Section Elts.
Variable A : Type.
(*****************************)
(** ** Nth element of a list *)
(*****************************)
Fixpoint nth (n:nat) (l:list A) (default:A) {struct l} : A :=
match n, l with
| O, x :: l' => x
| O, other => default
| S m, [] => default
| S m, x :: t => nth m t default
end.
Fixpoint nth_ok (n:nat) (l:list A) (default:A) {struct l} : bool :=
match n, l with
| O, x :: l' => true
| O, other => false
| S m, [] => false
| S m, x :: t => nth_ok m t default
end.
Fixpoint nth_error (l:list A) (n:nat) {struct n} : option A :=
match n, l with
| O, x :: _ => Some x
| S n, _ :: l => nth_error l n
| _, _ => None
end.
Definition nth_default (default:A) (l:list A) (n:nat) : A :=
match nth_error l n with
| Some x => x
| None => default
end.
(** Results about [nth] *)

Lemma nth_split n l d : n < length l ->
exists l1, exists l2, l = l1 ++ nth n l d :: l2 /\ length l1 = n.