Definition increasing (f : nat -> nat) :=
  forall x y : nat, x < y -> f x < f y.

(* Specifications: *)
Lemma compose_increasing : forall f g : nat -> nat, increasing f -> increasing g -> increasing (fun x : nat => f (g x)).
Lemma increasingbystep : forall f : nat -> nat, (forall n : nat, f n < f (S n)) -> increasing f.