Proof using.
intros f P Q extP extQ QP s.
apply eventually_map_conv_ext with (J := True_tl); auto.
- apply extensional_True_tl.
- intros. apply E0. apply QP. assumption.
Qed.

Lemma eventually_map_monotonic :
forall (f: A->B) (P Q J: infseq A->Prop) (K : infseq B -> Prop),
(forall x s, J (Cons x s) -> J s) ->
(forall x s, K (map f (Cons x s)) -> K (map f s)) ->
(forall s, J s -> K (map f s) -> Q s -> P s) ->
forall s, J s -> K (map f s) -> eventually Q s -> eventually P s.