(** ** 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_add:
forall a b c d, eqmod a b -> eqmod c d -> eqmod (a + c) (b + d).