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
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| docstring
stringclasses 1
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|---|---|---|---|---|---|---|
equivf:= (@FracField.equivf _).
#[global] Hint Resolve denom_ratioP : core.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
equivf
| |
Ratio_numden(x : {ratio R}) : Ratio \n_x \d_x = x.
Proof. exact: FracField.Ratio_numden. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
Ratio_numden
| |
tofrac_is_zmod_morphism: zmod_morphism tofrac.
Proof.
move=> p q /=; unlock tofrac.
rewrite -[X in _ = _ + X]pi_opp -[RHS]pi_add.
by rewrite /addf /oppf /= !numden_Ratio ?(oner_neq0, mul1r, mulr1).
Qed.
#[deprecated(since="mathcomp 2.5.0",
note="use `tofrac_is_zmod_morphism` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac_is_zmod_morphism
| |
tofrac_is_additive:= tofrac_is_zmod_morphism.
HB.instance Definition _ := GRing.isZmodMorphism.Build R {fraction R} tofrac
tofrac_is_zmod_morphism.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac_is_additive
| |
tofrac_is_monoid_morphism: monoid_morphism tofrac.
Proof.
split=> [//|p q]; unlock tofrac; rewrite -[RHS]pi_mul.
by rewrite /mulf /= !numden_Ratio ?(oner_neq0, mul1r, mulr1).
Qed.
#[deprecated(since="mathcomp 2.5.0",
note="use `tofrac_is_monoid_morphism` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac_is_monoid_morphism
| |
tofrac_is_multiplicative:= tofrac_is_monoid_morphism.
HB.instance Definition _ := GRing.isMonoidMorphism.Build R {fraction R} tofrac
tofrac_is_monoid_morphism.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac_is_multiplicative
| |
tofrac0: 0%:F = 0. Proof. exact: rmorph0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac0
| |
tofracN: {morph tofrac: x / - x}. Proof. exact: rmorphN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofracN
| |
tofracD: {morph tofrac: x y / x + y}. Proof. exact: rmorphD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofracD
| |
tofracB: {morph tofrac: x y / x - y}. Proof. exact: rmorphB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofracB
| |
tofracMnn : {morph tofrac: x / x *+ n}. Proof. exact: rmorphMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofracMn
| |
tofracMNnn : {morph tofrac: x / x *- n}. Proof. exact: rmorphMNn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofracMNn
| |
tofrac1: 1%:F = 1. Proof. exact: rmorph1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac1
| |
tofracM: {morph tofrac: x y / x * y}. Proof. exact: rmorphM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofracM
| |
tofracXnn : {morph tofrac: x / x ^+ n}. Proof. exact: rmorphXn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofracXn
| |
tofrac_eq(p q : R): (p%:F == q%:F) = (p == q).
Proof.
apply/eqP/eqP=> [|->//]; unlock tofrac=> /eqmodP /eqP /=.
by rewrite !numden_Ratio ?(oner_eq0, mul1r, mulr1).
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac_eq
| |
tofrac_eq0(p : R): (p%:F == 0) = (p == 0).
Proof. by rewrite tofrac_eq. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import ssrAC choice tuple bigop ssralg poly polydiv",
"From mathcomp Require Import generic_quotient"
] |
algebra/fraction.v
|
tofrac_eq0
| |
divz(m d : int) : int :=
let: (K, n) := match m with Posz n => (Posz, n) | Negz n => (Negz, n) end in
sgz d * K (n %/ `|d|)%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
|
divz
| |
modz(m d : int) : int := m - divz m d * d.
|
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
|
modz
| |
dvdzd m := (`|d| %| `|m|)%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
|
dvdz
| |
gcdzm n := (gcdn `|m| `|n|)%:Z.
|
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
|
gcdz
| |
lcmzm n := (lcmn `|m| `|n|)%:Z.
|
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
|
lcmz
| |
egcdzm n : int * int :=
if m == 0 then (0, (-1) ^+ (n < 0)%R) else
let: (u, v) := egcdn `|m| `|n| in (sgz m * u, - (-1) ^+ (n < 0)%R * v%:Z).
|
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
|
egcdz
| |
coprimezm n := (gcdz m n == 1).
