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 |
|---|---|---|---|---|---|---|
telescope_sumr_eqn m (f u : nat -> V) : n <= m ->
(forall k, (n <= k < m)%N -> u k = f k.+1 - f k) ->
\sum_(n <= k < m) u k = f m - f n.
Proof. exact: telescope_sumr_eq. 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
|
telescope_sumr_eq
| |
oppr_closed:= oppr_closed 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
|
oppr_closed
| |
subr_2closed:= subr_closed 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
|
subr_2closed
| |
zmod_closed:= zmod_closed 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
|
zmod_closed
| |
zmod_closedN: zmod_closed -> oppr_closed.
Proof. exact: zmod_closedN. 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
|
zmod_closedN
| |
zmod_closedD: zmod_closed -> nmod_closed S.
Proof. by move=> z; split; [case: z|apply/zmod_closedD]. 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
|
zmod_closedD
| |
RecordNmodule_isPzSemiRing R of Nmodule 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;
mul0r : left_zero zero mul;
mulr0 : right_zero zero mul;
}.
#[short(type="pzSemiRingType")]
HB.structure Definition PzSemiRing :=
{ R of Nmodule_isPzSemiRing R & Nmodule R }.
HB.factory Record isPzSemiRing R of Choice R := {
zero : R;
add : R -> R -> R;
one : R;
mul : R -> R -> R;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
mul0r : left_zero zero mul;
mulr0 : right_zero zero mul;
}.
HB.builders Context R of isPzSemiRing R.
HB.instance Definition _ := @isNmodule.Build R
zero add addrA addrC add0r.
HB.instance Definition _ := @Nmodule_isPzSemiRing.Build R
one mul mulrA mul1r mulr1 mulrDl mulrDr mul0r mulr0.
HB.end.
|
HB.mixin
|
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
| |
RecordPzSemiRing_isNonZero R of PzSemiRing R := {
oner_neq0 : @one R != 0
}.
#[short(type="nzSemiRingType")]
HB.structure Definition NzSemiRing :=
{ R of PzSemiRing_isNonZero R & PzSemiRing R }.
#[deprecated(since="mathcomp 2.4.0",
note="Use NzSemiRing instead.")]
|
HB.mixin
|
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
| |
SemiRingR := (NzSemiRing 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
|
SemiRing
| |
sort:= (NzSemiRing.sort) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use NzSemiRing.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 := (NzSemiRing.on R) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use NzSemiRing.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 := (NzSemiRing.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
| |
RecordNmodule_isNzSemiRing R of Nmodule 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;
mul0r : left_zero zero mul;
mulr0 : right_zero zero mul;
oner_neq0 : one != 0
}.
HB.builders Context R of Nmodule_isNzSemiRing R.
HB.instance Definition _ :=
Nmodule_isPzSemiRing.Build R mulrA mul1r mulr1 mulrDl mulrDr mul0r mulr0.
HB.instance Definition _ := PzSemiRing_isNonZero.Build R oner_neq0.
HB.end.
|
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 := (Nmodule_isNzSemiRing.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
| |
Nmodule_isSemiRingR := (Nmodule_isNzSemiRing R) (only parsing).
HB.factory Record isNzSemiRing R of Choice R := {
zero : R;
add : R -> R -> R;
one : R;
mul : R -> R -> R;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
mul0r : left_zero zero mul;
mulr0 : right_zero zero mul;
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
|
Nmodule_isSemiRing
| |
BuildR := (isNzSemiRing.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
| |
isSemiRingR := (isNzSemiRing R) (only parsing).
HB.builders Context R of isNzSemiRing R.
HB.instance Definition _ := @isNmodule.Build R
zero add addrA addrC add0r.
HB.instance Definition _ := @Nmodule_isNzSemiRing.Build R
one mul mulrA mul1r mulr1 mulrDl mulrDr mul0r mulr0 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
|
isSemiRing
| |
expR x n := iterop n (@mul R) x (@one R).
Arguments exp : simpl never.
|
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
|
exp
| |
commR x y := @mul R x y = mul y x.
|
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
|
comm
| |
lregR x := injective (@mul R x).
|
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
|
lreg
| |
rregR x := injective ((@mul R)^~ x).
Local Notation "1" := (@one _) : ring_scope.
Local Notation "n %:R" := (1 *+ n) : ring_scope.
Local Notation "*%R" := (@mul _) : function_scope.
