: {in real &, forall x y, (x != y) = (x < y) || (y < x)}.
Proof. by move=> * /=; case: real_ltgtP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_neqr_lt
| |
lerB_realx y : x <= y -> y - x \is real.
Proof. by move=> le_xy; rewrite ger0_real // 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
|
lerB_real
| |
gerB_realx y : x <= y -> x - y \is real.
Proof. by move=> le_xy; rewrite ler0_real // subr_le0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
gerB_real
| |
ler_realy x : x <= y -> (x \is real) = (y \is real).
Proof. by move=> le_xy; rewrite -(addrNK x y) rpredDl ?lerB_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler_real
| |
ger_realx y : y <= x -> (x \is real) = (y \is real).
Proof. by move=> le_yx; rewrite -(ler_real le_yx). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_real
| |
ger1_realx : 1 <= x -> x \is real. Proof. by move=> /ger_real->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ger1_real
| |
ler1_realx : x <= 1 -> x \is real. Proof. by move=> /ler_real->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ler1_real
| |
Nreal_leFx y : y \is real -> x \notin real -> (x <= y) = false.
Proof. by move=> yR; apply: contraNF=> /ler_real->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
Nreal_leF
| |
Nreal_geFx y : y \is real -> x \notin real -> (y <= x) = false.
Proof. by move=> yR; apply: contraNF=> /ger_real->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
Nreal_geF
| |
Nreal_ltFx y : y \is real -> x \notin real -> (x < y) = false.
Proof. by move=> yR xNR; rewrite lt_def Nreal_leF ?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
|
Nreal_ltF
| |
Nreal_gtFx y : y \is real -> x \notin real -> (y < x) = false.
Proof. by move=> yR xNR; rewrite lt_def Nreal_geF ?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
|
Nreal_gtF
| |
real_wlog_lerP :
(forall a b, P b a -> P a b) -> (forall a b, a <= b -> P a b) ->
forall a b : R, a \is real -> b \is real -> P a b.
Proof.
move=> sP hP a b ha hb; wlog: a b ha hb / a <= b => [hwlog|]; last exact: hP.
by case: (real_leP ha hb)=> [/hP //|/ltW hba]; apply/sP/hP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_wlog_ler
| |
real_wlog_ltrP :
(forall a, P a a) -> (forall a b, (P b a -> P a b)) ->
(forall a b, a < b -> P a b) ->
forall a b : R, a \is real -> b \is real -> P a b.
Proof.
move=> rP sP hP; apply: real_wlog_ler=> // a b.
by rewrite le_eqVlt; case: eqVneq => [->|] //= _ /hP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_wlog_ltr
| |
lerD2lx : {mono +%R x : y z / y <= z}.
Proof. by move=> y z; rewrite -subr_ge0 opprD addrAC addNKr addrC subr_ge0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerD2l
| |
lerD2rx : {mono +%R^~ x : y z / y <= z}.
Proof. by move=> y z; rewrite ![_ + x]addrC lerD2l. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerD2r
| |
ltrD2lx : {mono +%R x : y z / y < z}.
Proof. by move=> y z; rewrite (leW_mono (lerD2l _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrD2l
| |
ltrD2rx : {mono +%R^~ x : y z / y < z}.
Proof. by move=> y z /=; rewrite (leW_mono (lerD2r _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrD2r
| |
lerD2:= (lerD2l, lerD2r).
|
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
|
lerD2
| |
ltrD2:= (ltrD2l, ltrD2r).
|
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
|
ltrD2
| |
lterD2:= (lerD2, ltrD2).
|
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
|
lterD2
| |
lerDx y z t : x <= y -> z <= t -> x + z <= y + t.
Proof. by move=> lxy lzt; rewrite (@le_trans _ _ (y + z)) ?lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerD
| |
ler_ltDx y z t : x <= y -> z < t -> x + z < y + t.
Proof. by move=> lxy lzt; rewrite (@le_lt_trans _ _ (y + z)) ?lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_ltD
| |
ltr_leDx y z t : x < y -> z <= t -> x + z < y + t.
Proof. by move=> lxy lzt; rewrite (@lt_le_trans _ _ (y + z)) ?lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_leD
| |
ltrDx y z t : x < y -> z < t -> x + z < y + t.
Proof. by move=> lxy lzt; rewrite ltr_leD // 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
|
ltrD
| |
lerBx y z t : x <= y -> t <= z -> x - z <= y - t.
Proof. by move=> lxy ltz; rewrite lerD // lterN2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerB
| |
ler_ltBx y z t : x <= y -> t < z -> x - z < y - t.
Proof. by move=> lxy lzt; rewrite ler_ltD // lterN2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_ltB
| |
ltr_leBx y z t : x < y -> t <= z -> x - z < y - t.