Infix "%/" := divz : int_scope.
Infix "%%" := modz : int_scope.
|
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
|
coprimez
| |
divz_nat(n d : nat) : (n %/ d)%Z = (n %/ d)%N.
Proof. by case: d => // d; rewrite /divz /= mul1r. 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
|
divz_nat
| |
divzNm d : (m %/ - d)%Z = - (m %/ d)%Z.
Proof. by case: m => n; rewrite /divz /= sgzN abszN mulNr. 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
|
divzN
| |
divz_abs(m d : int) : (m %/ `|d|)%Z = (-1) ^+ (d < 0)%R * (m %/ d)%Z.
Proof.
by rewrite {3}[d]intEsign !mulr_sign; case: ifP => -> //; rewrite divzN opprK.
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
|
divz_abs
| |
div0zd : (0 %/ d)%Z = 0.
Proof.
by rewrite -(canLR (signrMK _) (divz_abs _ _)) (divz_nat 0) div0n mulr0.
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
|
div0z
| |
divNz_natm d : (d > 0)%N -> (Negz m %/ d)%Z = - (m %/ d).+1%:Z.
Proof. by case: d => // d _; apply: mul1r. 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
|
divNz_nat
| |
divz_eqm d : m = (m %/ d)%Z * d + (m %% d)%Z.
Proof. by rewrite addrC subrK. 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
|
divz_eq
| |
modzNm d : (m %% - d)%Z = (m %% d)%Z.
Proof. by rewrite /modz divzN mulrNN. 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
|
modzN
| |
modz_absm d : (m %% `|d|%N)%Z = (m %% d)%Z.
Proof. by rewrite {2}[d]intEsign mulr_sign; case: ifP; rewrite ?modzN. 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
|
modz_abs
| |
modz_nat(m d : nat) : (m %% d)%Z = (m %% d)%N.
Proof.
by apply: (canLR (addrK _)); rewrite addrC divz_nat {1}(divn_eq m d).
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
|
modz_nat
| |
modNz_natm d : (d > 0)%N -> (Negz m %% d)%Z = d%:Z - 1 - (m %% d)%:Z.
Proof.
rewrite /modz => /divNz_nat->; apply: (canLR (addrK _)).
rewrite -!addrA -!opprD -!PoszD -opprB mulnSr !addnA PoszD addrK.
by rewrite addnAC -addnA mulnC -divn_eq.
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
|
modNz_nat
| |
modz_ge0m d : d != 0 -> 0 <= (m %% d)%Z.
Proof.
rewrite -absz_gt0 -modz_abs => d_gt0.
case: m => n; rewrite ?modNz_nat ?modz_nat // -addrA -opprD subr_ge0.
by rewrite lez_nat ltn_mod.
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
|
modz_ge0
| |
divz0m : (m %/ 0)%Z = 0. Proof. by case: m. 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
|
divz0
| |
mod0zd : (0 %% d)%Z = 0. Proof. by rewrite /modz div0z mul0r subrr. 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
|
mod0z
| |
modz0m : (m %% 0)%Z = m. Proof. by rewrite /modz mulr0 subr0. 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
|
modz0
| |
divz_smallm d : 0 <= m < `|d|%:Z -> (m %/ d)%Z = 0.
Proof.
rewrite -(canLR (signrMK _) (divz_abs _ _)); case: m => // n /divn_small.
by rewrite divz_nat => ->; rewrite mulr0.
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
|
divz_small
| |
divzMDlq m d : d != 0 -> ((q * d + m) %/ d)%Z = q + (m %/ d)%Z.
