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 |
|---|---|---|---|---|---|---|
pFrobenius_autEx : x^f = x ^+ p. Proof. by []. Qed.
Local Notation f'E := pFrobenius_autE.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_autE
| |
pFrobenius_aut0: 0^f = 0.
Proof. by rewrite f'E -(prednK (prime_gt0 pcharf_prime)) exprS mul0r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_aut0
| |
pFrobenius_aut1: 1^f = 1.
Proof. by rewrite f'E expr1n. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_aut1
| |
pFrobenius_autD_commx y (cxy : comm x y) : (x + y)^f = x^f + y^f.
Proof.
have defp := prednK (prime_gt0 pcharf_prime).
rewrite !f'E exprDn_comm // big_ord_recr subnn -defp big_ord_recl /= defp.
rewrite subn0 mulr1 mul1r bin0 binn big1 ?addr0 // => i _.
by rewrite -mulr_natl bin_lt_pcharf_0 ?mul0r //= -{2}defp ltnS (valP i).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_autD_comm
| |
pFrobenius_autMnx n : (x *+ n)^f = x^f *+ n.
Proof.
elim: n => [|n IHn]; first exact: pFrobenius_aut0.
by rewrite !mulrS pFrobenius_autD_comm ?IHn //; apply: commrMn.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_autMn
| |
pFrobenius_aut_natn : (n%:R)^f = n%:R.
Proof. by rewrite pFrobenius_autMn pFrobenius_aut1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_aut_nat
| |
pFrobenius_autM_commx y : comm x y -> (x * y)^f = x^f * y^f.
Proof. exact: exprMn_comm. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_autM_comm
| |
pFrobenius_autXx n : (x ^+ n)^f = x^f ^+ n.
Proof. by rewrite !f'E -!exprM mulnC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_autX
| |
addrr_pchar2x : x + x = 0. Proof. by rewrite -mulr2n mulrn_pchar. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
addrr_pchar2
| |
DefinitionPzRing := { R of PzSemiRing R & Zmodule R }.
HB.factory Record Zmodule_isPzRing R of Zmodule R := {
one : R;
mul : R -> R -> R;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul +%R;
mulrDr : right_distributive mul +%R;
}.
HB.builders Context R of Zmodule_isPzRing R.
Local Notation "1" := one.
Local Notation "x * y" := (mul x y).
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
mul0r: @left_zero R R 0 mul.
Proof. by move=> x; apply: (addIr (1 * x)); rewrite -mulrDl !add0r mul1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mul0r
| |
mulr0: @right_zero R R 0 mul.
Proof. by move=> x; apply: (addIr (x * 1)); rewrite -mulrDr !add0r mulr1. Qed.
HB.instance Definition _ := Nmodule_isPzSemiRing.Build R
mulrA mul1r mulr1 mulrDl mulrDr mul0r mulr0.
HB.end.
HB.factory Record isPzRing R of Choice R := {
zero : R;
opp : R -> R;
add : R -> R -> R;
one : R;
mul : R -> R -> R;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
addNr : left_inverse zero opp add;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
}.
HB.builders Context R of isPzRing R.
HB.instance Definition _ := @isZmodule.Build R
zero opp add addrA addrC add0r addNr.
HB.instance Definition _ := @Zmodule_isPzRing.Build R
one mul mulrA mul1r mulr1 mulrDl mulrDr.
