(** We now state the (left and right) compatibility of the membership relation w.r.t. the bisimilarity relation.  Both compatibilities rely on the transitivity of the bisimilarity relation, and the right compatibility uses the EQV_shift lemma we proved above. *)

(* Specifications: *)
Lemma ELT_compat_l :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y)
(b : Y) (Z : Typ1) (C : Rel Z) (c : Z),
EQV X A a Y B b -> ELT Y B b Z C c -> ELT X A a Z C c.
Lemma ELT_compat_r :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y)
(b : Y) (Z : Typ1) (C : Rel Z) (c : Z),
ELT X A a Y B b -> EQV Y B b Z C c -> ELT X A a Z C c.