Proof. by move=> lxy lzt; rewrite ltr_leD // lterN2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_leB
| |
ltrBx y z t : x < y -> t < z -> x - z < y - t.
Proof. by move=> lxy lzt; rewrite ltrD // lterN2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrB
| |
lerBlDrx y z : (x - y <= z) = (x <= z + y).
Proof. by rewrite (monoLR (addrK _) (lerD2r _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerBlDr
| |
ltrBlDrx y z : (x - y < z) = (x < z + y).
Proof. by rewrite (monoLR (addrK _) (ltrD2r _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrBlDr
| |
lerBrDrx y z : (x <= y - z) = (x + z <= y).
Proof. by rewrite (monoLR (addrNK _) (lerD2r _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerBrDr
| |
ltrBrDrx y z : (x < y - z) = (x + z < y).
Proof. by rewrite (monoLR (addrNK _) (ltrD2r _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrBrDr
| |
lerBDr:= (lerBlDr, lerBrDr).
|
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
|
lerBDr
| |
ltrBDr:= (ltrBlDr, ltrBrDr).
|
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
|
ltrBDr
| |
lterBDr:= (lerBDr, ltrBDr).
|
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
|
lterBDr
| |
lerBlDlx y z : (x - y <= z) = (x <= y + z).
Proof. by rewrite lterBDr addrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerBlDl
| |
ltrBlDlx y z : (x - y < z) = (x < y + z).
Proof. by rewrite lterBDr addrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrBlDl
| |
lerBrDlx y z : (x <= y - z) = (z + x <= y).
Proof. by rewrite lerBrDr addrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerBrDl
| |
ltrBrDlx y z : (x < y - z) = (z + x < y).
Proof. by rewrite lterBDr addrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrBrDl
| |
lerBDl:= (lerBlDl, lerBrDl).
|
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
|
lerBDl
| |
ltrBDl:= (ltrBlDl, ltrBrDl).
|
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
|
ltrBDl
| |
lterBDl:= (lerBDl, ltrBDl).
|
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
|
lterBDl
| |
lerDlx y : (x <= x + y) = (0 <= y).
Proof. by rewrite -{1}[x]addr0 lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerDl
| |
ltrDlx y : (x < x + y) = (0 < y).
Proof. by rewrite -{1}[x]addr0 lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrDl
| |
lerDrx y : (x <= y + x) = (0 <= y).
Proof. by rewrite -{1}[x]add0r lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
lerDr
| |
ltrDrx y : (x < y + x) = (0 < y).
Proof. by rewrite -{1}[x]add0r lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ltrDr
| |
gerDlx y : (x + y <= x) = (y <= 0).
Proof. by rewrite -{2}[x]addr0 lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
gerDl
| |
gerBlx y : (x - y <= x) = (0 <= y).
Proof. by rewrite lerBlDl lerDr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
gerBl
| |
gtrDlx y : (x + y < x) = (y < 0).
Proof. by rewrite -{2}[x]addr0 lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
gtrDl
| |
gtrBlx y : (x - y < x) = (0 < y).
Proof. by rewrite ltrBlDl ltrDr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
gtrBl
| |
gerDrx y : (y + x <= x) = (y <= 0).
Proof. by rewrite -{2}[x]add0r lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
gerDr
| |
gtrDrx y : (y + x < x) = (y < 0).
Proof. by rewrite -{2}[x]add0r lterD2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
gtrDr
| |
cprD:= (lerDl, lerDr, gerDl, gerDl,
ltrDl, ltrDr, gtrDl, gtrDl).
|
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
|
cprD
| |
ler_wpDly x z : 0 <= x -> y <= z -> y <= x + z.
Proof. by move=> *; rewrite -[y]add0r 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_wpDl
| |
ltr_wpDly x z : 0 <= x -> y < z -> y < x + z.
Proof. by move=> *; rewrite -[y]add0r ler_ltD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_wpDl
| |
ltr_pwDly x z : 0 < x -> y <= z -> y < x + z.
Proof. by move=> *; rewrite -[y]add0r ltr_leD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_pwDl
| |
ltr_pDly x z : 0 < x -> y < z -> y < x + z.
Proof. by move=> *; rewrite -[y]add0r ltrD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_pDl
| |
ler_wnDly x z : x <= 0 -> y <= z -> x + y <= z.
Proof. by move=> *; rewrite -[z]add0r 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_wnDl
| |
ltr_wnDly x z : x <= 0 -> y < z -> x + y < z.
Proof. by move=> *; rewrite -[z]add0r ler_ltD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_wnDl
| |
ltr_nwDly x z : x < 0 -> y <= z -> x + y < z.
Proof. by move=> *; rewrite -[z]add0r ltr_leD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_nwDl
| |
ltr_nDly x z : x < 0 -> y < z -> x + y < z.
Proof. by move=> *; rewrite -[z]add0r ltrD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"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_nDl
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.