HB.end.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr0
| |
Frobenius_aut:= pFrobenius_aut (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pcharf0 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_aut
| |
charf0:= pcharf0 (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pcharf_prime instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
charf0
| |
charf_prime:= pcharf_prime (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use mulrn_pchar instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
charf_prime
| |
mulrn_char:= mulrn_pchar (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use natr_mod_pchar instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrn_char
| |
natr_mod_char:= natr_mod_pchar (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use dvdn_pcharf instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
natr_mod_char
| |
dvdn_charf:= dvdn_pcharf (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pcharf_eq instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
dvdn_charf
| |
charf_eq:= pcharf_eq (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use bin_lt_pcharf_0 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
charf_eq
| |
bin_lt_charf_0:= bin_lt_pcharf_0 (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_autE instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
bin_lt_charf_0
| |
Frobenius_autE:= pFrobenius_autE (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_aut0 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_autE
| |
Frobenius_aut0:= pFrobenius_aut0 (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_aut1 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_aut0
| |
Frobenius_aut1:= pFrobenius_aut1 (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_autD_comm instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_aut1
| |
Frobenius_autD_comm:= pFrobenius_autD_comm (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_autMn instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_autD_comm
| |
Frobenius_autMn:= pFrobenius_autMn (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_aut_nat instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_autMn
| |
Frobenius_aut_nat:= pFrobenius_aut_nat (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_autM_comm instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_aut_nat
| |
Frobenius_autM_comm:= pFrobenius_autM_comm (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_autX instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_autM_comm
| |
Frobenius_autX:= pFrobenius_autX (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use addrr_pchar2 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_autX
| |
addrr_char2:= addrr_pchar2 (only parsing).
#[short(type="nzRingType")]
HB.structure Definition NzRing := { R of NzSemiRing R & Zmodule R }.
#[deprecated(since="mathcomp 2.4.0",
note="Use NzRing instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
addrr_char2
| |
RingR := (NzRing R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Ring
| |
sort:= (NzRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use NzRing.on instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sort
| |
onR := (NzRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use NzRing.copy instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
on
| |
copyT U := (NzRing.copy T U) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
copy
| |
RecordZmodule_isNzRing R of Zmodule R := {
one : R;
mul : R -> R -> R;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul +%R;
mulrDr : right_distributive mul +%R;
oner_neq0 : one != 0
}.
|
HB.factory
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
BuildR := (Zmodule_isNzRing.Build R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Build
| |
Zmodule_isRingR := (Zmodule_isNzRing R) (only parsing).
HB.builders Context R of Zmodule_isNzRing R.
HB.instance Definition _ := Zmodule_isPzRing.Build R
mulrA mul1r mulr1 mulrDl mulrDr.
HB.instance Definition _ := PzSemiRing_isNonZero.Build R oner_neq0.
HB.end.
HB.factory Record isNzRing R of Choice R := {
zero : R;
opp : R -> R;
add : R -> R -> R;
one : R;
mul : R -> R -> R;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
addNr : left_inverse zero opp add;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
oner_neq0 : one != zero
}.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Zmodule_isRing
| |
BuildR := (isNzRing.Build R) (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Build
| |
isRingR := (isNzRing R) (only parsing).
HB.builders Context R of isNzRing R.
HB.instance Definition _ := @isZmodule.Build R
zero opp add addrA addrC add0r addNr.
HB.instance Definition _ := @Zmodule_isNzRing.Build R
one mul mulrA mul1r mulr1 mulrDl mulrDr oner_neq0.
HB.end.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
isRing
| |
signR b := (exp (- @one R) (nat_of_bool b)) (only parsing).
Local Notation "- 1" := (- (1)) : ring_scope.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sign
| |
mulrNx y : x * (- y) = - (x * y).
Proof. by apply: (addrI (x * y)); rewrite -mulrDr !subrr mulr0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrN
| |
mulNrx y : (- x) * y = - (x * y).
Proof. by apply: (addrI (x * y)); rewrite -mulrDl !subrr mul0r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulNr
| |
mulrNNx y : (- x) * (- y) = x * y.
Proof. by rewrite mulrN mulNr opprK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrNN
| |
mulN1rx : -1 * x = - x.
Proof. by rewrite mulNr mul1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulN1r
| |
mulrN1x : x * -1 = - x.
Proof. by rewrite mulrN mulr1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrN1
| |
mulrBlx y z : (y - z) * x = y * x - z * x.
Proof. by rewrite mulrDl mulNr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrBl
| |
mulrBrx y z : x * (y - z) = x * y - x * z.
