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 |
|---|---|---|---|---|---|---|
mulrzlx n : n%:~R * x = x *~ n. Proof. by rewrite mulrzAl mul1r. 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
|
mulrzl
| |
mulrzrx n : x * n%:~R = x *~ n. Proof. by rewrite mulrzAr mulr1. 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
|
mulrzr
| |
mulNrNzn x : (- x) *~ (- n) = x *~ n.
Proof. by rewrite mulNrz mulrNz opprK. 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
|
mulNrNz
| |
mulrbzx (b : bool) : x *~ b = (if b then x else 0).
Proof. by case: b. 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
|
mulrbz
| |
intrNn : (- n)%:~R = - n%:~R :> R. Proof. exact: mulrNz. 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
|
intrN
| |
intrDm n : (m + n)%:~R = m%:~R + n%:~R :> R. Proof. exact: mulrzDr. 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
|
intrD
| |
intrBm n : (m - n)%:~R = m%:~R - n%:~R :> R. Proof. exact: mulrzBr. 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
|
intrB
| |
intrMm n : (m * n)%:~R = m%:~R * n%:~R :> R.
Proof. by rewrite mulrzA -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
|
intrM
| |
intmul1_is_monoid_morphism: monoid_morphism ( *~%R (1 : R)).
Proof. by split; move=> // x y /=; rewrite ?intrD ?mulrNz ?intrM. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `intmul1_is_monoid_morphism` instead")]
|
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
|
intmul1_is_monoid_morphism
| |
intmul1_is_multiplicative:=
(fun g => (g.2,g.1)) intmul1_is_monoid_morphism.
HB.instance Definition _ := GRing.isMonoidMorphism.Build int R ( *~%R 1)
intmul1_is_monoid_morphism.
|
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
|
intmul1_is_multiplicative
| |
mulr2zn : n *~ 2 = n + n. Proof. exact: mulr2n. 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
|
mulr2z
| |
mulrzzm n : m *~ n = m * n. Proof. by rewrite -mulrzr 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
|
mulrzz
| |
mulz2n : n * 2%:Z = n + n. Proof. by rewrite -mulrzz. 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
|
mulz2
| |
mul2zn : 2%:Z * n = n + n. Proof. by rewrite mulrC -mulrzz. 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
|
mul2z
| |
scaler_intn v : n%:~R *: v = v *~ n.
Proof. by case: n => n; rewrite /intmul ?scaleNr scaler_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
|
scaler_int
| |
scalerMzla v n : (a *: v) *~ n = (a *~ n) *: v.
Proof. by rewrite -mulrzl -scaler_int scalerA. 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
|
scalerMzl
| |
scalerMzra v n : (a *: v) *~ n = a *: (v *~ n).
Proof. by rewrite -!scaler_int !scalerA mulrzr mulrzl. 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
|
scalerMzr
| |
mulrz_int(M : zmodType) (n : int) (x : M) : x *~ n%:~R = x *~ n.
Proof. by rewrite -scalezrE scaler_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
|
mulrz_int
| |
raddfMzn : {morph f : x / x *~ n}.
Proof. by case: n=> n x; rewrite 1?raddfN raddfMn. 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
|
raddfMz
| |
rmorphMz: forall n, {morph f : x / x *~ n}. Proof. exact: raddfMz. 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
|
rmorphMz
| |
rmorph_int: forall n, f n%:~R = n%:~R.
Proof. by move=> n; rewrite rmorphMz rmorph1. 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
|
rmorph_int
| |
linearMn: forall n, {morph f : x / x *~ n}. Proof. exact: raddfMz. 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
|
linearMn
| |
raddf_int_scalable(aV rV : lmodType int) (f : {additive aV -> rV}) :
scalable f.
Proof. by move=> z u; rewrite -[z]intz !scaler_int raddfMz. 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
|
raddf_int_scalable
| |
commrMz(x y : R) n : GRing.comm x y -> GRing.comm x (y *~ n).
Proof. by rewrite /GRing.comm=> com_xy; rewrite mulrzAr mulrzAl com_xy. 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
|
commrMz
| |
commr_int(x : R) n : GRing.comm x n%:~R.
Proof. exact/commrMz/commr1. 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
|
commr_int
| |
sumMz: forall I r (P : pred I) F,
(\sum_(i <- r | P i) F i)%N%:~R = \sum_(i <- r | P i) ((F i)%:~R) :> R.
