Definition relation_decreases (E : Set) (R : E -> E -> Prop)
  (f : E -> nat) := forall x y : E, R x y -> f x < f y.

(* Specifications: *)
Lemma Z_nat_business : forall x y : positive, nat_of_P y < nat_of_P x -> (Zpos y < Zpos x)%Z.
Lemma Z_nat_business1 : forall x y : positive, (Zpos y < Zpos x)%Z -> nat_of_P y < nat_of_P x.