Proof. by rewrite mulrDr mulrN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrBr
| |
natrBm n : n <= m -> (m - n)%:R = m%:R - n%:R :> R.
Proof. exact: mulrnBr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
natrB
| |
commrNx y : comm x y -> comm x (- y).
Proof. by move=> com_xy; rewrite /comm mulrN com_xy mulNr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
commrN
| |
commrN1x : comm x (-1). Proof. exact/commrN/commr1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
commrN1
| |
commrBx y z : comm x y -> comm x z -> comm x (y - z).
Proof. by move=> com_xy com_xz; apply: commrD => //; apply: commrN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
commrB
| |
commr_signx n : comm x ((-1) ^+ n).
Proof. exact: (commrX n (commrN1 x)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
commr_sign
| |
signr_oddn : (-1) ^+ (odd n) = (-1) ^+ n :> R.
Proof.
elim: n => //= n IHn; rewrite exprS -{}IHn.
by case/odd: n; rewrite !mulN1r ?opprK.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
signr_odd
| |
mulr_sign(b : bool) x : (-1) ^+ b * x = (if b then - x else x).
Proof. by case: b; rewrite ?mulNr mul1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr_sign
| |
signr_addbb1 b2 : (-1) ^+ (b1 (+) b2) = (-1) ^+ b1 * (-1) ^+ b2 :> R.
Proof. by rewrite mulr_sign; case: b1 b2 => [] []; rewrite ?opprK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
signr_addb
| |
signrE(b : bool) : (-1) ^+ b = 1 - b.*2%:R :> R.
Proof. by case: b; rewrite ?subr0 // opprD addNKr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
signrE
| |
signrNb : (-1) ^+ (~~ b) = - (-1) ^+ b :> R.
Proof. by case: b; rewrite ?opprK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
signrN
| |
mulr_signM(b1 b2 : bool) x1 x2 :
((-1) ^+ b1 * x1) * ((-1) ^+ b2 * x2) = (-1) ^+ (b1 (+) b2) * (x1 * x2).
Proof.
by rewrite signr_addb -!mulrA; congr (_ * _); rewrite !mulrA commr_sign.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr_signM
| |
exprNnx n : (- x) ^+ n = (-1) ^+ n * x ^+ n :> R.
Proof. by rewrite -mulN1r exprMn_comm // /comm mulN1r mulrN mulr1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
exprNn
| |
sqrrNx : (- x) ^+ 2 = x ^+ 2. Proof. exact: mulrNN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sqrrN
| |
sqrr_signn : ((-1) ^+ n) ^+ 2 = 1 :> R.
Proof. by rewrite exprAC sqrrN !expr1n. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sqrr_sign
| |
signrMKn : @involutive R ( *%R ((-1) ^+ n)).
Proof. by move=> x; rewrite mulrA -expr2 sqrr_sign mul1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
signrMK
| |
mulrI0_lregx : (forall y, x * y = 0 -> y = 0) -> lreg x.
Proof.
move=> reg_x y z eq_xy_xz; apply/eqP; rewrite -subr_eq0 [y - z]reg_x //.
by rewrite mulrBr eq_xy_xz subrr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulrI0_lreg
| |
lregNx : lreg x -> lreg (- x).
Proof. by move=> reg_x y z; rewrite !mulNr => /oppr_inj/reg_x. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
lregN
| |
lreg_signn : lreg ((-1) ^+ n : R). Proof. exact/lregX/lregN/lreg1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
lreg_sign
| |
prodrN(I : finType) (A : pred I) (F : I -> R) :
\prod_(i in A) - F i = (- 1) ^+ #|A| * \prod_(i in A) F i.
Proof.
rewrite -sum1_card; elim/big_rec3: _ => [|i x n _ _ ->]; first by rewrite mulr1.
by rewrite exprS !mulrA mulN1r !mulNr commrX //; apply: commrN1.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
prodrN
| |
exprBn_commx y n (cxy : comm x y) :
(x - y) ^+ n =
\sum_(i < n.+1) ((-1) ^+ i * x ^+ (n - i) * y ^+ i) *+ 'C(n, i).