Proof. exact: rmorph_sum. 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
|
sumMz
| |
prodMz: forall I r (P : pred I) F,
(\prod_(i <- r | P i) F i)%N%:~R = \prod_(i <- r | P i) ((F i)%:~R) :> R.
Proof. exact: rmorph_prod. 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
|
prodMz
| |
pFrobenius_autMzx n : (x *~ n)^f = x^f *~ n.
Proof.
case: n=> n /=; first exact: pFrobenius_autMn.
by rewrite !NegzE !mulrNz pFrobenius_autN pFrobenius_autMn.
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
|
pFrobenius_autMz
| |
pFrobenius_aut_intn : (n%:~R)^f = n%:~R.
Proof. by rewrite pFrobenius_autMz pFrobenius_aut1. 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
|
pFrobenius_aut_int
| |
Frobenius_autMz:= (pFrobenius_autMz) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_aut_int instead.")]
|
Notation
|
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
|
Frobenius_autMz
| |
Frobenius_aut_int:= (pFrobenius_aut_int) (only parsing).
|
Notation
|
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
|
Frobenius_aut_int
| |
rmorphzP(f : {rmorphism int -> R}) : f =1 ( *~%R 1).
Proof. by move=> n; rewrite -[n in LHS]intz rmorph_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
|
rmorphzP
| |
ler_pMz2rn (hn : 0 < n) : {mono *~%R^~ n :x y / x <= y :> R}.
Proof. by move=> x y; case: n hn=> [[]|] // n _; rewrite ler_pMn2r. 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_pMz2r
| |
ltr_pMz2rn (hn : 0 < n) : {mono *~%R^~ n : x y / x < y :> R}.
Proof. exact: leW_mono (ler_pMz2r _). 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_pMz2r
| |
ler_nMz2rn (hn : n < 0) : {mono *~%R^~ n : x y /~ x <= y :> R}.
Proof.
by move=> x y /=; rewrite -![_ *~ n]mulNrNz ler_pMz2r (oppr_cp0, lerN2).
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_nMz2r
| |
ltr_nMz2rn (hn : n < 0) : {mono *~%R^~ n : x y /~ x < y :> R}.
Proof. exact: leW_nmono (ler_nMz2r _). 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_nMz2r
| |
ler_wpMz2rn (hn : 0 <= n) : {homo *~%R^~ n : x y / x <= y :> R}.
Proof. by move=> x y xy; case: n hn=> [] // n _; rewrite ler_wMn2r. 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_wpMz2r
| |
ler_wnMz2rn (hn : n <= 0) : {homo *~%R^~ n : x y /~ x <= y :> R}.
Proof. by move=> x y xy /=; rewrite -lerN2 -!mulrNz ler_wpMz2r // oppr_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
|
ler_wnMz2r
| |
mulrz_ge0x n (x0 : 0 <= x) (n0 : 0 <= n) : 0 <= x *~ n.
Proof. by rewrite -(mul0rz _ n) ler_wpMz2r. 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
|
mulrz_ge0
| |
mulrz_le0x n (x0 : x <= 0) (n0 : n <= 0) : 0 <= x *~ n.
Proof. by rewrite -(mul0rz _ n) ler_wnMz2r. 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
|
mulrz_le0
| |
mulrz_ge0_le0x n (x0 : 0 <= x) (n0 : n <= 0) : x *~ n <= 0.
Proof. by rewrite -(mul0rz _ n) ler_wnMz2r. 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
|
mulrz_ge0_le0
| |
mulrz_le0_ge0x n (x0 : x <= 0) (n0 : 0 <= n) : x *~ n <= 0.
Proof. by rewrite -(mul0rz _ n) ler_wpMz2r. 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
|
mulrz_le0_ge0
| |
pmulrz_lgt0x n (n0 : 0 < n) : 0 < x *~ n = (0 < x).
Proof. by rewrite -(mul0rz _ n) ltr_pMz2r // mul0rz. 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
|
pmulrz_lgt0
| |
nmulrz_lgt0x n (n0 : n < 0) : 0 < x *~ n = (x < 0).
Proof. by rewrite -(mul0rz _ n) ltr_nMz2r // mul0rz. 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
|
nmulrz_lgt0
| |
pmulrz_llt0x n (n0 : 0 < n) : x *~ n < 0 = (x < 0).
Proof. by rewrite -(mul0rz _ n) ltr_pMz2r // mul0rz. 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
|
pmulrz_llt0
| |
nmulrz_llt0x n (n0 : n < 0) : x *~ n < 0 = (0 < x).
