Definition nat_directed_set {X : hSet} (PX : PartialOrder X) (D : ℕ → X) (HD : ∏ (i : ℕ), PX (D i) (D (S i))) : directed_set PX. Proof. use (nat_directed_set_monotone PX D). abstract (intros i j p ; pose (k := nat_le_diff p) ; induction k as [ k q ] ; rewrite <- q ; use nat_directed_set_help_monotone ; exact HD). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.Posets.Basics. Require Import UniMath.OrderTheory.Posets.MonotoneFunctions.