Local Notation "x * y" := (mul x y) : ring_scope.
Local Notation "x ^+ n" := (exp x n) : ring_scope.
Local Notation "\prod_ ( i <- r | P ) F" := (\big[*%R/1]_(i <- r | P) F).
Local Notation "\prod_ ( i | P ) F" := (\big[*%R/1]_(i | P) F).
Local Notation "\prod_ ( i 'in' A ) F" := (\big[*%R/1]_(i in A) F).
Local Notation "\prod_ ( m <= i < n ) F" := (\big[*%R/1%R]_(m <= i < n) F%R).
|
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
|
rreg
| |
pchar(R : nzSemiRingType) : nat_pred :=
[pred p | prime p & p%:R == 0 :> R].
#[deprecated(since="mathcomp 2.4.0", note="Use pchar instead.")]
|
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
|
pchar
| |
char:= pchar (only parsing).
Local Notation has_pchar0 L := (pchar L =i pred0).
#[deprecated(since="mathcomp 2.4.0", note="Use has_pchar0 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
|
char
| |
has_char0L := (has_pchar0 L) (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
|
has_char0
| |
converseR : Type := R.
Local Notation "R ^c" := (converse R) : type_scope.
|
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
|
converse
| |
Definition_ := Monoid.isLaw.Build R 1 *%R mulrA mul1r mulr1.
#[export]
HB.instance Definition _ := Monoid.isMulLaw.Build R 0 *%R mul0r mulr0.
#[export]
HB.instance Definition _ := Monoid.isAddLaw.Build R *%R +%R mulrDl mulrDr.
|
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
| |
mulr_sumlI r P (F : I -> R) x :
(\sum_(i <- r | P i) F i) * x = \sum_(i <- r | P i) F i * x.
Proof. exact: big_distrl. 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_suml
| |
mulr_sumrI r P (F : I -> R) x :
x * (\sum_(i <- r | P i) F i) = \sum_(i <- r | P i) x * F i.
Proof. exact: big_distrr. 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_sumr
| |
mulrnAlx y n : (x *+ n) * y = (x * y) *+ n.
Proof. by elim: n => [|n IHn]; rewrite ?mul0r // !mulrS mulrDl 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
|
mulrnAl
| |
mulrnArx y n : x * (y *+ n) = (x * y) *+ n.
Proof. by elim: n => [|n IHn]; rewrite ?mulr0 // !mulrS mulrDr 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
|
mulrnAr
| |
mulr_natlx n : n%:R * x = x *+ n.
Proof. by rewrite mulrnAl 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_natl
| |
mulr_natrx n : x * n%:R = x *+ n.
Proof. by rewrite mulrnAr 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
|
mulr_natr
| |
natrDm n : (m + n)%:R = m%:R + n%:R :> R. Proof. exact: mulrnDr. 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
|
natrD
| |
natr1n : n%:R + 1 = n.+1%:R :> R. Proof. by rewrite mulrSr. 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
|
natr1
| |
nat1rn : 1 + n%:R = n.+1%:R :> R. Proof. by rewrite mulrS. 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
|
nat1r
| |
natr_sum:= big_morph (natmul 1) natrD (mulr0n 1).
|
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
|
natr_sum
| |
natrMm n : (m * n)%:R = m%:R * n%:R :> R.
Proof. by rewrite mulrnA mulr_natr. 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
|
natrM
| |
expr0x : x ^+ 0 = 1. Proof. by []. 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
|
expr0
| |
expr1x : x ^+ 1 = x. Proof. by []. 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
|
expr1
| |
expr2x : x ^+ 2 = x * x. Proof. by []. 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
|
expr2
| |
exprSx n : x ^+ n.+1 = x * x ^+ n.
Proof. by case: n => //; rewrite 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
|
exprS
| |
expr0nn : 0 ^+ n = (n == 0%N)%:R :> R.
Proof. by case: n => // n; rewrite 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
|
expr0n
| |
expr1nn : 1 ^+ n = 1 :> R.
Proof. by elim: n => // n IHn; rewrite exprS 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
|
expr1n
| |
exprDx m n : x ^+ (m + n) = x ^+ m * x ^+ n.
Proof. by elim: m => [|m IHm]; rewrite ?mul1r // !exprS -mulrA -IHm. 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
|
exprD
| |
exprSrx n : x ^+ n.+1 = x ^+ n * x.
