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 |
|---|---|---|---|---|---|---|
divzAm n p : (p %| n -> n %| m * p -> m %/ (n %/ p)%Z = m * p %/ n)%Z.
Proof.
move/divzK=> p_dv_n; have [->|] := eqVneq n 0; first by rewrite div0z !divz0.
rewrite -{1 2}p_dv_n mulf_eq0 => /norP[pn_nz p_nz] /divzK; rewrite mulrA p_dv_n.
by move/mulIf=> {1} <- //; rewrite mulzK.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
divzA
| |
divzMAm n p : (n * p %| m -> m %/ (n * p) = (m %/ n)%Z %/ p)%Z.
Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite mulr0 !divz0.
have [-> | nz_n] := eqVneq n 0; first by rewrite mul0r !divz0 div0z.
by move/divzK=> {2} <-; rewrite mulrA mulrAC !mulzK.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
divzMA
| |
divzACm n p : (n * p %| m -> (m %/ n)%Z %/ p = (m %/ p)%Z %/ n)%Z.
Proof. by move=> np_dv_mn; rewrite -!divzMA // mulrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
divzAC
| |
divzMlp m d : p != 0 -> (d %| m -> p * m %/ (p * d) = m %/ d)%Z.
Proof.
have [-> | nz_d nz_p] := eqVneq d 0; first by rewrite mulr0 !divz0.
by move/divzK=> {1}<-; rewrite mulrCA mulzK ?mulf_neq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
divzMl
| |
divzMrp m d : p != 0 -> (d %| m -> m * p %/ (d * p) = m %/ d)%Z.
Proof. by rewrite -!(mulrC p); apply: divzMl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
divzMr
| |
dvdz_mul2lp d m : p != 0 -> (p * d %| p * m)%Z = (d %| m)%Z.
Proof. by rewrite !dvdzE -absz_gt0 !abszM; apply: dvdn_pmul2l. Qed.
Arguments dvdz_mul2l [p d m].
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_mul2l
| |
dvdz_mul2rp d m : p != 0 -> (d * p %| m * p)%Z = (d %| m)%Z.
Proof. by rewrite !dvdzE -absz_gt0 !abszM; apply: dvdn_pmul2r. Qed.
Arguments dvdz_mul2r [p d m].
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_mul2r
| |
dvdz_exp2lp m n : (m <= n)%N -> (p ^+ m %| p ^+ n)%Z.
Proof. by rewrite dvdzE !abszX; apply: dvdn_exp2l. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_exp2l
| |
dvdz_Pexp2lp m n : `|p| > 1 -> (p ^+ m %| p ^+ n)%Z = (m <= n)%N.
Proof. by rewrite dvdzE !abszX ltz_nat; apply: dvdn_Pexp2l. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_Pexp2l
| |
dvdz_exp2rm n k : (m %| n -> m ^+ k %| n ^+ k)%Z.
Proof. by rewrite !dvdzE !abszX; apply: dvdn_exp2r. Qed.
Fact dvdz_zmod_closed d : zmod_closed (dvdz d).
Proof.
split=> [|_ _ /dvdzP[p ->] /dvdzP[q ->]]; first exact: dvdz0.
by rewrite -mulrBl dvdz_mull.
Qed.
HB.instance Definition _ d := GRing.isZmodClosed.Build int (dvdz d)
(dvdz_zmod_closed d).
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_exp2r
| |
dvdz_expk d m : (0 < k)%N -> (d %| m -> d %| m ^+ k)%Z.
Proof. by case: k => // k _ d_dv_m; rewrite exprS dvdz_mulr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_exp
| |
eqz_mod_dvdd m n : (m == n %[mod d])%Z = (d %| m - n)%Z.
Proof.
apply/eqP/dvdz_mod0P=> eq_mn.
by rewrite -modzDml eq_mn modzDml subrr mod0z.
by rewrite -(subrK n m) -modzDml eq_mn add0r.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
eqz_mod_dvd
| |
divzDlm n d :
(d %| m)%Z -> ((m + n) %/ d)%Z = (m %/ d)%Z + (n %/ d)%Z.
