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 |
|---|---|---|---|---|---|---|
exprzACx m n : (x ^ m) ^ n = (x ^ n) ^ m.
Proof. by rewrite !exprz_exp mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprzAC
| |
exprz_outx n (nux : x \isn't a GRing.unit) (hn : 0 <= n) :
x ^ (- n) = x ^ n.
Proof. by case: (intP n) hn=> //= m; rewrite -exprnN -exprVn invr_out. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprz_out
| |
exprz_pMzlx m n : 0 <= n -> (x *~ m) ^ n = x ^ n *~ (m ^ n).
Proof.
by elim: n=> [|n ihn|n _] // _; rewrite !exprSz ihn // mulrzAr mulrzAl -mulrzA.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprz_pMzl
| |
exprz_pintlm n (hn : 0 <= n) : m%:~R ^ n = (m ^ n)%:~R :> R.
Proof. by rewrite exprz_pMzl // exp1rz. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprz_pintl
| |
exprzMzlx m n (ux : x \is a GRing.unit) (um : m%:~R \is a @GRing.unit R):
(x *~ m) ^ n = (m%:~R ^ n) * x ^ n :> R.
Proof. rewrite -[x *~ _]mulrzl exprMz_comm //; exact/commr_sym/commr_int. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprzMzl
| |
expNrzx n : (- x) ^ n = (-1) ^ n * x ^ n :> R.
Proof.
case: n=> [] n; rewrite ?NegzE; first exact: exprNn.
by rewrite -!exprz_inv !invrN invr1; apply: exprNn.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
expNrz
| |
unitr_n0expzx n :
n != 0 -> (x ^ n \is a GRing.unit) = (x \is a GRing.unit).
Proof.
by case: n => *; rewrite ?NegzE -?exprz_inv ?unitrX_pos ?unitrV ?lt0n.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
unitr_n0expz
| |
intrV(n : int) :
n \in [:: 0; 1; -1] -> n%:~R ^-1 = n%:~R :> R.
Proof.
by case: (intP n)=> // [|[]|[]] //; rewrite ?rmorphN ?invrN (invr0, invr1).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
intrV
| |
rmorphXz(R' : unitRingType) (f : {rmorphism R -> R'}) n :
{in GRing.unit, {morph f : x / x ^ n}}.
Proof. by case: n => n x Ux; rewrite ?rmorphV ?rpredX ?rmorphXn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
rmorphXz
| |
expfz_eq0x n : (x ^ n == 0) = (n != 0) && (x == 0).
Proof.
by case: n=> n; rewrite ?NegzE -?exprz_inv ?expf_eq0 ?lt0n ?invr_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
expfz_eq0
| |
expfz_neq0x n : x != 0 -> x ^ n != 0.
Proof. by move=> x_nz; rewrite expfz_eq0; apply/nandP; right. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
expfz_neq0
| |
exprzMlx y n (ux : x \is a GRing.unit) (uy : y \is a GRing.unit) :
(x * y) ^ n = x ^ n * y ^ n.
Proof. by rewrite exprMz_comm //; apply: mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprzMl
| |
expfV(x : R) (i : int) : (x ^ i) ^-1 = (x ^-1) ^ i.
Proof. by rewrite invr_expz exprz_inv. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
expfV
| |
expfzDrx m n : x != 0 -> x ^ (m + n) = x ^ m * x ^ n.
Proof. by move=> hx; rewrite exprzDr ?unitfE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
expfzDr
| |
expfz_n0addrx m n : m + n != 0 -> x ^ (m + n) = x ^ m * x ^ n.
Proof.
have [-> hmn|nx0 _] := eqVneq x 0; last exact: expfzDr.
rewrite !exp0rz (negPf hmn).
case: (eqVneq m 0) hmn => [->|]; rewrite (mul0r, mul1r) //.
by rewrite add0r=> /negPf->.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
expfz_n0addr
| |
expfzMlx y n : (x * y) ^ n = x ^ n * y ^ n.