Proof. by rewrite -addn1 exprD expr1. 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
|
exprSr
| |
expr_sumx (I : Type) (s : seq I) (P : pred I) F :
x ^+ (\sum_(i <- s | P i) F i) = \prod_(i <- s | P i) x ^+ F i :> R.
Proof. exact: (big_morph _ (exprD _)). 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
|
expr_sum
| |
commr_symx y : comm x y -> comm y x. Proof. by []. 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_sym
| |
commr_reflx : comm x x. Proof. by []. 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_refl
| |
commr0x : comm x 0.
Proof. by rewrite /comm mulr0 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
|
commr0
| |
commr1x : comm x 1.
Proof. by rewrite /comm mulr1 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
|
commr1
| |
commrDx y z : comm x y -> comm x z -> comm x (y + z).
Proof. by rewrite /comm mulrDl mulrDr => -> ->. 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
|
commrD
| |
commr_sum(I : Type) (s : seq I) (P : pred I) (F : I -> R) x :
(forall i, P i -> comm x (F i)) -> comm x (\sum_(i <- s | P i) F i).
Proof.
move=> comm_x_F; rewrite /comm mulr_suml mulr_sumr.
by apply: eq_bigr => i /comm_x_F.
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_sum
| |
commrMnx y n : comm x y -> comm x (y *+ n).
Proof.
rewrite /comm => com_xy.
by elim: n => [|n IHn]; rewrite ?commr0 // mulrS commrD.
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
|
commrMn
| |
commrMx y z : comm x y -> comm x z -> comm x (y * z).
Proof. by move=> com_xy; rewrite /comm mulrA com_xy -!mulrA => ->. 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
|
commrM
| |
commr_prod(I : Type) (s : seq I) (P : pred I) (F : I -> R) x :
(forall i, P i -> comm x (F i)) -> comm x (\prod_(i <- s | P i) F i).
Proof. exact: (big_ind _ (commr1 x) (@commrM 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_prod
| |
commr_natx n : comm x n%:R. Proof. exact/commrMn/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
|
commr_nat
| |
commrXx y n : comm x y -> comm x (y ^+ n).
Proof.
rewrite /comm => com_xy.
by elim: n => [|n IHn]; rewrite ?commr1 // exprS commrM.
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
|
commrX
| |
exprMn_commx y n : comm x y -> (x * y) ^+ n = x ^+ n * y ^+ n.
Proof.
move=> com_xy; elim: n => /= [|n IHn]; first by rewrite mulr1.
by rewrite !exprS IHn !mulrA; congr (_ * _); rewrite -!mulrA -commrX.
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
|
exprMn_comm
| |
exprMn_nx m n : (x *+ m) ^+ n = x ^+ n *+ (m ^ n) :> R.
Proof.
elim: n => [|n IHn]; first by rewrite mulr1n.
by rewrite exprS IHn mulrnAl mulrnAr -mulrnA exprS -expnSr.
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
|
exprMn_n
| |
exprMx m n : x ^+ (m * n) = x ^+ m ^+ n.
Proof.
elim: m => [|m IHm]; first by rewrite expr1n.
by rewrite mulSn exprD IHm exprS exprMn_comm //; apply: commrX.
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
|
exprM
| |
exprACx m n : (x ^+ m) ^+ n = (x ^+ n) ^+ m.
Proof. by rewrite -!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
|
exprAC
| |
expr_modn x i : x ^+ n = 1 -> x ^+ (i %% n) = x ^+ i.
Proof.
move=> xn1; rewrite {2}(divn_eq i n) exprD mulnC exprM xn1.
by 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
|
expr_mod
| |
expr_dvdn x i : x ^+ n = 1 -> n %| i -> x ^+ i = 1.
Proof.
by move=> xn1 dvd_n_i; rewrite -(expr_mod i xn1) (eqnP dvd_n_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
|
expr_dvd
| |
natrXn k : (n ^ k)%:R = n%:R ^+ k :> R.
Proof. by rewrite exprMn_n 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
|
natrX
| |
mulrI_eq0x y : lreg x -> (x * y == 0) = (y == 0).
Proof. by move=> reg_x; rewrite -{1}(mulr0 x) (inj_eq 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
|
mulrI_eq0
| |
lreg1: lreg (1 : R).
Proof. by move=> x y; rewrite !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
|
lreg1
| |
lregMx y : lreg x -> lreg y -> lreg (x * y).
