(* ratInverse only works correctly when the number is positive *)
Definition ratInverse (r : Rat) :=
  match r with
    | RatIntro n d =>
      match n with
        | O => RatIntro d (pos (S O))
        | S n' => RatIntro d (pos (S n'))
      end
  end.Fixpoint expRat r n :=
  match n with
    | O => rat1 
    | S n' => r * (expRat r n')
  end.

Lemma rat_ge_1 : forall n (d : posnat), n >= d -> 1 <= RatIntro n d.