Section is_RInt.
(** * Definition of Riemann integral *)
Context {V : NormedModule R_AbsRing}.
Definition is_RInt (f : R -> V) (a b : R) (If : V) :=
filterlim (fun ptd => scal (sign (b-a)) (Riemann_sum f ptd)) (Riemann_fine a b) (locally If).
Definition ex_RInt (f : R -> V) (a b : R) :=
exists If : V, is_RInt f a b If.
(** ** Usual properties *)
(** The integral between a and a is null *)
(** Swapping bounds *)
(** Integrable implies bounded *)
(** Extensionality *)
(** ** Constant functions *)
(** ** Composition *)
(** ** Chasles *)
(** ** Operations *)
End is_RInt.

Lemma ex_RInt_inside {V : CompleteNormedModule R_AbsRing} :
forall (f : R -> V) (a b x e : R),
ex_RInt f (x-e) (x+e) -> Rabs (a-x) <= e -> Rabs (b-x) <= e ->
ex_RInt f a b.