Section Elts.
Variable A : Type.
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(** ** Nth element of a list *)
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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] *)
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(** ** Remove    *)
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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.

Theorem remove_In : forall (l : list A) (x : A), ~ In x (remove x l).