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.
(** ** Exchange limit and integral *)

Lemma filterlim_RInt {U} {V : CompleteNormedModule R_AbsRing} :
forall (f : U -> R -> V) (a b : R) F (FF : ProperFilter F)
g h,
(forall x, is_RInt (f x) a b (h x))
-> (filterlim f F (locally g))
-> exists If, filterlim h F (locally If) /\ is_RInt g a b If.