(** ** Operations *)

(* Specifications: *)
Lemma is_RInt_scal :
forall (f : R -> V) (a b : R) (k : R) (If : V),
is_RInt f a b If ->
is_RInt (fun y => scal k (f y)) a b (scal k If).
Lemma ex_RInt_scal :
forall (f : R -> V) (a b : R) (k : R),
ex_RInt f a b ->
ex_RInt (fun y => scal k (f y)) a b.
Lemma is_RInt_opp :
forall (f : R -> V) (a b : R) (If : V),
is_RInt f a b If ->
is_RInt (fun y => opp (f y)) a b (opp If).
Lemma ex_RInt_opp :
forall (f : R -> V) (a b : R),
ex_RInt f a b ->
ex_RInt (fun x => opp (f x)) a b.
Lemma is_RInt_plus :
forall (f g : R -> V) (a b : R) (If Ig : V),
is_RInt f a b If ->
is_RInt g a b Ig ->
is_RInt (fun y => plus (f y) (g y)) a b (plus If Ig).
Lemma ex_RInt_plus :
forall (f g : R -> V) (a b : R),
ex_RInt f a b ->
ex_RInt g a b ->
ex_RInt (fun y => plus (f y) (g y)) a b.
Lemma is_RInt_minus :
forall (f g : R -> V) (a b : R) (If Ig : V),
is_RInt f a b If ->
is_RInt g a b Ig ->
is_RInt (fun y => minus (f y) (g y)) a b (minus If Ig).
Lemma ex_RInt_minus :
forall (f g : R -> V) (a b : R),
ex_RInt f a b ->
ex_RInt g a b ->
ex_RInt (fun y => minus (f y) (g y)) a b.
Lemma filterdiff_ext_lin {F} {FF : Filter F} (f : U -> V) (l1 l2 : U -> V) :
filterdiff f F l1 -> (forall y, l1 y = l2 y) -> filterdiff f F l2.
Lemma filterdiff_ext_loc {F} {FF : Filter F} (f g : U -> V) (l : U -> V) :
F (fun y => f y = g y) -> (forall x, is_filter_lim F x -> f x = g x)
-> filterdiff f F l -> filterdiff g F l.
Lemma ex_filterdiff_ext_loc {F} {FF : Filter F} (f g : U -> V) :
F (fun y => f y = g y) -> (forall x, is_filter_lim F x -> f x = g x)
-> ex_filterdiff f F -> ex_filterdiff g F.
Lemma filterdiff_ext_locally (f g : U -> V) (x : U) (l : U -> V) :
locally x (fun y => f y = g y)
-> filterdiff f (locally x) l -> filterdiff g (locally x) l.
Lemma ex_filterdiff_ext_locally (f g : U -> V) x :
locally x (fun y => f y = g y)
-> ex_filterdiff f (locally x) -> ex_filterdiff g (locally x).
Lemma filterdiff_ext {F} {FF : Filter F} (f g : U -> V) (l : U -> V) :
(forall y , f y = g y)
-> filterdiff f F l -> filterdiff g F l.
Lemma ex_filterdiff_ext {F} {FF : Filter F} (f g : U -> V) :
(forall y , f y = g y)
-> ex_filterdiff f F -> ex_filterdiff g F.
Lemma filterdiff_const {F} {FF : Filter F} (a : V) :
filterdiff (fun _ => a) F (fun _ => zero).
Lemma ex_filterdiff_const {F} {FF : Filter F} (a : V) :
ex_filterdiff (fun _ => a) F.
Lemma filterdiff_linear {F} (l : U -> V) :
is_linear l -> filterdiff l F l.
Lemma ex_filterdiff_linear {F} (l : U -> V) :
is_linear l -> ex_filterdiff l F.