Proof.
rewrite neq_lt -oppr_gt0 => nz_d.
wlog{nz_d} d_gt0: q d / d > 0; last case: d => // d in d_gt0 *.
move=> IH; case/orP: nz_d => /IH// /(_ (- q)).
by rewrite mulrNN !divzN -opprD => /oppr_inj.
wlog q_gt0: q m / q >= 0; last case: q q_gt0 => // q _.
move=> IH; case: q => n; first exact: IH; rewrite NegzE mulNr.
by apply: canRL (addKr _) _; rewrite -IH ?addNKr.
case: m => n; first by rewrite !divz_nat divnMDl.
have [le_qd_n | lt_qd_n] := leqP (q * d) n.
rewrite divNz_nat // NegzE -(subnKC le_qd_n) divnMDl //.
by rewrite -!addnS !PoszD !opprD !addNKr divNz_nat.
rewrite divNz_nat // NegzE -PoszM subzn // divz_nat.
apply: canRL (addrK _) _; congr _%:Z; rewrite addnC -divnMDl // mulSnr.
rewrite -{3}(subnKC (ltn_pmod n d_gt0)) addnA addnS -divn_eq addnAC.
by rewrite subnKC // divnMDl // divn_small ?addn0 // subnSK ?ltn_mod ?leq_subr.
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
|
divzMDl
| |
mulzKm d : d != 0 -> (m * d %/ d)%Z = m.
Proof. by move=> d_nz; rewrite -[m * d]addr0 divzMDl // div0z addr0. 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
|
mulzK
| |
mulKzm d : d != 0 -> (d * m %/ d)%Z = m.
Proof. by move=> d_nz; rewrite mulrC 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
|
mulKz
| |
expzBp m n : p != 0 -> (m >= n)%N -> p ^+ (m - n) = (p ^+ m %/ p ^+ n)%Z.
Proof. by move=> p_nz /subnK{2}<-; rewrite exprD mulzK // expf_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
|
expzB
| |
modz1m : (m %% 1)%Z = 0.
Proof. by case: m => n; rewrite (modNz_nat, modz_nat) ?modn1. 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
|
modz1
| |
divz1m : (m %/ 1)%Z = m. Proof. by rewrite -{1}[m]mulr1 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
|
divz1
| |
divzzd : (d %/ d)%Z = (d != 0).
Proof. by have [-> // | d_nz] := eqVneq; rewrite -{1}[d]mul1r 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
|
divzz
| |
ltz_pmodm d : d > 0 -> (m %% d)%Z < d.
Proof.
case: m d => n [] // d d_gt0; first by rewrite modz_nat ltz_nat ltn_pmod.
by rewrite modNz_nat // -lezD1 addrAC subrK gerDl oppr_le0.
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
|
ltz_pmod
| |
ltz_modm d : d != 0 -> (m %% d)%Z < `|d|.
Proof. by rewrite -absz_gt0 -modz_abs => d_gt0; apply: ltz_pmod. 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
|
ltz_mod
| |
divzMplp m d : p > 0 -> (p * m %/ (p * d) = m %/ d)%Z.
Proof.
case: p => // p p_gt0; wlog d_gt0: d / d > 0; last case: d => // d in d_gt0 *.
by move=> IH; case/intP: d => [|d|d]; rewrite ?mulr0 ?divz0 ?mulrN ?divzN ?IH.
rewrite {1}(divz_eq m d) mulrDr mulrCA divzMDl ?mulf_neq0 ?gt_eqF // addrC.
rewrite divz_small ?add0r // PoszM pmulr_rge0 ?modz_ge0 ?gt_eqF //=.
by rewrite ltr_pM2l ?ltz_pmod.
Qed.
Arguments divzMpl [p m 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
|
divzMpl
| |
divzMprp m d : p > 0 -> (m * p %/ (d * p) = m %/ d)%Z.
Proof. by move=> p_gt0; rewrite -!(mulrC p) divzMpl. Qed.
Arguments divzMpr [p m 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
|
divzMpr
| |
lez_floorm d : d != 0 -> (m %/ d)%Z * d <= m.