Proof.
rewrite exprDn_comm; last exact: commrN.
by apply: eq_bigr => i _; congr (_ *+ _); rewrite -commr_sign -mulrA -exprNn.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
exprBn_comm
| |
subrXX_commx y n (cxy : comm x y) :
x ^+ n - y ^+ n = (x - y) * (\sum_(i < n) x ^+ (n.-1 - i) * y ^+ i).
Proof.
case: n => [|n]; first by rewrite big_ord0 mulr0 subrr.
rewrite mulrBl !big_distrr big_ord_recl big_ord_recr /= subnn mulr1 mul1r.
rewrite subn0 -!exprS opprD -!addrA; congr (_ + _); rewrite addrA -sumrB.
rewrite big1 ?add0r // => i _; rewrite !mulrA -exprS -subSn ?(valP i) //.
by rewrite subSS (commrX _ (commr_sym cxy)) -mulrA -exprS subrr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subrXX_comm
| |
subrX1x n : x ^+ n - 1 = (x - 1) * (\sum_(i < n) x ^+ i).
Proof.
rewrite -!(opprB 1) mulNr -{1}(expr1n _ n).
rewrite (subrXX_comm _ (commr_sym (commr1 x))); congr (- (_ * _)).
by apply: eq_bigr => i _; rewrite expr1n mul1r.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subrX1
| |
sqrrB1x : (x - 1) ^+ 2 = x ^+ 2 - x *+ 2 + 1.
Proof. by rewrite -sqrrN opprB addrC sqrrD1 sqrrN mulNrn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
sqrrB1
| |
subr_sqr_1x : x ^+ 2 - 1 = (x - 1) * (x + 1).
Proof. by rewrite subrX1 !big_ord_recr big_ord0 /= addrAC add0r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subr_sqr_1
| |
smulr_closed:= -1 \in S /\ mulr_2closed S.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
smulr_closed
| |
subring_closed:= [/\ 1 \in S, subr_2closed S & mulr_2closed S].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subring_closed
| |
smulr_closedM: smulr_closed -> mulr_closed S.
Proof. by case=> SN1 SM; split=> //; rewrite -[1]mulr1 -mulrNN SM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
smulr_closedM
| |
smulr_closedN: smulr_closed -> oppr_closed S.
Proof. by case=> SN1 SM x Sx; rewrite -mulN1r SM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
smulr_closedN
| |
subring_closedB: subring_closed -> zmod_closed S.
Proof. by case=> S1 SB _; split; rewrite // -(subrr 1) SB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subring_closedB
| |
subring_closedM: subring_closed -> smulr_closed.
Proof.
by case=> S1 SB SM; split; rewrite ?(zmod_closedN (subring_closedB _)).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subring_closedM
| |
subring_closed_semi: subring_closed -> semiring_closed S.
Proof.
by move=> ringS; split; [apply/zmod_closedD/subring_closedB | case: ringS].
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subring_closed_semi
| |
signr_eq0n : ((-1) ^+ n == 0 :> R) = false.
Proof. by rewrite -signr_odd; case: odd; rewrite ?oppr_eq0 oner_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
signr_eq0
| |
pFrobenius_autNx : (- x)^f = - x^f.
Proof.
apply/eqP; rewrite -subr_eq0 opprK addrC.
by rewrite -(pFrobenius_autD_comm _ (commrN _)) // subrr pFrobenius_aut0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_autN
| |
pFrobenius_autB_commx y : comm x y -> (x - y)^f = x^f - y^f.
Proof.
by move/commrN/pFrobenius_autD_comm->; rewrite pFrobenius_autN.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
pFrobenius_autB_comm
| |
exprNn_pcharx n : (pchar R).-nat n -> (- x) ^+ n = - (x ^+ n).
