(** Some preliminary results  **)
Instance not_symmetric (A : Type) (R: relation A) `{Symmetric A R} : Symmetric (fun x y => ~R x y).Instance InA_compat {A : Type} : Proper (subrelation ==> eq ==> eq ==> impl) (@InA A).Definition full_relation {A : Type} : relation A := fun x y : A => True.Definition t := O.t.Definition eq := O.eq.Definition lt := O.lt.Definition eq_refl : forall x, eq x x := reflexivity.

Lemma lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z.