Inductive ct: comparison -> comparison -> comparison -> Prop :=
| ct_xxx: forall x, ct x  x  x
| ct_xex: forall x, ct x  Eq x
| ct_exx: forall x, ct Eq x  x
| ct_glx: forall x, ct Gt Lt x
| ct_lgx: forall x, ct Lt Gt x.

Lemma compare_x_empty: forall a, is_empty a = true ->
forall b, compare b a = if is_empty b then Eq else Gt.