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 ct_compare:
forall a b c, ct (compare a b) (compare b c) (compare a c).