Proof. by rewrite -(mul0rz _ n) ltr_nMz2r // mul0rz. 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
|
nmulrz_llt0
| |
pmulrz_lge0x n (n0 : 0 < n) : 0 <= x *~ n = (0 <= x).
Proof. by rewrite -(mul0rz _ n) ler_pMz2r // mul0rz. 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
|
pmulrz_lge0
| |
nmulrz_lge0x n (n0 : n < 0) : 0 <= x *~ n = (x <= 0).
Proof. by rewrite -(mul0rz _ n) ler_nMz2r // mul0rz. 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
|
nmulrz_lge0
| |
pmulrz_lle0x n (n0 : 0 < n) : x *~ n <= 0 = (x <= 0).
Proof. by rewrite -(mul0rz _ n) ler_pMz2r // mul0rz. 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
|
pmulrz_lle0
| |
nmulrz_lle0x n (n0 : n < 0) : x *~ n <= 0 = (0 <= x).
Proof. by rewrite -(mul0rz _ n) ler_nMz2r // mul0rz. 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
|
nmulrz_lle0
| |
ler_wpMz2lx (hx : 0 <= x) : {homo *~%R x : x y / x <= y}.
Proof.
by move=> m n /= hmn; rewrite -subr_ge0 -mulrzBr mulrz_ge0 // subr_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
|
ler_wpMz2l
| |
ler_wnMz2lx (hx : x <= 0) : {homo *~%R x : x y /~ x <= y}.
Proof.
by move=> m n /= hmn; rewrite -subr_ge0 -mulrzBr mulrz_le0 // subr_le0.
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_wnMz2l
| |
ler_pMz2lx (hx : 0 < x) : {mono *~%R x : x y / x <= y}.
Proof.
move=> m n /=; rewrite real_mono ?num_real // => {m n}.
by move=> m n /= hmn; rewrite -subr_gt0 -mulrzBr pmulrz_lgt0 // subr_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
|
ler_pMz2l
| |
ler_nMz2lx (hx : x < 0) : {mono *~%R x : x y /~ x <= y}.
Proof.
move=> m n /=; rewrite real_nmono ?num_real // => {m n}.
by move=> m n /= hmn; rewrite -subr_gt0 -mulrzBr nmulrz_lgt0 // subr_lt0.
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_nMz2l
| |
ltr_pMz2lx (hx : 0 < x) : {mono *~%R x : x y / x < y}.
Proof. exact: leW_mono (ler_pMz2l _). 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_pMz2l
| |
ltr_nMz2lx (hx : x < 0) : {mono *~%R x : x y /~ x < y}.
Proof. exact: leW_nmono (ler_nMz2l _). 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_nMz2l
| |
pmulrz_rgt0x n (x0 : 0 < x) : 0 < x *~ n = (0 < n).
Proof. by rewrite -(mulr0z x) ltr_pMz2l. 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
|
pmulrz_rgt0
| |
nmulrz_rgt0x n (x0 : x < 0) : 0 < x *~ n = (n < 0).
Proof. by rewrite -(mulr0z x) ltr_nMz2l. 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
|
nmulrz_rgt0
| |
pmulrz_rlt0x n (x0 : 0 < x) : x *~ n < 0 = (n < 0).
Proof. by rewrite -(mulr0z x) ltr_pMz2l. 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
|
pmulrz_rlt0
| |
nmulrz_rlt0x n (x0 : x < 0) : x *~ n < 0 = (0 < n).
Proof. by rewrite -(mulr0z x) ltr_nMz2l. 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
|
nmulrz_rlt0
| |
pmulrz_rge0x n (x0 : 0 < x) : 0 <= x *~ n = (0 <= n).
Proof. by rewrite -(mulr0z x) ler_pMz2l. 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
|
pmulrz_rge0
| |
nmulrz_rge0x n (x0 : x < 0) : 0 <= x *~ n = (n <= 0).
Proof. by rewrite -(mulr0z x) ler_nMz2l. 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
|
nmulrz_rge0
| |
pmulrz_rle0x n (x0 : 0 < x) : x *~ n <= 0 = (n <= 0).
Proof. by rewrite -(mulr0z x) ler_pMz2l. 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
|
pmulrz_rle0
| |
nmulrz_rle0x n (x0 : x < 0) : x *~ n <= 0 = (0 <= n).
