Definition idealPosetMixin :=
  PosetMixin ideal_botP ideal_refl ideal_asym ideal_trans.Canonical idealPoset := Eval hnf in Poset (ideal P) idealPosetMixin.Definition prop_bot := False.Definition prop_leq (p1 p2 : Prop) := p1 -> p2.

(* Specifications: *)
Lemma prop_botP x : prop_leq prop_bot x.
Lemma prop_refl x : prop_leq x x.
Lemma prop_asym x y : prop_leq x y -> prop_leq y x -> x = y.
Lemma prop_trans x y z : prop_leq x y -> prop_leq y z -> prop_leq x z.