Definition DecRel (A : Set) (R : Rel A) :=
  forall x y : A, {R x y} + {~ R x y}.

(* Specifications: *)
Theorem nat_double_ind_set : forall R : nat -> nat -> Set, (forall n : nat, R 0 n) -> (forall n : nat, R (S n) 0) -> (forall n m : nat, R n m -> R (S n) (S m)) -> forall n m : nat, R n m.
Lemma dec_lt : DecRel nat lt.