End RInt.
(** ** Order *)

(* Specifications: *)
Lemma is_RInt_ge_0 (f : R -> R) (a b If : R) :
a <= b -> is_RInt f a b If ->
(forall x, a < x < b -> 0 <= f x) -> 0 <= If.
Lemma RInt_ge_0 (f : R -> R) (a b : R) :
a <= b -> ex_RInt f a b
-> (forall x, a < x < b -> 0 <= f x) -> 0 <= RInt f a b.
Lemma is_RInt_le (f g : R -> R) (a b If Ig : R) :
a <= b ->
is_RInt f a b If -> is_RInt g a b Ig ->
(forall x, a < x < b -> f x <= g x) ->
If <= Ig.
Lemma RInt_le (f g : R -> R) (a b : R) :
a <= b ->
ex_RInt f a b -> ex_RInt g a b->
(forall x, a < x < b -> f x <= g x) ->
RInt f a b <= RInt g a b.
Lemma RInt_gt_0 (g : R -> R) (a b : R) :
(a < b) -> (forall x, a < x < b -> (0 < g x)) ->
(forall x, a <= x <= b -> continuous g x) ->
0 < RInt g a b.
Lemma RInt_lt (f g : R -> R) (a b : R) :
a < b ->
(forall x : R, a <= x <= b ->continuous g x) ->
(forall x : R, a <= x <= b ->continuous f x) ->
(forall x : R, a < x < b -> f x < g x) ->
RInt f a b < RInt g a b.
Lemma abs_RInt_le_const :
forall (f : R -> R) a b M,
a <= b -> ex_RInt f a b ->
(forall t, a <= t <= b -> Rabs (f t) <= M) ->
Rabs (RInt f a b) <= (b - a) * M.