Proof.
have [->|/negPf n0] := eqVneq n 0; first by rewrite !expr0z mulr1.
case: (boolP ((x * y) == 0)); rewrite ?mulf_eq0.
by case/pred2P=> ->; rewrite ?(mul0r, mulr0, exp0rz, n0).
by case/norP=> x0 y0; rewrite exprzMl ?unitfE.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
expfzMl
| |
fmorphXz(R : unitRingType) (f : {rmorphism F -> R}) n :
{morph f : x / x ^ n}.
Proof. by case: n => n x; rewrite ?fmorphV rmorphXn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
fmorphXz
| |
exprz_ge0n x (hx : 0 <= x) : (0 <= x ^ n).
Proof. by case: n => n; rewrite ?invr_ge0 ?exprn_ge0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprz_ge0
| |
exprz_gt0n x (hx : 0 < x) : (0 < x ^ n).
Proof. by case: n => n; rewrite ?invr_gt0 ?exprn_gt0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprz_gt0
| |
exprz_gte0:= (exprz_ge0, exprz_gt0).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
exprz_gte0
| |
ler_wpiXz2lx (x0 : 0 <= x) (x1 : x <= 1) :
{in >= 0 &, {homo exprz x : x y /~ x <= y}}.
Proof.
move=> [] m [] n; rewrite -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
by rewrite lez_nat -?exprnP => /ler_wiXn2l; apply.
Qed.
Fact ler_wpeXz2l x (x1 : 1 <= x) : {in >= 0 &, {homo exprz x : x y / x <= y}}.
Proof.
move=> [] m [] n; rewrite -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
by rewrite lez_nat -?exprnP=> /ler_weXn2l; apply.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_wpiXz2l
| |
pexprz_eq1x n (x0 : 0 <= x) : (x ^ n == 1) = ((n == 0) || (x == 1)).
Proof.
case: n=> n; rewrite ?NegzE -?exprz_inv ?oppr_eq0 pexprn_eq1 // ?invr_eq1 //.
by rewrite invr_ge0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
pexprz_eq1
| |
ler_wpXz2rn (hn : 0 <= n) :
{in >= 0 & , {homo (@exprz R)^~ n : x y / x <= y}}.
Proof. by case: n hn=> // n _; exact: lerXn2r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_wpXz2r
| |
ler_wniXz2lx (x0 : 0 <= x) (x1 : x <= 1) :
{in < 0 &, {homo exprz x : x y /~ x <= y}}.
Proof.
move=> [] m [] n; rewrite ?NegzE -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
rewrite lerN2 lez_nat -?invr_expz=> hmn; have := x0.
rewrite le0r=> /predU1P [->|lx0]; first by rewrite !exp0rz invr0.
by rewrite lef_pV2 -?topredE /= ?exprz_gt0 // ler_wiXn2l.
Qed.
Fact ler_wneXz2l x (x1 : 1 <= x) : {in <= 0 &, {homo exprz x : x y / x <= y}}.
Proof.
move=> m n hm hn /= hmn.
rewrite -lef_pV2 -?topredE /= ?exprz_gt0 ?(lt_le_trans ltr01) //.
by rewrite !invr_expz ler_wpeXz2l ?lerN2 -?topredE //= oppr_cp0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_wniXz2l
| |
ler_weXz2lx (x1 : 1 <= x) : {homo exprz x : x y / x <= y}.
Proof.
move=> m n /= hmn; case: (lerP 0 m)=> [|/ltW] hm.
by rewrite ler_wpeXz2l // [_ \in _](le_trans hm).
case: (lerP n 0)=> [|/ltW] hn.
by rewrite ler_wneXz2l // [_ \in _](le_trans hmn).
apply: (@le_trans _ _ (x ^ 0)); first by rewrite ler_wneXz2l.
by rewrite ler_wpeXz2l.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_weXz2l
| |
ieexprIzx (x0 : 0 < x) (nx1 : x != 1) : injective (exprz x).
Proof.
apply: wlog_lt=> // m n hmn; first by move=> hmn'; rewrite hmn.
move=> /(f_equal ( *%R^~ (x ^ (- n)))).
rewrite -!expfzDr ?gt_eqF // subrr expr0z=> /eqP.
by rewrite pexprz_eq1 ?(ltW x0) // (negPf nx1) subr_eq0 orbF=> /eqP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ieexprIz
| |
ler_piXz2lx (x0 : 0 < x) (x1 : x < 1) :
{in >= 0 &, {mono exprz x : x y /~ x <= y}}.