Proof. by rewrite -subr_ge0; apply: modz_ge0. 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
|
lez_floor
| |
lez_divm d : (`|(m %/ d)%Z| <= `|m|)%N.
Proof.
wlog d_gt0: d / d > 0; last case: d d_gt0 => // d d_gt0.
by move=> IH; case/intP: d => [|n|n]; rewrite ?divz0 ?divzN ?abszN // IH.
case: m => n; first by rewrite divz_nat leq_div.
by rewrite divNz_nat // NegzE !abszN ltnS leq_div.
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
|
lez_div
| |
ltz_ceilm d : d > 0 -> m < ((m %/ d)%Z + 1) * d.
Proof.
by case: d => // d d_gt0; rewrite mulrDl mul1r -ltrBlDl ltz_mod ?gt_eqF.
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
|
ltz_ceil
| |
ltz_divLRm n d : d > 0 -> ((m %/ d)%Z < n) = (m < n * d).
Proof.
move=> d_gt0; apply/idP/idP.
by rewrite -[_ < n]lezD1 -(ler_pM2r d_gt0); exact/lt_le_trans/ltz_ceil.
by rewrite -(ltr_pM2r d_gt0 _ n); apply/le_lt_trans/lez_floor; rewrite gt_eqF.
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
|
ltz_divLR
| |
lez_divRLm n d : d > 0 -> (m <= (n %/ d)%Z) = (m * d <= n).
Proof. by move=> d_gt0; rewrite !leNgt ltz_divLR. 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
|
lez_divRL
| |
lez_pdiv2rd : 0 <= d -> {homo divz^~ d : m n / m <= n}.
Proof.
by case: d => [[|d]|]// _ [] m [] n //; rewrite /divz !mul1r; apply: leq_div2r.
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
|
lez_pdiv2r
| |
divz_ge0m d : d > 0 -> ((m %/ d)%Z >= 0) = (m >= 0).
Proof. by case: d m => // d [] n d_gt0; rewrite (divz_nat, divNz_nat). 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
|
divz_ge0
| |
divzMA_ge0m n p : n >= 0 -> (m %/ (n * p) = (m %/ n)%Z %/ p)%Z.
Proof.
case: n => // [[|n]] _; first by rewrite mul0r !divz0 div0z.
wlog p_gt0: p / p > 0; last case: p => // p in p_gt0 *.
by case/intP: p => [|p|p] IH; rewrite ?mulr0 ?divz0 ?mulrN ?divzN // IH.
rewrite {2}(divz_eq m (n.+1%:Z * p)) mulrA mulrAC !divzMDl // ?gt_eqF //.
rewrite [rhs in _ + rhs]divz_small ?addr0 // ltz_divLR // divz_ge0 //.
by rewrite mulrC ltz_pmod ?modz_ge0 ?gt_eqF ?pmulr_lgt0.
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_ge0
| |
modz_smallm d : 0 <= m < d -> (m %% d)%Z = m.
Proof. by case: m d => //= m [] // d; rewrite modz_nat => /modn_small->. 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
|
modz_small
| |
modz_modm d : ((m %% d)%Z = m %[mod d])%Z.
Proof.
rewrite -!(modz_abs _ d); case: {d}`|d|%N => [|d]; first by rewrite !modz0.
by rewrite modz_small ?modz_ge0 ?ltz_mod.
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
|
modz_mod
| |
modzMDlp m d : (p * d + m = m %[mod d])%Z.
Proof.
have [-> | d_nz] := eqVneq d 0; first by rewrite mulr0 add0r.
by rewrite /modz divzMDl // mulrDl opprD addrACA subrr 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
|
modzMDl
| |
mulz_modr{p m d} : 0 < p -> p * (m %% d)%Z = ((p * m) %% (p * d))%Z.
Proof.
case: p => // p p_gt0; rewrite mulrBr; apply: canLR (addrK _) _.
by rewrite mulrCA -(divzMpl p_gt0) subrK.
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_modr
| |
mulz_modl{p m d} : 0 < p -> (m %% d)%Z * p = ((m * p) %% (d * p))%Z.