Proof. by rewrite -(mulr0z x) ler_nMz2l. 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
|
nmulrz_rle0
| |
mulrIzx (hx : x != 0) : injective ( *~%R x).
Proof.
move=> y z; rewrite -![x *~ _]mulrzr => /(mulfI hx).
by apply: inc_inj y z; exact: ler_pMz2l.
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
|
mulrIz
| |
ler_intm n : (m%:~R <= n%:~R :> R) = (m <= n).
Proof. by rewrite ler_pMz2l. 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_int
| |
ltr_intm n : (m%:~R < n%:~R :> R) = (m < n).
Proof. by rewrite ltr_pMz2l. 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_int
| |
eqr_intm n : (m%:~R == n%:~R :> R) = (m == n).
Proof. by rewrite (inj_eq (mulrIz _)) ?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
|
eqr_int
| |
ler0zn : (0 <= n%:~R :> R) = (0 <= n).
Proof. by rewrite pmulrz_rge0. 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
|
ler0z
| |
ltr0zn : (0 < n%:~R :> R) = (0 < n).
Proof. by rewrite pmulrz_rgt0. 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
|
ltr0z
| |
lerz0n : (n%:~R <= 0 :> R) = (n <= 0).
Proof. by rewrite pmulrz_rle0. 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
|
lerz0
| |
ltrz0n : (n%:~R < 0 :> R) = (n < 0).
Proof. by rewrite pmulrz_rlt0. 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
|
ltrz0
| |
ler1z(n : int) : (1 <= n%:~R :> R) = (1 <= n).
Proof. by rewrite -[1]/(1%:~R) ler_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
|
ler1z
| |
ltr1z(n : int) : (1 < n%:~R :> R) = (1 < n).
Proof. by rewrite -[1]/(1%:~R) ltr_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
|
ltr1z
| |
lerz1n : (n%:~R <= 1 :> R) = (n <= 1).
Proof. by rewrite -[1]/(1%:~R) ler_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
|
lerz1
| |
ltrz1n : (n%:~R < 1 :> R) = (n < 1).
Proof. by rewrite -[1]/(1%:~R) ltr_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
|
ltrz1
| |
intr_eq0n : (n%:~R == 0 :> R) = (n == 0).
Proof. by rewrite -(mulr0z 1) (inj_eq (mulrIz _)) // 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
|
intr_eq0
| |
mulrz_eq0x n : (x *~ n == 0) = ((n == 0) || (x == 0)).
Proof. by rewrite -mulrzl mulf_eq0 intr_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
|
mulrz_eq0
| |
mulrz_neq0x n : x *~ n != 0 = ((n != 0) && (x != 0)).
Proof. by rewrite mulrz_eq0 negb_or. 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
|
mulrz_neq0
| |
realzn : (n%:~R : R) \in Num.real.
Proof. by rewrite -topredE /Num.real /= ler0z lerz0 le_total. Qed.
Hint Resolve realz : core.
|
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
|
realz
| |
intr_inj:= @mulrIz 1 (oner_neq0 R).
|
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
|
intr_inj
| |
exprz(R : unitRingType) (x : R) (n : int) :=
match n with
| Posz n => x ^+ n
| Negz n => x ^- (n.+1)
end.
Arguments exprz : simpl never.
|
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
| |
exprnPx (n : nat) : x ^+ n = x ^ 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
|
exprnP
| |
exprnNx (n : nat) : x ^- n = x ^ (-n%:Z).
Proof. by case: n=> //; rewrite oppr0 expr0 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
|
exprnN
| |
expr0zx : x ^ 0 = 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
|
expr0z
| |
expr1zx : x ^ 1 = x. 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
|
expr1z
| |
exprN1x : x ^ (-1) = x^-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
|
exprN1
| |
invr_expzx n : (x ^ n)^-1 = x ^ (- n).
Proof. by case: (intP n)=> // [|m]; rewrite ?opprK ?expr0z ?invr1 // invrK. 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
|
invr_expz
| |
exprz_invx n : (x^-1) ^ n = x ^ (- n).
Proof. by case: (intP n)=> // m; rewrite -[_ ^ (- _)]exprVn ?opprK ?invrK. 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_inv
| |
exp1rzn : 1 ^ n = 1 :> R.
Proof. by case: (intP n)=> // m; rewrite -?exprz_inv ?invr1; apply: expr1n. 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
|
exp1rz
| |
exprSzx (n : nat) : x ^ n.+1 = x * x ^ n. Proof. exact: exprS. 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
|
exprSz
| |
exprSzrx (n : nat) : x ^ n.+1 = x ^ n * x. Proof. exact: exprSr. Qed.
