End is_RInt_gen.
Section RInt_gen.
Context {V : CompleteNormedModule R_AbsRing}.
Definition RInt_gen (f : R -> V) (a b : (R -> Prop) -> Prop) :=
iota (is_RInt_gen f a b).

(* Specifications: *)
Lemma is_RInt_gen_unique {Fa Fb : (R -> Prop) -> Prop}
{FFa : ProperFilter' Fa} {FFb : ProperFilter' Fb} (f : R -> V) (l : V) :
is_RInt_gen f Fa Fb l -> RInt_gen f Fa Fb = l.
Lemma RInt_gen_correct {Fa Fb : (R -> Prop) -> Prop}
{FFa : ProperFilter' Fa} {FFb : ProperFilter' Fb} (f : R -> V) :
ex_RInt_gen f Fa Fb -> is_RInt_gen f Fa Fb (RInt_gen f Fa Fb).
Lemma RInt_gen_norm {Fa Fb : (R -> Prop) -> Prop}
{FFa : ProperFilter Fa} {FFb : ProperFilter Fb} (f : R -> V) (g : R -> R) (lf : V) (lg : R) :
filter_prod Fa Fb (fun ab => fst ab <= snd ab) ->
filter_prod Fa Fb (fun ab => forall x, fst ab <= x <= snd ab -> norm (f x) <= g x) ->
is_RInt_gen f Fa Fb lf -> is_RInt_gen g Fa Fb lg ->
norm lf <= lg.
Lemma RInt_gen_ext {Fa Fb : (R -> Prop) -> Prop}
{FFa : ProperFilter Fa} {FFb : ProperFilter Fb} (f g : R -> V) :
filter_prod Fa Fb (fun ab => forall x, Rmin (fst ab) (snd ab) < x < Rmax (fst ab) (snd ab)
-> f x = g x) ->
ex_RInt_gen f Fa Fb -> RInt_gen f Fa Fb = RInt_gen g Fa Fb.
Lemma RInt_gen_ext_eq {Fa Fb : (R -> Prop) -> Prop}
{FFa : ProperFilter Fa} {FFb : ProperFilter Fb} (f g : R -> V):
(forall x, f x = g x) ->
ex_RInt_gen f Fa Fb -> RInt_gen f Fa Fb = RInt_gen g Fa Fb.