Proof.
apply: (le_nmono_in (inj_nhomo_lt_in _ _)).
by move=> n m hn hm /=; apply: ieexprIz; rewrite // lt_eqF.
by apply: ler_wpiXz2l; rewrite ?ltW.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_piXz2l
| |
ltr_piXz2lx (x0 : 0 < x) (x1 : x < 1) :
{in >= 0 &, {mono exprz x : x y /~ x < y}}.
Proof. exact: (leW_nmono_in (ler_piXz2l _ _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ltr_piXz2l
| |
ler_niXz2lx (x0 : 0 < x) (x1 : x < 1) :
{in < 0 &, {mono exprz x : x y /~ x <= y}}.
Proof.
apply: (le_nmono_in (inj_nhomo_lt_in _ _)).
by move=> n m hn hm /=; apply: ieexprIz; rewrite // lt_eqF.
by apply: ler_wniXz2l; rewrite ?ltW.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_niXz2l
| |
ltr_niXz2lx (x0 : 0 < x) (x1 : x < 1) :
{in < 0 &, {mono (exprz x) : x y /~ x < y}}.
Proof. exact: (leW_nmono_in (ler_niXz2l _ _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ltr_niXz2l
| |
ler_eXz2lx (x1 : 1 < x) : {mono exprz x : x y / x <= y}.
Proof.
apply: (le_mono (inj_homo_lt _ _)).
by apply: ieexprIz; rewrite ?(lt_trans ltr01) // gt_eqF.
by apply: ler_weXz2l; rewrite ?ltW.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_eXz2l
| |
ltr_eXz2lx (x1 : 1 < x) : {mono exprz x : x y / x < y}.
Proof. exact: (leW_mono (ler_eXz2l _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ltr_eXz2l
| |
ler_wnXz2rn (hn : n <= 0) :
{in > 0 & , {homo (@exprz R)^~ n : x y /~ x <= y}}.
Proof.
move=> x y /= hx hy hxy; rewrite -lef_pV2 ?[_ \in _]exprz_gt0 //.
by rewrite !invr_expz ler_wpXz2r ?[_ \in _]ltW // oppr_cp0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_wnXz2r
| |
pexpIrzn (n0 : n != 0) : {in >= 0 &, injective ((@exprz R)^~ n)}.
Proof.
move=> x y; rewrite ![_ \in _]le0r=> /predU1P [-> _ /eqP|hx].
by rewrite exp0rz ?(negPf n0) eq_sym expfz_eq0=> /andP [_ /eqP->].
case/predU1P=> [-> /eqP|hy].
by rewrite exp0rz ?(negPf n0) expfz_eq0=> /andP [_ /eqP].
move=> /(f_equal ( *%R^~ (y ^ (- n)))) /eqP.
rewrite -expfzDr ?(gt_eqF hy) // subrr expr0z -exprz_inv -expfzMl.
rewrite pexprz_eq1 ?(negPf n0) /= ?mulr_ge0 ?invr_ge0 ?ltW //.
by rewrite (can2_eq (mulrVK _) (mulrK _)) ?unitfE ?(gt_eqF hy) // mul1r=> /eqP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
pexpIrz
| |
nexpIrzn (n0 : n != 0) : {in <= 0 &, injective ((@exprz R)^~ n)}.
Proof.
move=> x y; rewrite ![_ \in _]le_eqVlt => /predU1P [-> _ /eqP|hx].
by rewrite exp0rz ?(negPf n0) eq_sym expfz_eq0=> /andP [_ /eqP->].
case/predU1P=> [-> /eqP|hy].
by rewrite exp0rz ?(negPf n0) expfz_eq0=> /andP [_ /eqP].
move=> /(f_equal ( *%R^~ (y ^ (- n)))) /eqP.
rewrite -expfzDr ?(lt_eqF hy) // subrr expr0z -exprz_inv -expfzMl.
rewrite pexprz_eq1 ?(negPf n0) /= ?mulr_le0 ?invr_le0 ?ltW //.
by rewrite (can2_eq (mulrVK _) (mulrK _)) ?unitfE ?(lt_eqF hy) // mul1r=> /eqP.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
nexpIrz
| |
ler_pXz2rn (hn : 0 < n) :
{in >= 0 & , {mono ((@exprz R)^~ n) : x y / x <= y}}.
