(* The converse seems to require much more work *)

Definition fstAB := fst (A:=A) (B:=B).

Lemma eventually_map_conv : forall (f: A->B) (P: infseq A->Prop) (Q: infseq B->Prop), extensional P -> extensional Q -> (forall s, Q (map f s) -> P s) -> forall s, eventually Q (map f s) -> eventually P s.