Proof. by rewrite -!(mulrC p); apply: mulz_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
|
mulz_modl
| |
modzDlm d : (d + m = m %[mod d])%Z.
Proof. by rewrite -{1}[d]mul1r 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
|
modzDl
| |
modzDrm d : (m + d = m %[mod d])%Z.
Proof. by rewrite addrC modzDl. 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
|
modzDr
| |
modzzd : (d %% d)%Z = 0.
Proof. by rewrite -{1}[d]addr0 modzDl mod0z. 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
|
modzz
| |
modzMlp d : (p * d %% d)%Z = 0.
Proof. by rewrite -[p * d]addr0 modzMDl mod0z. 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
|
modzMl
| |
modzMrp d : (d * p %% d)%Z = 0.
Proof. by rewrite mulrC modzMl. 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
|
modzMr
| |
modzDmlm n d : ((m %% d)%Z + n = m + n %[mod d])%Z.
Proof. by rewrite {2}(divz_eq m d) -[_ * d + _ + n]addrA 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
|
modzDml
| |
modzDmrm n d : (m + (n %% d)%Z = m + n %[mod d])%Z.
Proof. by rewrite !(addrC m) modzDml. 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
|
modzDmr
| |
modzDmm n d : ((m %% d)%Z + (n %% d)%Z = m + n %[mod d])%Z.
Proof. by rewrite modzDml modzDmr. 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
|
modzDm
| |
eqz_modDlp m n d : (p + m == p + n %[mod d])%Z = (m == n %[mod d])%Z.
Proof.
have [-> | d_nz] := eqVneq d 0; first by rewrite !modz0 (inj_eq (addrI p)).
apply/eqP/eqP=> eq_mn; last by rewrite -modzDmr eq_mn modzDmr.
by rewrite -(addKr p m) -modzDmr eq_mn modzDmr addKr.
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_modDl
| |
eqz_modDrp m n d : (m + p == n + p %[mod d])%Z = (m == n %[mod d])%Z.
Proof. by rewrite -!(addrC p) eqz_modDl. 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_modDr
| |
modzMmlm n d : ((m %% d)%Z * n = m * n %[mod d])%Z.
Proof. by rewrite {2}(divz_eq m d) [in RHS]mulrDl mulrAC 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
|
modzMml
| |
modzMmrm n d : (m * (n %% d)%Z = m * n %[mod d])%Z.
Proof. by rewrite !(mulrC m) modzMml. 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
|
modzMmr
| |
modzMmm n d : ((m %% d)%Z * (n %% d)%Z = m * n %[mod d])%Z.
Proof. by rewrite modzMml modzMmr. 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
|
modzMm
| |
modzXmk m d : ((m %% d)%Z ^+ k = m ^+ k %[mod d])%Z.
Proof. by elim: k => // k IHk; rewrite !exprS -modzMmr IHk modzMm. 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
|
modzXm
| |
modzNmm d : (- (m %% d)%Z = - m %[mod d])%Z.
Proof. by rewrite -mulN1r modzMmr mulN1r. 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
|
modzNm
| |
modz_absmm d : ((-1) ^+ (m < 0)%R * (m %% d)%Z = `|m|%:Z %[mod d])%Z.
Proof. by rewrite modzMmr -abszEsign. 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
|
modz_absm
| |
dvdzEd m : (d %| m)%Z = (`|d| %| `|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
|
dvdzE
| |
dvdz0d : (d %| 0)%Z. Proof. exact: dvdn0. 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
|
dvdz0
| |
dvd0zn : (0 %| n)%Z = (n == 0). Proof. by rewrite -absz_eq0 -dvd0n. 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
|
dvd0z
| |
dvdz1d : (d %| 1)%Z = (`|d|%N == 1). Proof. exact: dvdn1. 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
|
dvdz1
| |
dvd1zm : (1 %| m)%Z. Proof. exact: dvd1n. 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
|
dvd1z
| |
dvdzzm : (m %| m)%Z. Proof. exact: dvdnn. 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
|
dvdzz
| |
dvdz_mulld m n : (d %| n)%Z -> (d %| m * n)%Z.
