Section is_RInt_gen.
Context {V : NormedModule R_AbsRing}.
Definition is_RInt_gen (f : R -> V) (Fa Fb : (R -> Prop) -> Prop) (l : V) :=
filterlimi (fun ab => is_RInt f (fst ab) (snd ab)) (filter_prod Fa Fb) (locally l).
Definition ex_RInt_gen (f : R -> V) (Fa Fb : (R -> Prop) -> Prop) :=
exists l : V, is_RInt_gen f Fa Fb l.
Definition is_RInt_gen_at_point f a b l :
is_RInt_gen f (at_point a) (at_point b) l
<-> is_RInt f a b l.
(** * Basic properties of integrals *)
(** * Composition *)
(* Lemma is_RInt_gen_comp_opp {Fa Fb : (R -> Prop) -> Prop} {FFa : Filter Fa} {FFb : Filter Fb} (f : R -> V) (l : V) : is_RInt_gen f (filtermap opp Fa) (filtermap opp Fb) l -> is_RInt_gen (fun y => opp (f (- y))) Fa Fb l.  Lemma is_RInt_gen_comp_lin {Fa Fb : (R -> Prop) -> Prop} {FFa : Filter Fa} {FFb : Filter Fb} (f : R -> V) (u v : R) (l : V) : is_RInt_gen f (filtermap (fun a => u * a + v) Fa) (filtermap (fun b => u * b + v) Fb) l -> is_RInt_gen (fun y => scal u (f (u * y + v))) Fa Fb l. *)
(** * Operations *)

Lemma is_RInt_gen_plus {Fa Fb : (R -> Prop) -> Prop}
{FFa : Filter Fa} {FFb : Filter Fb} (f g : R -> V) (lf lg : V) :
is_RInt_gen f Fa Fb lf ->
is_RInt_gen g Fa Fb lg ->
is_RInt_gen (fun y => plus (f y) (g y)) Fa Fb (plus lf lg).