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|].

Theorem Zdiv_lt_upper_bound: forall a b q, 0 < b -> a < q*b -> a/b < q.