Proof.
have [-> | d_nz] := eqVneq d 0; first by rewrite !divz0.
by move/divzK=> {1}<-; rewrite divzMDl.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
divzDl
| |
divzDrm n d :
(d %| n)%Z -> ((m + n) %/ d)%Z = (m %/ d)%Z + (n %/ d)%Z.
Proof. by move=> dv_n; rewrite addrC divzDl // addrC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
divzDr
| |
dvdz_pcharf(R : nzRingType) p : p \in [pchar R] ->
forall n : int, (p %| n)%Z = (n%:~R == 0 :> R).
Proof.
move=> pcharRp [] n; rewrite [LHS](dvdn_pcharf pcharRp)//.
by rewrite NegzE abszN rmorphN// oppr_eq0.
Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use dvdz_pcharf instead.")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_pcharf
| |
dvdz_charfchRp := (dvdz_pcharf chRp).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_charf
| |
gcdzzm : gcdz m m = `|m|%:Z. Proof. by rewrite /gcdz gcdnn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzz
| |
gcdzC: commutative gcdz. Proof. by move=> m n; rewrite /gcdz gcdnC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzC
| |
gcd0zm : gcdz 0 m = `|m|%:Z. Proof. by rewrite /gcdz gcd0n. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcd0z
| |
gcdz0m : gcdz m 0 = `|m|%:Z. Proof. by rewrite /gcdz gcdn0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdz0
| |
gcd1z: left_zero 1 gcdz. Proof. by move=> m; rewrite /gcdz gcd1n. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcd1z
| |
gcdz1: right_zero 1 gcdz. Proof. by move=> m; rewrite /gcdz gcdn1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdz1
| |
dvdz_gcdrm n : (gcdz m n %| n)%Z. Proof. exact: dvdn_gcdr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_gcdr
| |
dvdz_gcdlm n : (gcdz m n %| m)%Z. Proof. exact: dvdn_gcdl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_gcdl
| |
gcdz_eq0m n : (gcdz m n == 0) = (m == 0) && (n == 0).
Proof. by rewrite -absz_eq0 eqn0Ngt gcdn_gt0 !negb_or -!eqn0Ngt !absz_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdz_eq0
| |
gcdNzm n : gcdz (- m) n = gcdz m n. Proof. by rewrite /gcdz abszN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdNz
| |
gcdzNm n : gcdz m (- n) = gcdz m n. Proof. by rewrite /gcdz abszN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzN
| |
gcdz_modrm n : gcdz m (n %% m)%Z = gcdz m n.
Proof.
rewrite -modz_abs /gcdz; move/absz: m => m.
have [-> | m_gt0] := posnP m; first by rewrite modz0.
case: n => n; first by rewrite modz_nat gcdn_modr.
rewrite modNz_nat // NegzE abszN {2}(divn_eq n m) -addnS gcdnMDl.
rewrite -addrA -opprD -intS /=; set m1 := _.+1.
have le_m1m: (m1 <= m)%N by apply: ltn_pmod.
by rewrite subzn // !(gcdnC m) -{2 3}(subnK le_m1m) gcdnDl gcdnDr gcdnC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdz_modr
| |
gcdz_modlm n : gcdz (m %% n)%Z n = gcdz m n.
Proof. by rewrite -!(gcdzC n) gcdz_modr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdz_modl
| |
gcdzMDlq m n : gcdz m (q * m + n) = gcdz m n.
Proof. by rewrite -gcdz_modr modzMDl gcdz_modr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzMDl
| |
gcdzDlm n : gcdz m (m + n) = gcdz m n.
Proof. by rewrite -{2}(mul1r m) gcdzMDl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzDl
| |
gcdzDrm n : gcdz m (n + m) = gcdz m n.
Proof. by rewrite addrC gcdzDl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzDr
| |
gcdzMln m : gcdz n (m * n) = `|n|%:Z.
Proof. by rewrite -[m * n]addr0 gcdzMDl gcdz0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzMl
| |
gcdzMrn m : gcdz n (n * m) = `|n|%:Z.
Proof. by rewrite mulrC gcdzMl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzMr
| |
gcdz_idPl{m n} : reflect (gcdz m n = `|m|%:Z) (m %| n)%Z.
