Inductive P_nth (init:nat) : nat -> nat -> Prop :=
| nth_O : P_nth init init 0
| nth_S :
forall k l (n:nat),
P_nth init k n -> between (S k) l -> Q l -> P_nth init l (S n).

(* Specifications: *)
Lemma nth_le : forall (init:nat) l (n:nat), P_nth init l n -> init <= l.
Definition eventually (n:nat) :=  exists2 k : nat, k <= n & Q k.
Lemma event_O : eventually 0 -> Q 0.
End Between.