End NatChain.
Section Kleene.
Variables (D : cpo) (f : D -> D) (C : continuous f).
Fixpoint pow m := if m is n.+1 then f (pow n) else bot.

(* Specifications: *)
Lemma pow_mono : monotone pow.
Definition pow_chain := [pow ^^ nat_chain by pow_mono].
Lemma reindex : pow_chain =p lift_chain [f ^^ pow_chain by cont_mono C].
Definition kleene_lfp := lim pow_chain.
Lemma kleene_lfp_fixed : f kleene_lfp = kleene_lfp.
End Kleene.