Proof using.
intros P Q J is_inv JPQ s Js ev.
apply (eventually_trans P Q J is_inv); try assumption.
intros; constructor 1. apply JPQ; assumption.
Qed.

(* corollary which turns out to be too weak in practice *)
Lemma eventually_monotonic_simple :
forall (P Q: infseq T -> Prop),
(forall s, P s -> Q s) ->
forall s, eventually P s -> eventually Q s.