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 .