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.

Lemma prop_refl x : prop_leq x x.