(* --------------------------------------------------------------------------- *)
(* Infinite subsequences *)
Section sec_subseq.
Variable T: Type.
(* suff s s'  means  s is a suffix of s' *)
Inductive suff (s : infseq T) : infseq T -> Prop :=
| sp_eq : suff s s
| sp_next : forall x s0, suff s s0 -> suff s (Cons x s0).
(* simple but not the most useful -- useless indeed *)
CoInductive subseq : infseq T -> infseq T -> Prop :=
| Subseq : forall s s0 s1,
suff s1 s0 -> subseq s (tl s1) -> subseq (Cons (hd s1) s) s0.
CoInductive subseqs' : infseq (infseq T) -> infseq T -> Prop :=
| Subseqs' : forall si s0 s1,
suff s1 s0 -> subseqs' si (tl s1) -> subseqs' (Cons s1 si) s0.
CoInductive subseqs : infseq (infseq T) -> infseq T -> Prop :=
| Subseqs : forall si s,
suff (hd si) s -> subseqs (tl si) (tl (hd si)) -> subseqs si s.

Lemma subseqs_subseqs' : forall si s, subseqs si s -> subseqs' si s.
Proof using.