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 Zdiv_le_compat_l: forall p q r, 0 <= p -> 0 < q < r -> p / r <= p / q.