Ltac zero_or_not a :=
  destruct (Z.eq_dec a 0);
  [subst; rewrite ?Zmod_0_l, ?Zdiv_0_l, ?Zmod_0_r, ?Zdiv_0_r;
   auto with zarith|].

Lemma Zplus_mod_idemp_l: forall a b n, (a mod n + b) mod n = (a + b) mod n.