End LiftChain.
(****************************)
(* common cpo constructions *)
(****************************)
(* pairs *)
Section PairCPO.
Variables (A B : cpo).
Local Notation tp := [poset of A * B].
Definition pair_bot : tp := (bot, bot).
Definition pair_lim (s : chain tp) : tp :=
(lim (proj1_chain s), lim (proj2_chain s)).

Lemma chain_clos_idemp (s : Pred T) :
chain_closed s -> chain_closure s =p s.