Definition R_noet (x y : nat * nat) : Prop :=
  ((fst x) + (snd x) < (fst y) + (snd y))%nat.Definition f (x : nat * nat) := ((fst x) + (snd x))%nat.

Lemma infinite_descent_nat : forall P : nat * nat -> Prop, (forall x : nat * nat, (P x -> exists y : nat * nat, (fst(y) + snd(y) < fst(x) + snd(x))%nat /\ P y)) -> forall x : nat * nat, ~(P x).