Proof. by apply: (iffP gcdn_idPl) => [<- | []]. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdz_idPl
| |
gcdz_idPr{m n} : reflect (gcdz m n = `|n|%:Z) (n %| m)%Z.
Proof. by rewrite gcdzC; apply: gcdz_idPl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdz_idPr
| |
expz_mine m n : e >= 0 -> e ^+ minn m n = gcdz (e ^+ m) (e ^+ n).
Proof.
by case: e => // e _; rewrite /gcdz !abszX -expn_min -natz -natrX !natz.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
expz_min
| |
dvdz_gcdp m n : (p %| gcdz m n)%Z = (p %| m)%Z && (p %| n)%Z.
Proof. exact: dvdn_gcd. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_gcd
| |
gcdzAC: right_commutative gcdz.
Proof. by move=> m n p; rewrite /gcdz gcdnAC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzAC
| |
gcdzA: associative gcdz.
Proof. by move=> m n p; rewrite /gcdz gcdnA. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzA
| |
gcdzCA: left_commutative gcdz.
Proof. by move=> m n p; rewrite /gcdz gcdnCA. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzCA
| |
gcdzACA: interchange gcdz gcdz.
Proof. by move=> m n p q; rewrite /gcdz gcdnACA. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
gcdzACA
| |
mulz_gcdrm n p : `|m|%:Z * gcdz n p = gcdz (m * n) (m * p).
Proof. by rewrite -PoszM muln_gcdr -!abszM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
mulz_gcdr
| |
mulz_gcdlm n p : gcdz m n * `|p|%:Z = gcdz (m * p) (n * p).
Proof. by rewrite -PoszM muln_gcdl -!abszM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
mulz_gcdl
| |
mulz_divCA_gcdn m : n * (m %/ gcdz n m)%Z = m * (n %/ gcdz n m)%Z.
Proof. by rewrite mulz_divCA ?dvdz_gcdl ?dvdz_gcdr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
mulz_divCA_gcd
| |
dvdz_lcmrm n : (n %| lcmz m n)%Z.
Proof. exact: dvdn_lcmr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_lcmr
| |
dvdz_lcmlm n : (m %| lcmz m n)%Z.
Proof. exact: dvdn_lcml. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_lcml
| |
dvdz_lcmd1 d2 m : ((lcmn d1 d2 %| m) = (d1 %| m) && (d2 %| m))%Z.
Proof. exact: dvdn_lcm. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_lcm
| |
lcmzC: commutative lcmz.
Proof. by move=> m n; rewrite /lcmz lcmnC. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
lcmzC
| |
lcm0z: left_zero 0 lcmz.
Proof. by move=> x; rewrite /lcmz absz0 lcm0n. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
lcm0z
| |
lcmz0: right_zero 0 lcmz.
Proof. by move=> x; rewrite /lcmz absz0 lcmn0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
lcmz0
| |
lcmz_ge0m n : 0 <= lcmz m n.
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
lcmz_ge0
| |
lcmz_neq0m n : (lcmz m n != 0) = (m != 0) && (n != 0).
Proof.
have [->|m_neq0] := eqVneq m 0; first by rewrite lcm0z.
have [->|n_neq0] := eqVneq n 0; first by rewrite lcmz0.
by rewrite gt_eqF// [0 < _]lcmn_gt0 !absz_gt0 m_neq0 n_neq0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
lcmz_neq0
| |
coprimezEm n : coprimez m n = coprime `|m| `|n|. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimezE
| |
coprimez_sym: symmetric coprimez.
Proof. by move=> m n; apply: coprime_sym. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimez_sym
| |
coprimeNzm n : coprimez (- m) n = coprimez m n.
Proof. by rewrite coprimezE abszN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimeNz
| |
coprimezNm n : coprimez m (- n) = coprimez m n.
Proof. by rewrite coprimezE abszN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimezN
| |
egcdz_specm n : int * int -> Type :=
EgcdzSpec u v of u * m + v * n = gcdz m n & coprimez u v
: egcdz_spec m n (u, v).
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
egcdz_spec
| |
egcdzPm n : egcdz_spec m n (egcdz m n).