Proof.
apply: le_mono_in (inj_homo_lt_in _ _).
by move=> x y hx hy /=; apply: pexpIrz; rewrite // gt_eqF.
by apply: ler_wpXz2r; rewrite ltW.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_pXz2r
| |
ltr_pXz2rn (hn : 0 < n) :
{in >= 0 & , {mono ((@exprz R)^~ n) : x y / x < y}}.
Proof. exact: leW_mono_in (ler_pXz2r _). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ltr_pXz2r
| |
ler_nXz2rn (hn : n < 0) :
{in > 0 & , {mono ((@exprz R)^~ n) : x y /~ x <= y}}.
Proof.
apply: le_nmono_in (inj_nhomo_lt_in _ _); last first.
by apply: ler_wnXz2r; rewrite ltW.
by move=> x y hx hy /=; apply: pexpIrz; rewrite ?[_ \in _]ltW ?lt_eqF.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ler_nXz2r
| |
ltr_nXz2rn (hn : n < 0) :
{in > 0 & , {mono ((@exprz R)^~ n) : x y /~ x < y}}.
Proof. exact: leW_nmono_in (ler_nXz2r _). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ltr_nXz2r
| |
eqrXz2n x y : n != 0 -> 0 <= x -> 0 <= y -> (x ^ n == y ^ n) = (x == y).
Proof. by move=> *; rewrite (inj_in_eq (pexpIrz _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
eqrXz2
| |
sgzx : int := if x == 0 then 0 else if x < 0 then -1 else 1.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz
| |
sgz_defx : sgz x = (-1) ^+ (x < 0)%R *+ (x != 0).
Proof. by rewrite /sgz; case: (_ == _); case: (_ < _). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_def
| |
sgrEzx : sgr x = (sgz x)%:~R. Proof. by rewrite !(fun_if intr). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgrEz
| |
gtr0_sgzx : 0 < x -> sgz x = 1.
Proof. by move=> x_gt0; rewrite /sgz lt_neqAle andbC eq_le lt_geF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
gtr0_sgz
| |
ltr0_sgzx : x < 0 -> sgz x = -1.
Proof. by move=> x_lt0; rewrite /sgz eq_sym eq_le x_lt0 lt_geF. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ltr0_sgz
| |
sgz0: sgz (0 : R) = 0. Proof. by rewrite /sgz eqxx. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz0
| |
sgz1: sgz (1 : R) = 1. Proof. by rewrite gtr0_sgz // ltr01. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz1
| |
sgzN1: sgz (-1 : R) = -1. Proof. by rewrite ltr0_sgz // ltrN10. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgzN1
| |
sgzE:= (sgz0, sgz1, sgzN1).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgzE
| |
sgz_sgrx : sgz (sgr x) = sgz x.
Proof. by rewrite !(fun_if sgz) !sgzE. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_sgr
| |
normr_sgzx : `|sgz x| = (x != 0).
Proof. by rewrite sgz_def -mulr_natr normrMsign normr_nat natz. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
normr_sgz
| |
normr_sgx : `|sgr x| = (x != 0)%:~R.
Proof. by rewrite sgr_def -mulr_natr normrMsign normr_nat. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
normr_sg
| |
sgz_intm : sgz (m%:~R : R) = sgz m.
Proof. by rewrite /sgz intr_eq0 ltrz0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_int
| |
sgrz(n : int) : sgr n = sgz n. Proof. by rewrite sgrEz intz. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgrz
| |
intr_sgm : (sgr m)%:~R = sgr (m%:~R) :> R.
Proof. by rewrite sgrz -sgz_int -sgrEz. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
intr_sg
| |
sgz_id(x : R) : sgz (sgz x) = sgz x.
