(*****************)
(** ** Remove    *)
(*****************)
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).