(** Notations *)

Definition Sup_seq (u : nat -> Rbar) := proj1_sig (ex_sup_seq u).Definition Inf_seq (u : nat -> Rbar) := proj1_sig (ex_inf_seq u).

(* Specifications: *)
Lemma is_sup_seq_unique (u : nat -> Rbar) (l : Rbar) : is_sup_seq u l -> Sup_seq u = l.
Lemma Sup_seq_correct (u : nat -> Rbar) : is_sup_seq u (Sup_seq u).
Lemma is_inf_seq_unique (u : nat -> Rbar) (l : Rbar) : is_inf_seq u l -> Inf_seq u = l.
Lemma Inf_seq_correct (u : nat -> Rbar) : is_inf_seq u (Inf_seq u).
Lemma Sup_seq_ext (u v : nat -> Rbar) : (forall n, (u n) = (v n)) -> Sup_seq u = Sup_seq v.
Lemma Inf_seq_ext (u v : nat -> Rbar) : (forall n, (u n) = (v n)) -> Inf_seq u = Inf_seq v.
Lemma Rbar_sup_eq_lub (u : nat -> Rbar) : Sup_seq u = Rbar_lub (fun x => exists n, x = u n).
Lemma Inf_eq_glb (u : nat -> Rbar) : Inf_seq u = Rbar_glb (fun x => exists n, x = u n).
Lemma Sup_opp_inf (u : nat -> Rbar) : Sup_seq u = Rbar_opp (Inf_seq (fun n => Rbar_opp (u n))).
Lemma Inf_opp_sup (u : nat -> Rbar) : Inf_seq u = Rbar_opp (Sup_seq (fun n => Rbar_opp (u n))).
Lemma Sup_seq_scal_l (a : R) (u : nat -> Rbar) : 0 <= a -> Sup_seq (fun n => Rbar_mult a (u n)) = Rbar_mult a (Sup_seq u).
Lemma Inf_seq_scal_l (a : R) (u : nat -> Rbar) : 0 <= a -> Inf_seq (fun n => Rbar_mult a (u n)) = Rbar_mult a (Inf_seq u).
Lemma is_sup_seq_le (u v : nat -> Rbar) {l1 l2 : Rbar} : (forall n, Rbar_le (u n) (v n)) -> (is_sup_seq u l1) -> (is_sup_seq v l2) -> Rbar_le l1 l2.
Lemma is_inf_seq_le (u v : nat -> Rbar) {l1 l2 : Rbar} : (forall n, Rbar_le (u n) (v n)) -> (is_inf_seq u l1) -> (is_inf_seq v l2) -> Rbar_le l1 l2.
Lemma Sup_seq_le (u v : nat -> Rbar) : (forall n, Rbar_le (u n) (v n)) -> Rbar_le (Sup_seq u) (Sup_seq v).
Lemma Inf_seq_le (u v : nat -> Rbar) : (forall n, Rbar_le (u n) (v n)) -> Rbar_le (Inf_seq u) (Inf_seq v).
Lemma Inf_le_sup (u : nat -> Rbar) : Rbar_le (Inf_seq u) (Sup_seq u).
Lemma is_sup_seq_major (u : nat -> Rbar) (l : Rbar) (n : nat) : is_sup_seq u l -> Rbar_le (u n) l.
Lemma Sup_seq_minor_lt (u : nat -> Rbar) (M : R) : Rbar_lt M (Sup_seq u) <-> exists n, Rbar_lt M (u n).
Lemma Sup_seq_minor_le (u : nat -> Rbar) (M : R) (n : nat) : Rbar_le M (u n) -> Rbar_le M (Sup_seq u).