Proof. by rewrite !dvdzE abszM; apply: dvdn_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
|
dvdz_mull
| |
dvdz_mulrd m n : (d %| m)%Z -> (d %| m * n)%Z.
Proof. by move=> d_m; rewrite mulrC dvdz_mull. Qed.
#[global] Hint Resolve dvdz0 dvd1z dvdzz dvdz_mull dvdz_mulr : core.
|
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_mulr
| |
dvdz_muld1 d2 m1 m2 : (d1 %| m1 -> d2 %| m2 -> d1 * d2 %| m1 * m2)%Z.
Proof. by rewrite !dvdzE !abszM; apply: dvdn_mul. 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_mul
| |
dvdz_transn d m : (d %| n -> n %| m -> d %| m)%Z.
Proof. by rewrite !dvdzE; apply: dvdn_trans. 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_trans
| |
dvdzPd m : reflect (exists q, m = q * d) (d %| m)%Z.
Proof.
apply: (iffP dvdnP) => [] [q Dm]; last by exists `|q|%N; rewrite Dm abszM.
exists ((-1) ^+ (m < 0)%R * q%:Z * (-1) ^+ (d < 0)%R).
by rewrite -!mulrA -abszEsign -PoszM -Dm -intEsign.
Qed.
Arguments dvdzP {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
|
dvdzP
| |
dvdz_mod0Pd m : reflect (m %% d = 0)%Z (d %| m)%Z.
Proof.
apply: (iffP dvdzP) => [[q ->] | md0]; first by rewrite modzMl.
by rewrite (divz_eq m d) md0 addr0; exists (m %/ d)%Z.
Qed.
Arguments dvdz_mod0P {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_mod0P
| |
dvdz_eqd m : (d %| m)%Z = ((m %/ d)%Z * d == m).
Proof. by rewrite (sameP dvdz_mod0P eqP) subr_eq0 eq_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
|
dvdz_eq
| |
divzKd m : (d %| m)%Z -> (m %/ d)%Z * d = m.
Proof. by rewrite dvdz_eq => /eqP. 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
|
divzK
| |
lez_divLRd m n : 0 < d -> (d %| m)%Z -> ((m %/ d)%Z <= n) = (m <= n * d).
Proof. by move=> /ler_pM2r <- /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
|
lez_divLR
| |
ltz_divRLd m n : 0 < d -> (d %| m)%Z -> (n < m %/ d)%Z = (n * d < m).
Proof. by move=> /ltr_pM2r/(_ n)<- /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
|
ltz_divRL
| |
eqz_divd m n : d != 0 -> (d %| m)%Z -> (n == m %/ d)%Z = (n * d == m).
Proof. by move=> /mulIf/inj_eq <- /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
|
eqz_div
| |
eqz_muld m n : d != 0 -> (d %| m)%Z -> (m == n * d) = (m %/ d == n)%Z.
Proof. by move=> d_gt0 dv_d_m; rewrite eq_sym -eqz_div // eq_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
|
eqz_mul
| |
divz_mulACd m n : (d %| m)%Z -> (m %/ d)%Z * n = (m * n %/ d)%Z.
Proof.
have [-> | d_nz] := eqVneq d 0; first by rewrite !divz0 mul0r.
by move/divzK=> {2} <-; rewrite 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
|
divz_mulAC
| |
mulz_divAd m n : (d %| n)%Z -> m * (n %/ d)%Z = (m * n %/ d)%Z.
Proof. by move=> dv_d_m; rewrite !(mulrC m) divz_mulAC. 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_divA
| |
mulz_divCAd m n :
(d %| m)%Z -> (d %| n)%Z -> m * (n %/ d)%Z = n * (m %/ d)%Z.
Proof. by move=> dv_d_m dv_d_n; rewrite mulrC divz_mulAC ?mulz_divA. 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
|
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