Proof.
rewrite /egcdz; have [-> | m_nz] := eqVneq.
by split; [rewrite -abszEsign gcd0z | rewrite coprimezE absz_sign].
have m_gt0 : (`|m| > 0)%N by rewrite absz_gt0.
case: egcdnP (coprime_egcdn `|n| m_gt0) => //= u v Duv _ co_uv; split.
rewrite !mulNr -!mulrA mulrCA -abszEsg mulrCA -abszEsign.
by rewrite -!PoszM Duv addnC PoszD addrK.
by rewrite coprimezE abszM absz_sg m_nz mul1n mulNr abszN abszMsign.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
egcdzP
| |
Bezoutzm n : {u : int & {v : int | u * m + v * n = gcdz m n}}.
Proof. by exists (egcdz m n).1, (egcdz m n).2; case: egcdzP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
Bezoutz
| |
coprimezPm n :
reflect (exists uv, uv.1 * m + uv.2 * n = 1) (coprimez m n).
Proof.
apply: (iffP eqP) => [<-| [[u v] /= Duv]].
by exists (egcdz m n); case: egcdzP.
congr _%:Z; apply: gcdn_def; rewrite ?dvd1n // => d dv_d_n dv_d_m.
by rewrite -(dvdzE d 1) -Duv [m]intEsg [n]intEsg rpredD ?dvdz_mull.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimezP
| |
Gauss_dvdzm n p :
coprimez m n -> (m * n %| p)%Z = (m %| p)%Z && (n %| p)%Z.
Proof. by move/Gauss_dvd <-; rewrite -abszM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
Gauss_dvdz
| |
Gauss_dvdzrm n p : coprimez m n -> (m %| n * p)%Z = (m %| p)%Z.
Proof. by rewrite dvdzE abszM => /Gauss_dvdr->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
Gauss_dvdzr
| |
Gauss_dvdzlm n p : coprimez m p -> (m %| n * p)%Z = (m %| n)%Z.
Proof. by rewrite mulrC; apply: Gauss_dvdzr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
Gauss_dvdzl
| |
Gauss_gcdzrp m n : coprimez p m -> gcdz p (m * n) = gcdz p n.
Proof. by rewrite /gcdz abszM => /Gauss_gcdr->. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
Gauss_gcdzr
| |
Gauss_gcdzlp m n : coprimez p n -> gcdz p (m * n) = gcdz p m.
Proof. by move=> co_pn; rewrite mulrC Gauss_gcdzr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
Gauss_gcdzl
| |
coprimezMrp m n : coprimez p (m * n) = coprimez p m && coprimez p n.
Proof. by rewrite -coprimeMr -abszM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimezMr
| |
coprimezMlp m n : coprimez (m * n) p = coprimez m p && coprimez n p.
Proof. by rewrite -coprimeMl -abszM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimezMl
| |
coprimez_pexplk m n : (0 < k)%N -> coprimez (m ^+ k) n = coprimez m n.
Proof. by rewrite /coprimez /gcdz abszX; apply: coprime_pexpl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimez_pexpl
| |
coprimez_pexprk m n : (0 < k)%N -> coprimez m (n ^+ k) = coprimez m n.
Proof. by move=> k_gt0; rewrite !(coprimez_sym m) coprimez_pexpl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimez_pexpr
| |
coprimezXlk m n : coprimez m n -> coprimez (m ^+ k) n.
Proof. by rewrite /coprimez /gcdz abszX; apply: coprimeXl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimezXl
| |
coprimezXrk m n : coprimez m n -> coprimez m (n ^+ k).
Proof. by rewrite !(coprimez_sym m); apply: coprimezXl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimezXr
| |
coprimez_dvdlm n p : (m %| n)%N -> coprimez n p -> coprimez m p.
Proof. exact: coprime_dvdl. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimez_dvdl
| |
coprimez_dvdrm n p : (m %| n)%N -> coprimez p n -> coprimez p m.
Proof. exact: coprime_dvdr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
coprimez_dvdr
| |
dvdz_pexp2rm n k : (k > 0)%N -> (m ^+ k %| n ^+ k)%Z = (m %| n)%Z.
Proof. by rewrite dvdzE !abszX; apply: dvdn_pexp2r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_pexp2r
| |
zchinese_remainderx y :
(x == y %[mod m1 * m2])%Z = (x == y %[mod m1])%Z && (x == y %[mod m2])%Z.
