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).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).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).

Lemma nth_le : forall (init:nat) l (n:nat), P_nth init l n -> init <= l.