Proof.
pose p := pdiv n; have [|n_gt1 pcharRn] := leqP n 1; first by case: (n) => [|[]].
have pcharRp: p \in pchar R by rewrite (pnatPpi pcharRn) // pi_pdiv.
have /p_natP[e ->]: p.-nat n by rewrite -(eq_pnat _ (pcharf_eq pcharRp)).
elim: e => // e IHe; rewrite expnSr !exprM {}IHe.
by rewrite -pFrobenius_autE pFrobenius_autN.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
exprNn_pchar
| |
oppr_pchar2x : - x = x.
Proof. by apply/esym/eqP; rewrite -addr_eq0 addrr_pchar2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
oppr_pchar2
| |
subr_pchar2x y : x - y = x + y. Proof. by rewrite oppr_pchar2. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subr_pchar2
| |
addrK_pchar2x : involutive (+%R^~ x).
Proof. by move=> y; rewrite /= -subr_pchar2 addrK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
addrK_pchar2
| |
addKr_pchar2x : involutive (+%R x).
Proof. by move=> y; rewrite -{1}[x]oppr_pchar2 addKr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
addKr_pchar2
| |
Frobenius_autN:= pFrobenius_autN (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_autB_comm instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_autN
| |
Frobenius_autB_comm:= pFrobenius_autB_comm (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use exprNn_pchar instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Frobenius_autB_comm
| |
exprNn_char:= exprNn_pchar (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use oppr_pchar2 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
exprNn_char
| |
oppr_char2:= oppr_pchar2 (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use subr_pchar2 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
oppr_char2
| |
subr_char2:= subr_pchar2 (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use addrK_pchar2 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subr_char2
| |
addrK_char2:= addrK_pchar2 (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use addKr_pchar2 instead.")]
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
addrK_char2
| |
addKr_char2:= addKr_pchar2 (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
addKr_char2
| |
Definition_ (T : eqType) := Equality.on T^c.
#[export]
HB.instance Definition _ (T : choiceType) := Choice.on T^c.
#[export]
HB.instance Definition _ (U : nmodType) := Nmodule.on U^c.
#[export]
HB.instance Definition _ (U : zmodType) := Zmodule.on U^c.
#[export]
HB.instance Definition _ (R : pzSemiRingType) :=
let mul' (x y : R) := y * x in
let mulrA' x y z := esym (mulrA z y x) in
let mulrDl' x y z := mulrDr z x y in
let mulrDr' x y z := mulrDl y z x in
Nmodule_isPzSemiRing.Build R^c
mulrA' mulr1 mul1r mulrDl' mulrDr' mulr0 mul0r.
#[export]
HB.instance Definition _ (R : pzRingType) := PzSemiRing.on R^c.
#[export]
HB.instance Definition _ (R : nzSemiRingType) :=
PzSemiRing_isNonZero.Build R^c oner_neq0.
#[export]
HB.instance Definition _ (R : nzRingType) := NzSemiRing.on R^c.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
rev_prodr(R : pzSemiRingType)
(I : Type) (r : seq I) (P : pred I) (E : I -> R) :
\prod_(i <- r | P i) (E i : R^c) = \prod_(i <- rev r | P i) E i.
Proof. by rewrite rev_big_rev. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rev_prodr
| |
mulIr_eq0x y : rreg x -> (y * x == 0) = (y == 0).
Proof. exact: (@mulrI_eq0 R^c). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulIr_eq0
| |
rreg1: rreg (1 : R).
Proof. exact: (@lreg1 R^c). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rreg1
| |
rregMx y : rreg x -> rreg y -> rreg (x * y).
Proof. by move=> reg_x reg_y; apply: (@lregM R^c). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rregM
| |
revrXx n : (x : R^c) ^+ n = (x : R) ^+ n.
Proof. by elim: n => // n IHn; rewrite exprS exprSr IHn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
revrX
| |
rregXx n : rreg x -> rreg (x ^+ n).
Proof. by move/(@lregX R^c x n); rewrite revrX. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rregX
| |
rreg_neq0(R : nzSemiRingType) (x : R) : rreg x -> x != 0.
Proof. exact: (@lreg_neq0 R^c). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rreg_neq0
|
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