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] *)
(** Results about [nth_error] *)
(*****************)
(** ** Remove    *)
(*****************)
Hypothesis eq_dec : forall x y : A, {x = y}+{x <> y}.
Fixpoint remove (x : A) (l : list A) : list A :=
match l with
| [] => []
| y::tl => if (eq_dec x y) then remove x tl else y::(remove x tl)
end.
(******************************)
(** ** Last element of a list *)
(******************************)
(** [last l d] returns the last element of the list [l], or the default value [d] if [l] is empty. *)
Fixpoint last (l:list A) (d:A) : A :=
match l with
| [] => d
| [a] => a
| a :: l => last l d
end.
(** [removelast l] remove the last element of [l] *)
Fixpoint removelast (l:list A) : list A :=
match l with
| [] =>  []
| [a] => []
| a :: l => a :: removelast l
end.
(******************************************)
(** ** Counting occurrences of an element *)
(******************************************)
Fixpoint count_occ (l : list A) (x : A) : nat :=
match l with
| [] => 0
| y :: tl =>
let n := count_occ tl x in
if eq_dec y x then S n else n
end.

Theorem count_occ_inv_nil l :
(forall x:A, count_occ l x = 0) <-> l = [].