Proof. by rewrite !(fun_if (@sgz _)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_id
| |
sgz_cp0x :
((sgz x == 1) = (0 < x)) *
((sgz x == -1) = (x < 0)) *
((sgz x == 0) = (x == 0)).
Proof. by rewrite /sgz; case: ltrgtP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_cp0
| |
sgz_valx : bool -> bool -> bool -> bool -> bool -> bool
-> bool -> bool -> bool -> bool -> bool -> bool
-> bool -> bool -> bool -> bool -> bool -> bool
-> R -> R -> int -> Set :=
| SgzNull of x = 0 : sgz_val x true true true true false false
true false false true false false true false false true false false 0 0 0
| SgzPos of x > 0 : sgz_val x false false true false false true
false false true false false true false false true false false true x 1 1
| SgzNeg of x < 0 : sgz_val x false true false false true false
false true false false true false false true false false true false (-x) (-1) (-1).
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_val
| |
sgzPx :
sgz_val x (0 == x) (x <= 0) (0 <= x) (x == 0) (x < 0) (0 < x)
(0 == sgr x) (-1 == sgr x) (1 == sgr x)
(sgr x == 0) (sgr x == -1) (sgr x == 1)
(0 == sgz x) (-1 == sgz x) (1 == sgz x)
(sgz x == 0) (sgz x == -1) (sgz x == 1) `|x| (sgr x) (sgz x).
Proof.
rewrite ![_ == sgz _]eq_sym ![_ == sgr _]eq_sym !sgr_cp0 !sgz_cp0.
by rewrite /sgz; case: sgrP; constructor.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgzP
| |
sgzNx : sgz (- x) = - sgz x.
Proof. by rewrite /sgz oppr_eq0 oppr_lt0; case: ltrgtP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgzN
| |
mulz_sgx : sgz x * sgz x = (x != 0)%:~R.
Proof. by case: sgzP; rewrite ?(mulr0, mulr1, mulrNN). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
mulz_sg
| |
mulz_sg_eq1x y : (sgz x * sgz y == 1) = (x != 0) && (sgz x == sgz y).
Proof.
do 2?case: sgzP=> _; rewrite ?(mulr0, mulr1, mulrN1, opprK, oppr0, eqxx);
by rewrite ?[0 == 1]eq_sym ?oner_eq0 //= eqr_oppLR oppr0 oner_eq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
mulz_sg_eq1
| |
mulz_sg_eqN1x y : (sgz x * sgz y == -1) = (x != 0) && (sgz x == - sgz y).
Proof. by rewrite -eqr_oppLR -mulrN -sgzN mulz_sg_eq1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
mulz_sg_eqN1
| |
sgzMx y : sgz (x * y) = sgz x * sgz y.
Proof.
rewrite -sgz_sgr -(sgz_sgr x) -(sgz_sgr y) sgrM.
by case: sgrP; case: sgrP; rewrite /sgz ?(mulNr, mul0r, mul1r);
rewrite ?(oppr_eq0, oppr_cp0, eqxx, ltxx, ltr01, ltr10, oner_eq0).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgzM
| |
sgzX(n : nat) x : sgz (x ^+ n) = (sgz x) ^+ n.
Proof. by elim: n => [|n IHn]; rewrite ?sgz1 // !exprS sgzM IHn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgzX
| |
sgz_eq0x : (sgz x == 0) = (x == 0).
Proof. by rewrite sgz_cp0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_eq0
| |
sgz_odd(n : nat) x : x != 0 -> (sgz x) ^+ n = (sgz x) ^+ (odd n).
Proof. by case: sgzP => //=; rewrite ?expr1n // signr_odd. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_odd
| |
sgz_gt0x : (sgz x > 0) = (x > 0).
Proof. by case: sgzP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_gt0
| |
sgz_lt0x : (sgz x < 0) = (x < 0).
Proof. by case: sgzP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_lt0
| |
sgz_ge0x : (sgz x >= 0) = (x >= 0).
Proof. by case: sgzP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_ge0
| |
sgz_le0x : (sgz x <= 0) = (x <= 0).
