Definition pairPosetMixin :=
  PosetMixin pair_botP pair_refl pair_asym pair_trans.Canonical pairPoset := Eval hnf in Poset tp pairPosetMixin.Definition fun_bot : tp := fun x => bot.

(* Specifications: *)
Lemma fun_botP x : fun_leq fun_bot x.
Lemma fun_refl x : fun_leq x x.
Lemma fun_asym x y : fun_leq x y -> fun_leq y x -> x = y.
Lemma fun_trans x y z : fun_leq x y -> fun_leq y z -> fun_leq x z.