(** ** Modulo arithmetic *)
(** We define and state properties of equality and arithmetic modulo a positive integer. *)
Section EQ_MODULO.
Variable modul: Z.
Hypothesis modul_pos: modul > 0.
Definition eqmod (x y: Z) : Prop := exists k, x = k * modul + y.

Lemma eqmod_small_eq:
forall x y, eqmod x y -> 0 <= x < modul -> 0 <= y < modul -> x = y.