Proof. by move=> reg_x reg_y z t; rewrite -!mulrA => /reg_x/reg_y. 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
|
lregM
| |
lregMl(a b: R) : lreg (a * b) -> lreg b.
Proof. by move=> rab c c' eq_bc; apply/rab; rewrite -!mulrA eq_bc. 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
|
lregMl
| |
rregMr(a b: R) : rreg (a * b) -> rreg a.
Proof. by move=> rab c c' eq_ca; apply/rab; rewrite !mulrA eq_ca. 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
|
rregMr
| |
lregXx n : lreg x -> lreg (x ^+ n).
Proof.
by move=> reg_x; elim: n => [|n]; [apply: lreg1 | rewrite exprS; apply: lregM].
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
|
lregX
| |
iter_mulrn x y : iter n ( *%R x) y = x ^+ n * y.
Proof. by elim: n => [|n ih]; rewrite ?expr0 ?mul1r //= ih exprS -mulrA. 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
|
iter_mulr
| |
iter_mulr_1n x : iter n ( *%R x) 1 = x ^+ n.
Proof. by rewrite iter_mulr 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
|
iter_mulr_1
| |
prodr_const(I : finType) (A : pred I) x : \prod_(i in A) x = x ^+ #|A|.
Proof. by rewrite big_const -iteropE. 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
|
prodr_const
| |
prodr_const_natn m x : \prod_(n <= i < m) x = x ^+ (m - n).
Proof. by rewrite big_const_nat -iteropE. 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
|
prodr_const_nat
| |
prodrXrx I r P (F : I -> nat) :
\prod_(i <- r | P i) x ^+ F i = x ^+ (\sum_(i <- r | P i) F i).
Proof. by rewrite (big_morph _ (exprD _) (erefl _)). 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
|
prodrXr
| |
prodrM_comm{I : eqType} r (P : pred I) (F G : I -> R) :
(forall i j, P i -> P j -> comm (F i) (G j)) ->
\prod_(i <- r | P i) (F i * G i) =
\prod_(i <- r | P i) F i * \prod_(i <- r | P i) G i.
Proof.
move=> FG; elim: r => [|i r IHr]; rewrite !(big_nil, big_cons) ?mulr1//.
case: ifPn => // Pi; rewrite IHr !mulrA; congr (_ * _); rewrite -!mulrA.
by rewrite commr_prod // => j Pj; apply/commr_sym/FG.
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
|
prodrM_comm
| |
prodrMl_comm{I : finType} (A : pred I) (x : R) F :
(forall i, A i -> comm x (F i)) ->
\prod_(i in A) (x * F i) = x ^+ #|A| * \prod_(i in A) F i.
Proof. by move=> xF; rewrite prodrM_comm ?prodr_const// => i j _ /xF. 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
|
prodrMl_comm
| |
prodrMr_comm{I : finType} (A : pred I) (x : R) F :
(forall i, A i -> comm x (F i)) ->
\prod_(i in A) (F i * x) = \prod_(i in A) F i * x ^+ #|A|.
Proof. by move=> xF; rewrite prodrM_comm ?prodr_const// => i j /xF. 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
|
prodrMr_comm
| |
prodrMn(I : Type) (s : seq I) (P : pred I) (F : I -> R) (g : I -> nat) :
\prod_(i <- s | P i) (F i *+ g i) =
\prod_(i <- s | P i) (F i) *+ \prod_(i <- s | P i) g i.
Proof.
by elim/big_rec3: _ => // i y1 y2 y3 _ ->; rewrite mulrnAr mulrnAl -mulrnA.
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
|
prodrMn
| |
prodrMn_constn (I : finType) (A : pred I) (F : I -> R) :
\prod_(i in A) (F i *+ n) = \prod_(i in A) F i *+ n ^ #|A|.
Proof. by rewrite prodrMn prod_nat_const. 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
|
prodrMn_const
| |
natr_prodI r P (F : I -> nat) :
(\prod_(i <- r | P i) F i)%:R = \prod_(i <- r | P i) (F i)%:R :> R.
Proof. exact: (big_morph _ natrM). 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
|
natr_prod
| |
exprDn_commx y n (cxy : comm x y) :
(x + y) ^+ n = \sum_(i < n.+1) (x ^+ (n - i) * y ^+ i) *+ 'C(n, i).
