Fixpoint fin_compare (n : nat) :
  forall (x y : fin n), { cmp : comparison | CompSpec eq fin_lt x y cmp } :=
  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 n' x' y' with
                   | exist _ Lt Hc => exist _ Lt (CompLt _ _ (fin_lt_Some_intro (CompSpec_Lt Hc)))
                   | exist _ Eq Hc => exist _ Eq (CompEq _ _ (CompSpec_Eq_Some Hc))
                   | exist _ Gt Hc => exist _ Gt (CompGt _ _ (fin_lt_Some_intro (CompSpec_Gt Hc)))
                 end
               | Some x', None => exist _ Gt (CompGt _ _ (None_lt_Some n' x'))
               | None, Some y' => exist _ Lt (CompLt _ _ (None_lt_Some n' y'))
               | None, None => exist _ Eq (CompEq _ _ eq_refl)
             end
  end.Definition t := fin n.Definition eq := @eq (fin n).Definition eq_equiv := @eq_equivalence (fin n).Definition lt := @fin_lt n.Definition lt_strorder := fin_lt_strorder n.Definition lt_compat := fin_lt_lt_compat n.Definition compare := fun x y => proj1_sig (fin_compare n x y).Definition compare_spec := fun x y => proj2_sig (fin_compare n x y).Definition eq_dec := fin_eq_dec n.Fixpoint fin_of_nat (m n : nat) : fin n + {exists p, m = n + p} :=
  match n with
    | 0 => inright (ex_intro _ _ eq_refl)
    | S n' =>
      match m with
        | 0 => inleft None
        | S m' =>
          match fin_of_nat m' n' with
            | inleft f => inleft (Some f)
            | inright pf =>
              inright (match pf with
                         | ex_intro _ x H => ex_intro _ x (f_equal _ H)
                       end)
          end
    end
  end.

(* Specifications: *)
Lemma fin_of_nat_fin_to_nat : forall (n : nat) (a : fin n), fin_of_nat (fin_to_nat a) n = inleft a.