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.