Proof.
elim: n => [|n IHn]; rewrite big_ord_recl mulr1 ?big_ord0 ?addr0 //=.
rewrite exprS {}IHn /= mulrDl !big_distrr /= big_ord_recl mulr1 subn0.
rewrite !big_ord_recr /= !binn !subnn !mul1r !subn0 bin0 !exprS -addrA.
congr (_ + _); rewrite addrA -big_split /=; congr (_ + _).
apply: eq_bigr => i _; rewrite !mulrnAr !mulrA -exprS -subSn ?(valP i) //.
by rewrite subSS (commrX _ (commr_sym cxy)) -mulrA -exprS -mulrnDr.
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
|
exprDn_comm
| |
exprD1nx n : (x + 1) ^+ n = \sum_(i < n.+1) x ^+ i *+ 'C(n, i).
Proof.
rewrite addrC (exprDn_comm n (commr_sym (commr1 x))).
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
|
exprD1n
| |
sqrrD1x : (x + 1) ^+ 2 = x ^+ 2 + x *+ 2 + 1.
Proof.
rewrite exprD1n !big_ord_recr big_ord0 /= add0r.
by rewrite addrC addrA addrAC.
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
|
sqrrD1
| |
mulr_2closed:= {in S &, forall u v, u * v \in 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
|
mulr_2closed
| |
mulr_closed:= 1 \in S /\ mulr_2closed.
|
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
|
mulr_closed
| |
semiring_closed:= nmod_closed S /\ mulr_closed.
|
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
|
semiring_closed
| |
semiring_closedD: semiring_closed -> nmod_closed S. Proof. by case. 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
|
semiring_closedD
| |
semiring_closedM: semiring_closed -> mulr_closed. Proof. by case. 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
|
semiring_closedM
| |
oner_eq0: (1 == 0 :> R) = false. Proof. exact: negbTE oner_neq0. 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
|
oner_eq0
| |
lastr_eq0(s : seq R) x : x != 0 -> (last x s == 0) = (last 1 s == 0).
Proof. by case: s => [|y s] /negPf // ->; rewrite 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
|
lastr_eq0
| |
lreg_neq0x : lreg x -> x != 0.
Proof. by move=> reg_x; rewrite -[x]mulr1 mulrI_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
|
lreg_neq0
| |
pFrobenius_autp of p \in pchar R := fun x => x ^+ p.
|
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
|
pFrobenius_aut
| |
pcharf0: p%:R = 0 :> R. Proof. by apply/eqP; case/andP: pcharFp. 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
|
pcharf0
| |
pcharf_prime: prime p. Proof. by case/andP: pcharFp. Qed.
Hint Resolve pcharf_prime : core.
|
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
|
pcharf_prime
| |
mulrn_pcharx : x *+ p = 0. Proof. by rewrite -mulr_natl pcharf0 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
|
mulrn_pchar
| |
natr_mod_pcharn : (n %% p)%:R = n%:R :> R.
Proof. by rewrite {2}(divn_eq n p) natrD mulrnA mulrn_pchar 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
|
natr_mod_pchar
| |
dvdn_pcharfn : (p %| n)%N = (n%:R == 0 :> R).
Proof.
apply/idP/eqP=> [/dvdnP[n' ->]|n0]; first by rewrite natrM pcharf0 mulr0.
apply/idPn; rewrite -prime_coprime // => /eqnP pn1.
have [a _ /dvdnP[b]] := Bezoutl n (prime_gt0 pcharf_prime).
move/(congr1 (fun m => m%:R : R))/eqP.
by rewrite natrD !natrM pcharf0 n0 !mulr0 pn1 addr0 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
|
dvdn_pcharf
| |
pcharf_eq: pchar R =i (p : nat_pred).
Proof.
move=> q; apply/andP/eqP=> [[q_pr q0] | ->]; last by rewrite pcharf0.
by apply/eqP; rewrite eq_sym -dvdn_prime2 // dvdn_pcharf.
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
|
pcharf_eq
| |
bin_lt_pcharf_0k : 0 < k < p -> 'C(p, k)%:R = 0 :> R.
Proof. by move=> lt0kp; apply/eqP; rewrite -dvdn_pcharf prime_dvd_bin. Qed.
Local Notation "x ^f" := (pFrobenius_aut pcharFp x).
|
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
|
bin_lt_pcharf_0
|
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