Fixpoint fin_to_nat {n : nat} : fin n -> nat :=
  match n with
  | 0 => fun x : fin 0 => match x with end
  | S n' => fun x : fin (S n') =>
             match x with
             | None => 0
             | Some y => S (fin_to_nat y)
             end
  end.Definition fin_lt {n : nat} (a b : fin n) : Prop := lt (fin_to_nat a) (fin_to_nat b).

Lemma fin_lt_not_eq : forall n (x y : fin n), fin_lt x y -> x <> y.