(** *** Order *)

(* Specifications: *)
Lemma filterlim_le :
forall {T F} {FF : ProperFilter' F} (f g : T -> R) (lf lg : Rbar),
F (fun x => f x <= g x) ->
filterlim f F (Rbar_locally lf) ->
filterlim g F (Rbar_locally lg) ->
Rbar_le lf lg.
Lemma is_lim_seq_le_loc (u v : nat -> R) (l1 l2 : Rbar) :
eventually (fun n => u n <= v n) ->
is_lim_seq u l1 -> is_lim_seq v l2 ->
Rbar_le l1 l2.
Lemma Lim_seq_le_loc (u v : nat -> R) :
eventually (fun n => u n <= v n) ->
Rbar_le (Lim_seq u) (Lim_seq v).
Lemma is_lim_seq_le (u v : nat -> R) (l1 l2 : Rbar) :
(forall n, u n <= v n) -> is_lim_seq u l1 -> is_lim_seq v l2 -> Rbar_le l1 l2.
Lemma filterlim_ge_p_infty :
forall {T F} {FF : Filter F} (f g : T -> R),
F (fun x => f x <= g x) ->
filterlim f F (Rbar_locally p_infty) ->
filterlim g F (Rbar_locally p_infty).
Lemma filterlim_le_m_infty :
forall {T F} {FF : Filter F} (f g : T -> R),
F (fun x => g x <= f x) ->
filterlim f F (Rbar_locally m_infty) ->
filterlim g F (Rbar_locally m_infty).
Lemma filterlim_le_le :
forall {T F} {FF : Filter F} (f g h : T -> R) (l : Rbar),
F (fun x => f x <= g x <= h x) ->
filterlim f F (Rbar_locally l) ->
filterlim h F (Rbar_locally l) ->
filterlim g F (Rbar_locally l).
Lemma is_lim_seq_le_le_loc (u v w : nat -> R) (l : Rbar) :
eventually (fun n => u n <= v n <= w n) -> is_lim_seq u l -> is_lim_seq w l -> is_lim_seq v l.
Lemma is_lim_seq_le_le (u v w : nat -> R) (l : Rbar) :
(forall n, u n <= v n <= w n) -> is_lim_seq u l -> is_lim_seq w l -> is_lim_seq v l.
Lemma is_lim_seq_le_p_loc (u v : nat -> R) :
eventually (fun n => u n <= v n) ->
is_lim_seq u p_infty ->
is_lim_seq v p_infty.
Lemma is_lim_seq_le_m_loc (u v : nat -> R) :
eventually (fun n => v n <= u n) ->
is_lim_seq u m_infty ->
is_lim_seq v m_infty.
Lemma is_lim_seq_decr_compare (u : nat -> R) (l : R) :
is_lim_seq u l
-> (forall n, (u (S n)) <= (u n))
-> forall n, l <= u n.
Lemma is_lim_seq_incr_compare (u : nat -> R) (l : R) :
is_lim_seq u l
-> (forall n, (u n) <= (u (S n)))
-> forall n, u n <= l.
Lemma ex_lim_seq_decr (u : nat -> R) :
(forall n, (u (S n)) <= (u n))
-> ex_lim_seq u.
Lemma ex_lim_seq_incr (u : nat -> R) :
(forall n, (u n) <= (u (S n)))
-> ex_lim_seq u.
Lemma ex_finite_lim_seq_decr (u : nat -> R) (M : R) :
(forall n, (u (S n)) <= (u n)) -> (forall n, M <= u n)
-> ex_finite_lim_seq u.
Lemma ex_finite_lim_seq_incr (u : nat -> R) (M : R) :
(forall n, (u n) <= (u (S n))) -> (forall n, u n <= M)
-> ex_finite_lim_seq u.
Lemma is_lim_le_loc (f g : R -> R) (x lf lg : Rbar) :
Rbar_locally' x (fun y => f y <= g y) ->
is_lim f x lf -> is_lim g x lg ->
Rbar_le lf lg.
Lemma is_lim_le_p_loc (f g : R -> R) (x : Rbar) :
Rbar_locally' x (fun y => f y <= g y) ->
is_lim f x p_infty ->
is_lim g x p_infty.
Lemma is_lim_le_m_loc (f g : R -> R) (x : Rbar) :
Rbar_locally' x (fun y => g y <= f y) ->
is_lim f x m_infty ->
is_lim g x m_infty.
Lemma is_lim_le_le_loc (f g h : R -> R) (x : Rbar) (l : Rbar) :
Rbar_locally' x (fun y => f y <= h y <= g y) ->
is_lim f x l -> is_lim g x l ->
is_lim h x l.