Proof. by rewrite !eqz_mod_dvd Gauss_dvdz. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zchinese_remainder
| |
zchineser1 r2 :=
r1 * m2 * (egcdz m1 m2).2 + r2 * m1 * (egcdz m1 m2).1.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zchinese
| |
zchinese_modlr1 r2 : (zchinese r1 r2 = r1 %[mod m1])%Z.
Proof.
rewrite /zchinese; have [u v /= Duv _] := egcdzP m1 m2.
rewrite -{2}[r1]mulr1 -((gcdz _ _ =P 1) co_m12) -Duv.
by rewrite mulrDr mulrAC addrC (mulrAC r2) !mulrA !modzMDl.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zchinese_modl
| |
zchinese_modrr1 r2 : (zchinese r1 r2 = r2 %[mod m2])%Z.
Proof.
rewrite /zchinese; have [u v /= Duv _] := egcdzP m1 m2.
rewrite -{2}[r2]mulr1 -((gcdz _ _ =P 1) co_m12) -Duv.
by rewrite mulrAC modzMDl mulrAC addrC mulrDr !mulrA modzMDl.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zchinese_modr
| |
zchinese_modx : (x = zchinese (x %% m1)%Z (x %% m2)%Z %[mod m1 * m2])%Z.
Proof.
apply/eqP; rewrite zchinese_remainder //.
by rewrite zchinese_modl zchinese_modr !modz_mod !eqxx.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zchinese_mod
| |
zcontents(p : {poly int}) : int :=
sgz (lead_coef p) * \big[gcdn/0]_(i < size p) `|(p`_i)%R|%N.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zcontents
| |
sgz_contentsp : sgz (zcontents p) = sgz (lead_coef p).
Proof.
rewrite /zcontents mulrC sgzM sgz_id; set d := _%:Z.
have [-> | nz_p] := eqVneq p 0; first by rewrite lead_coef0 mulr0.
rewrite gtr0_sgz ?mul1r // ltz_nat polySpred ?big_ord_recr //= -lead_coefE.
by rewrite gcdn_gt0 orbC absz_gt0 lead_coef_eq0 nz_p.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
sgz_contents
| |
zcontents_eq0p : (zcontents p == 0) = (p == 0).
Proof. by rewrite -sgz_eq0 sgz_contents sgz_eq0 lead_coef_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zcontents_eq0
| |
zcontents0: zcontents 0 = 0.
Proof. by apply/eqP; rewrite zcontents_eq0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zcontents0
| |
zcontentsZa p : zcontents (a *: p) = a * zcontents p.
Proof.
have [-> | nz_a] := eqVneq a 0; first by rewrite scale0r mul0r zcontents0.
rewrite {2}[a]intEsg mulrCA -mulrA -PoszM big_distrr /= mulrCA mulrA -sgzM.
rewrite -lead_coefZ; congr (_ * _%:Z); rewrite size_scale //.
by apply: eq_bigr => i _; rewrite coefZ abszM.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zcontentsZ
| |
zcontents_monicp : p \is monic -> zcontents p = 1.
Proof.
move=> mon_p; rewrite /zcontents polySpred ?monic_neq0 //.
by rewrite big_ord_recr /= -lead_coefE (monicP mon_p) gcdn1.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zcontents_monic
| |
dvdz_contentsa p : (a %| zcontents p)%Z = (p \is a polyOver (dvdz a)).
Proof.
rewrite dvdzE abszM absz_sg lead_coef_eq0.
have [-> | nz_p] := eqVneq; first by rewrite mul0n dvdn0 rpred0.
rewrite mul1n; apply/dvdn_biggcdP/(all_nthP 0)=> a_dv_p i ltip /=.
exact: (a_dv_p (Ordinal ltip)).
exact: a_dv_p.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
dvdz_contents
| |
map_poly_divzK{a} p :
p \is a polyOver (dvdz a) -> a *: map_poly (divz^~ a) p = p.
