Proof using.
intros P si s su. destruct su as [si s sf _].
induction sf as [ | x s0 _ Hrec]; intro a.
- constructor 1. case a; simpl. intros; assumption.
- constructor 2. apply Hrec. apply a.
Qed.

(* Specifications: *)
Lemma subseqs_tl : forall si s, subseqs si (tl s) -> subseqs si s.
Proof using.