Section SCOTT_FIX.
Context {D : Type}.
Fixpoint iterS (n : nat) (f : D -> D) (x : D) {struct n} : D :=
match n with
| O => x
| S n' => f (iterS n' f x)
end.
Variant IterS (f : D -> D) (x : D) : ensemble D :=
| In_IterS (n : nat) : iterS n f x \in IterS f x.
#[local] Hint Constructors IterS : core.
Context {PROSET : isProset D} {CPO : isCpo D}.
Proof with eauto with *.
assert (claim1 : forall n : nat, iterS n f bottom_cpo =< iterS (S n) f bottom_cpo).
{ induction n as [ | n IH]; [eapply bottom_cpo_spec | simpl]... }
intros n1 n2 n1_le_n2. induction n1_le_n2 as [ | n2 n1_le_n2 IH].
- reflexivity.
- rewrite IH...
Qed.

Lemma IterS_f_bottom_isDirected_if_f_isMonotonic (f : D -> D)
(f_isMonotonic : isMonotonic1 f)
: isDirected (IterS f bottom_cpo).