(******************************************)
  (** ** 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_not_In l x : ~ In x l <-> count_occ l x = 0.