Fact exprzD_nat x (m n : nat) : x ^ (m%:Z + n) = x ^ m * x ^ n.
Proof. exact: exprD. Qed.
Fact exprzD_Nnat x (m n : nat) : x ^ (-m%:Z + -n%:Z) = x ^ (-m%:Z) * x ^ (-n%:Z).
Proof. by rewrite -opprD -!exprz_inv exprzD_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
|
exprSzr
| |
exprzD_ssx m n : (0 <= m) && (0 <= n) || (m <= 0) && (n <= 0)
-> x ^ (m + n) = x ^ m * x ^ n.
Proof.
case: (intP m)=> {m} [|m|m]; case: (intP n)=> {n} [|n|n] //= _;
by rewrite ?expr0z ?mul1r ?exprzD_nat ?exprzD_Nnat ?sub0r ?addr0 ?mulr1.
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
|
exprzD_ss
| |
exp0rzn : 0 ^ n = (n == 0)%:~R :> R.
Proof. by case: (intP n)=> // m; rewrite -?exprz_inv ?invr0 exprSz mul0r. 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
|
exp0rz
| |
commrXzx y n : GRing.comm x y -> GRing.comm x (y ^ n).
Proof.
rewrite /GRing.comm; elim: n x y=> [|n ihn|n ihn] x y com_xy //=.
* by rewrite expr0z mul1r mulr1.
* by rewrite -exprnP commrX //.
rewrite -exprz_inv -exprnP commrX //.
case: (boolP (y \is a GRing.unit))=> uy; last by rewrite invr_out.
by apply/eqP; rewrite (can2_eq (mulrVK _) (mulrK _)) // -mulrA com_xy mulKr.
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
|
commrXz
| |
exprMz_commx y n : x \is a GRing.unit -> y \is a GRing.unit ->
GRing.comm x y -> (x * y) ^ n = x ^ n * y ^ n.
Proof.
move=> ux uy com_xy; elim: n => [|n _|n _]; first by rewrite expr0z mulr1.
by rewrite -!exprnP exprMn_comm.
rewrite -!exprnN -!exprVn com_xy -exprMn_comm ?invrM//.
exact/commrV/commr_sym/commrV.
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
|
exprMz_comm
| |
commrXz_wmullsx y n :
0 <= n -> GRing.comm x y -> (x * y) ^ n = x ^ n * y ^ n.
Proof.
move=> n0 com_xy; elim: n n0 => [|n _|n _] //; first by rewrite expr0z mulr1.
by rewrite -!exprnP exprMn_comm.
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
|
commrXz_wmulls
| |
unitrXzx n (ux : x \is a GRing.unit) : x ^ n \is a GRing.unit.
Proof.
case: (intP n)=> {n} [|n|n]; rewrite ?expr0z ?unitr1 ?unitrX //.
by rewrite -invr_expz unitrV unitrX.
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
|
unitrXz
| |
exprzDrx (ux : x \is a GRing.unit) m n : x ^ (m + n) = x ^ m * x ^ n.
Proof.
move: n m; apply: wlog_le=> n m hnm.
by rewrite addrC hnm commrXz //; exact/commr_sym/commrXz.
case: (intP m) hnm=> {m} [|m|m]; rewrite ?mul1r ?add0r //;
case: (intP n)=> {n} [|n|n _]; rewrite ?mulr1 ?addr0 //;
do ?by rewrite exprzD_ss.
rewrite -invr_expz subzSS !exprSzr invrM ?unitrX // -mulrA mulVKr //.
case: (leqP n m)=> [|/ltnW] hmn; rewrite -{2}(subnK hmn) exprzD_nat -subzn //.
by rewrite mulrK ?unitrX.
by rewrite invrM ?unitrXz // mulVKr ?unitrXz // -opprB -invr_expz.
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
|
exprzDr
| |
exprz_expx m n : (x ^ m) ^ n = (x ^ (m * n)).
Proof.
wlog: n / 0 <= n.
by case: n=> [n -> //|n]; rewrite ?NegzE mulrN -?invr_expz=> -> /=.
elim: n x m=> [|n ihn|n ihn] x m // _; first by rewrite mulr0 !expr0z.
rewrite exprSz ihn // intS mulrDr mulr1 exprzD_ss //.
by case: (intP m)=> // m'; rewrite ?oppr_le0 //.
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_exp
|
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