(** Quantifications on both sides of a relation *)
    Definition bisimulation R := simulation  TX TY R /\ simulation TY TX (trans R).Definition expansion    R := expansion1  TX TY R /\ simulation TY TX (trans R).Definition wexpansion   R := wexpansion1 TX TY R /\ simulation TY TX (trans R).(** Definition of the co-inductive relations as `greatest relations such that' *)
    Definition bisim   := union_st bisimulation.Definition expand  := union_st expansion.Definition wexpand := union_st wexpansion.

Lemma expansion1_comp: expansion1 TX TY R -> expansion1 TY TZ S -> expansion1 TX TZ (comp R S).