Section EqualityModulo.
Variable N:Z.
Definition eqm a b := (a mod N = b mod N).
Infix "==" := eqm (at level 70).
Instance Zplus_eqm : Proper (eqm ==> eqm ==> eqm) Z.add.
Instance Zminus_eqm : Proper (eqm ==> eqm ==> eqm) Z.sub.
Instance Zmult_eqm : Proper (eqm ==> eqm ==> eqm) Z.mul.
Instance Zopp_eqm : Proper (eqm ==> eqm) Z.opp.
(* NB: Z.modulo and Z.div are not morphisms with respect to eqm. For instance, let (==) be (eqm 2). Then we have (3 == 1) but: ~ (3 mod 3 == 1 mod 3) ~ (1 mod 3 == 1 mod 1) ~ (3/3 == 1/3) ~ (1/3 == 1/1) *)
End EqualityModulo.
(** Unfortunately, the previous result isn't always true on negative numbers. For instance: 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) *)
(** A last inequality: *)
(** Particular case : dividing by 2 is related with parity *)
Proof Z.div2_div.
(** * Compatibility *)
(** Weaker results kept only for compatibility *)
(** * A direct way to compute Z.modulo *)
Fixpoint Zmod_POS (a : positive) (b : Z) : Z  :=
match a with
| xI a' =>
let r := Zmod_POS a' b in
let r' := (2 * r + 1) in
if r' <? b then r' else (r' - b)
| xO a' =>
let r := Zmod_POS a' b in
let r' := (2 * r) in
if r' <? b then r' else (r' - b)
| xH => if 2 <=? b then 1 else 0
end.
Definition Zmod' a b :=
match a with
| Z0 => 0
| Zpos a' =>
match b with
| Z0 => 0
| Zpos _ => Zmod_POS a' b
| Zneg b' =>
let r := Zmod_POS a' (Zpos b') in
match r  with Z0 =>  0 |  _  =>   b + r end
end
| Zneg a' =>
match b with
| Z0 => 0
| Zpos _ =>
let r := Zmod_POS a' b in
match r with Z0 =>  0 | _  =>  b - r end
| Zneg b' => - (Zmod_POS a' (Zpos b'))
end
end.
(** Another convention is possible for division by negative numbers: * quotient is always the biggest integer smaller than or equal to a/b * remainder is hence always positive or null. *)

Theorem Zdiv_eucl_extended :
forall b:Z,
b <> 0 ->
forall a:Z,
{qr : Z * Z | let (q, r) := qr in a = b * q + r /\ 0 <= r < Z.abs b}.