(** * Riemann integral and differentiability *)
Section Derive.
Context {V : NormedModule R_AbsRing}.

(* Specifications: *)
Lemma is_derive_RInt_0 (f If : R -> V) (a : R) :
locally a (fun b => is_RInt f a b (If b))
-> continuous f a
-> is_derive If a (f a).
Lemma is_derive_RInt (f If : R -> V) (a b : R) :
locally b (fun b => is_RInt f a b (If b))
-> continuous f b
-> is_derive If b (f b).
Lemma is_derive_RInt' (f If : R -> V) (a b : R) :
locally a (fun a => is_RInt f a b (If a))
-> continuous f a
-> is_derive If a (opp (f a)).
Lemma filterdiff_RInt (f : R -> V) (If : R -> R -> V) (a b : R) :
locally (a,b) (fun u : R * R => is_RInt f (fst u) (snd u) (If (fst u) (snd u)))
-> continuous f a -> continuous f b
-> filterdiff (fun u : R * R => If (fst u) (snd u)) (locally (a,b))
(fun u : R * R => minus (scal (snd u) (f b)) (scal (fst u) (f a))).