Definition propPosetMixin :=
  PosetMixin prop_botP prop_refl prop_asym prop_trans.Canonical propPoset := Eval hnf in Poset Prop propPosetMixin.Definition predPosetMixin : Poset.mixin_of (Pred A) :=
  funPosetMixin A propPoset.Canonical predPoset := Eval hnf in Poset (Pred A) predPosetMixin.

Lemma nat_botP x : 0 <= x.