Fixpoint fin_compare_compat (n : nat) : forall (x y : fin n), Compare fin_lt eq x y :=
  match n with
    | 0 => fun x y : fin 0 => match x with end
    | S n' => fun x y : fin (S n') =>
               match x, y with
                 | Some x', Some y' =>
                   match fin_compare_compat n' x' y' with
                     | LT pf => LT (fin_lt_Some_intro pf)
                     | EQ pf => EQ (f_equal _ pf)
                     | GT pf => GT (fin_lt_Some_intro pf)
                   end
                 | Some x', None => GT (None_lt_Some n' x')
                 | None, Some y' => LT (None_lt_Some n' y')
                 | None, None => EQ eq_refl
               end
  end.Definition t := fin n.Definition eq := @eq (fin n).Definition lt := @fin_lt n.Definition eq_refl := @eq_refl (fin n).Definition eq_sym := @eq_sym (fin n).Definition eq_trans := @eq_trans (fin n).Definition lt_trans := @fin_lt_trans n.Definition lt_not_eq := @fin_lt_not_eq n.Definition compare := fin_compare_compat n.Definition eq_dec := fin_eq_dec n.

Lemma fin_lt_irrefl : forall n, Irreflexive (@fin_lt n).