Proof.
move/polyOverP=> a_dv_p; apply/polyP=> i.
by rewrite coefZ coef_map_id0 ?div0z // mulrC divzK.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
map_poly_divzK
| |
polyOver_dvdzPa p :
reflect (exists q, p = a *: q) (p \is a polyOver (dvdz a)).
Proof.
apply: (iffP idP) => [/map_poly_divzK | [q ->]].
by exists (map_poly (divz^~ a) p).
by apply/polyOverP=> i; rewrite coefZ dvdz_mulr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
polyOver_dvdzP
| |
zprimitivep := map_poly (divz^~ (zcontents p)) p.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zprimitive
| |
zpolyEprimp : p = zcontents p *: zprimitive p.
Proof. by rewrite map_poly_divzK // -dvdz_contents. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zpolyEprim
| |
zprimitive0: zprimitive 0 = 0.
Proof.
by apply/polyP=> i; rewrite coef0 coef_map_id0 ?div0z // zcontents0 divz0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zprimitive0
| |
zprimitive_eq0p : (zprimitive p == 0) = (p == 0).
Proof.
apply/idP/idP=> /eqP p0; first by rewrite [p]zpolyEprim p0 scaler0.
by rewrite p0 zprimitive0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zprimitive_eq0
| |
size_zprimitivep : size (zprimitive p) = size p.
Proof.
have [-> | ] := eqVneq p 0; first by rewrite zprimitive0.
by rewrite {1 3}[p]zpolyEprim scale_poly_eq0 => /norP[/size_scale-> _].
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
size_zprimitive
| |
sgz_lead_primitivep : sgz (lead_coef (zprimitive p)) = (p != 0).
Proof.
have [-> | nz_p] := eqVneq; first by rewrite zprimitive0 lead_coef0.
apply: (@mulfI _ (sgz (zcontents p))); first by rewrite sgz_eq0 zcontents_eq0.
by rewrite -sgzM mulr1 -lead_coefZ -zpolyEprim sgz_contents.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
sgz_lead_primitive
| |
zcontents_primitivep : zcontents (zprimitive p) = (p != 0).
Proof.
have [-> | nz_p] := eqVneq; first by rewrite zprimitive0 zcontents0.
apply: (@mulfI _ (zcontents p)); first by rewrite zcontents_eq0.
by rewrite mulr1 -zcontentsZ -zpolyEprim.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zcontents_primitive
| |
zprimitive_idp : zprimitive (zprimitive p) = zprimitive p.
Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite !zprimitive0.
by rewrite {2}[zprimitive p]zpolyEprim zcontents_primitive nz_p scale1r.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zprimitive_id
| |
zprimitive_monicp : p \in monic -> zprimitive p = p.
Proof. by move=> mon_p; rewrite {2}[p]zpolyEprim zcontents_monic ?scale1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zprimitive_monic
| |
zprimitiveZa p : a != 0 -> zprimitive (a *: p) = zprimitive p.
Proof.
have [-> | nz_p nz_a] := eqVneq p 0; first by rewrite scaler0.
apply: (@mulfI _ (a * zcontents p)%:P).
by rewrite polyC_eq0 mulf_neq0 ?zcontents_eq0.
by rewrite -{1}zcontentsZ !mul_polyC -zpolyEprim -scalerA -zpolyEprim.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zprimitiveZ
| |
zprimitive_minp a q :
p != 0 -> p = a *: q ->
{b | sgz b = sgz (lead_coef q) & q = b *: zprimitive p}.
Proof.
move=> nz_p Dp; have /dvdzP/sig_eqW[b Db]: (a %| zcontents p)%Z.
by rewrite dvdz_contents; apply/polyOver_dvdzP; exists q.
suffices ->: q = b *: zprimitive p.
by rewrite lead_coefZ sgzM sgz_lead_primitive nz_p mulr1; exists b.
apply: (@mulfI _ a%:P).
by apply: contraNneq nz_p; rewrite Dp -mul_polyC => ->; rewrite mul0r.
by rewrite !mul_polyC -Dp scalerA mulrC -Db -zpolyEprim.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple prime order",
"From mathcomp Require Import ssralg poly ssrnum ssrint matrix",
"From mathcomp Require Import polydiv perm zmodp bigop"
] |
algebra/intdiv.v
|
zprimitive_min
|
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