Definition lub (Un:nat -> R) (pr:has_ub Un) : R :=
  let (a,_) := ub_to_lub Un pr in a.Definition glb (Un:nat -> R) (pr:has_lb Un) : R :=
  let (a,_) := lb_to_glb Un pr in - a.(* Compatibility with previous unappropriate terminology *)
Notation maj_sup := ub_to_lub (only parsing).Notation min_inf := lb_to_glb (only parsing).Notation majorant := lub (only parsing).Notation minorant := glb (only parsing).

(* Specifications: *)
Lemma maj_ss : forall (Un:nat -> R) (k:nat), has_ub Un -> has_ub (fun i:nat => Un (k + i)%nat).
Lemma min_ss : forall (Un:nat -> R) (k:nat), has_lb Un -> has_lb (fun i:nat => Un (k + i)%nat).