Definition converges_to (a: A) (b: option B) : Prop :=
  exists n, forall m, (n <= m)%nat -> iter m a = b.

(* Specifications: *)
Lemma converges_to_Some: forall a n b, iter n a = Some b -> converges_to a (Some b).
Lemma converges_to_exists: forall a, exists b, converges_to a b.
Lemma converges_to_unique: forall a b, converges_to a b -> forall b', converges_to a b' -> b = b'.
Lemma converges_to_exists_uniquely: forall a, exists! b, converges_to a b .