Proof. by case: sgzP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_le0
| |
sgz_smulx y : sgz (y *~ (sgz x)) = (sgz x) * (sgz y).
Proof. by rewrite -mulrzl sgzM -sgrEz sgz_sgr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_smul
| |
sgrMzm x : sgr (x *~ m) = sgr x *~ sgr m.
Proof. by rewrite -mulrzr sgrM -intr_sg mulrzr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgrMz
| |
sgz_eq(R R' : realDomainType) (x : R) (y : R') :
(sgz x == sgz y) = ((x == 0) == (y == 0)) && ((0 < x) == (0 < y)).
Proof. by do 2!case: sgzP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
sgz_eq
| |
intr_sign(R : pzRingType) s : ((-1) ^+ s)%:~R = (-1) ^+ s :> R.
Proof. exact: rmorph_sign. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
intr_sign
| |
absz_nat(n : nat) : `|n| = n. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
absz_nat
| |
abszE(m : int) : `|m| = `|m|%R :> int. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
abszE
| |
absz0: `|0%R| = 0. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
absz0
| |
abszNm : `|- m| = `|m|. Proof. by case: (normrN m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
abszN
| |
absz_eq0m : (`|m| == 0) = (m == 0%R). Proof. by case: (intP m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
absz_eq0
| |
absz_gt0m : (`|m| > 0) = (m != 0%R). Proof. by case: (intP m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
absz_gt0
| |
absz1: `|1%R| = 1. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
absz1
| |
abszN1: `|-1%R| = 1. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
abszN1
| |
absz_idm : `|(`|m|)| = `|m|. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
absz_id
| |
abszMm1 m2 : `|(m1 * m2)%R| = `|m1| * `|m2|.
Proof. by case: m1 m2 => [[|m1]|m1] [[|m2]|m2] //=; rewrite ?mulnS mulnC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
abszM
| |
abszX(n : nat) m : `|m ^+ n| = `|m| ^ n.
Proof. by elim: n => // n ihn; rewrite exprS expnS abszM ihn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
abszX
| |
absz_sgm : `|sgr m| = (m != 0%R). Proof. by case: (intP m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
absz_sg
| |
gez0_absm : (0 <= m)%R -> `|m| = m :> int.
Proof. by case: (intP m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
gez0_abs
| |
gtz0_absm : (0 < m)%R -> `|m| = m :> int.
Proof. by case: (intP m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
gtz0_abs
| |
lez0_absm : (m <= 0)%R -> `|m| = - m :> int.
Proof. by case: (intP m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
lez0_abs
| |
ltz0_absm : (m < 0)%R -> `|m| = - m :> int.
Proof. by case: (intP m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
ltz0_abs
| |
lez_absm : m <= `|m|%N :> int.
Proof. by case: (intP m). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
lez_abs
| |
absz_signs : `|(-1) ^+ s| = 1.
Proof. by rewrite abszX exp1n. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
absz_sign
| |
abszMsigns m : `|((-1) ^+ s * m)%R| = `|m|.
Proof. by rewrite abszM absz_sign mul1n. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
abszMsign
| |
mulz_sign_absm : ((-1) ^+ (m < 0)%R * `|m|%:Z)%R = m.
Proof. by rewrite abszE mulr_sign_norm. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
mulz_sign_abs
| |
mulz_Nsign_absm : ((-1) ^+ (0 < m)%R * `|m|%:Z)%R = - m.
Proof. by rewrite abszE mulr_Nsign_norm. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
mulz_Nsign_abs
| |
intEsignm : m = ((-1) ^+ (m < 0)%R * `|m|%:Z)%R.
Proof. exact: numEsign. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
intEsign
| |
abszEsignm : `|m|%:Z = ((-1) ^+ (m < 0)%R * m)%R.
Proof. exact: normrEsign. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
abszEsign
| |
intEsgm : m = (sgz m * `|m|%:Z)%R.
Proof. by rewrite -sgrz -numEsg. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
intEsg
| |
abszEsgm : (`|m|%:Z = sgz m * m)%R.
Proof. by rewrite -sgrz -normrEsg. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] |
algebra/ssrint.v
|
abszEsg
|
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