(** Multiplication by a scalar *)

(* Specifications: *)
Lemma filterlim_Rbar_mult_l :
forall (a : R) (l : Rbar),
filterlim (Rmult a) (Rbar_locally l) (Rbar_locally (Rbar_mult a l)).
Lemma filterlim_Rbar_mult_r :
forall (a : R) (l : Rbar),
filterlim (fun x => Rmult x a) (Rbar_locally l) (Rbar_locally (Rbar_mult l a)).
Lemma is_lim_seq_scal_l (u : nat -> R) (a : R) (lu : Rbar) :
is_lim_seq u lu ->
is_lim_seq (fun n => a * u n) (Rbar_mult a lu).
Lemma ex_lim_seq_scal_l (u : nat -> R) (a : R) :
ex_lim_seq u -> ex_lim_seq (fun n => a * u n).
Lemma Lim_seq_scal_l (u : nat -> R) (a : R) :
Lim_seq (fun n => a * u n) = Rbar_mult a (Lim_seq u).
Lemma is_lim_seq_scal_r (u : nat -> R) (a : R) (lu : Rbar) :
is_lim_seq u lu ->
is_lim_seq (fun n => u n * a) (Rbar_mult lu a).
Lemma ex_lim_seq_scal_r (u : nat -> R) (a : R) :
ex_lim_seq u -> ex_lim_seq (fun n => u n * a).
Lemma Lim_seq_scal_r (u : nat -> R) (a : R) :
Lim_seq (fun n => u n * a) = Rbar_mult (Lim_seq u) a.
Lemma Riemann_integrable_scal : forall (f : R -> R) (a b c : R),
Riemann_integrable f a b ->
Riemann_integrable (fun x => c * f x) a b.
Lemma RiemannInt_scal : forall (f : R -> R) (a b c : R)
(pr_f : Riemann_integrable f a b)
(pr : Riemann_integrable (fun x => c * f x) a b),
RiemannInt pr = c * RiemannInt pr_f.