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 Z_div_plus_full_l: forall a b c : Z, b <> 0 -> (a * b + c) / b = a + c / b.