Definition R_CompleteSpace_mixin :=
CompleteSpace.Mixin _ R_complete_lim R_complete R_complete_close.
Canonical R_CompleteSpace :=
CompleteSpace.Pack R (CompleteSpace.Class _ _ R_CompleteSpace_mixin) R.
Definition R_ModuleSpace_mixin :=
AbsRing_ModuleSpace_mixin R_AbsRing.
Canonical R_ModuleSpace :=
ModuleSpace.Pack _ R (ModuleSpace.Class _ _ _ R_ModuleSpace_mixin) R.
Canonical R_NormedModuleAux :=
NormedModuleAux.Pack _ R (NormedModuleAux.Class _ _ (ModuleSpace.class _ R_ModuleSpace) (UniformSpace.class R_UniformSpace)) R.
Definition R_NormedModule_mixin :=
AbsRing_NormedModule_mixin R_AbsRing.
Canonical R_NormedModule :=
NormedModule.Pack _ R (NormedModule.Class _ _ _ R_NormedModule_mixin) R.
Canonical R_CompleteNormedModule :=
CompleteNormedModule.Pack _ R (CompleteNormedModule.Class R_AbsRing _ (NormedModule.class _ R_NormedModule) R_CompleteSpace_mixin) R.
Definition at_left x := within (fun u : R => Rlt u x) (locally x).
Definition at_right x := within (fun u : R => Rlt x u) (locally x).
Global Instance at_right_proper_filter : forall (x : R),
ProperFilter (at_right x).
Global Instance at_left_proper_filter : forall (x : R),
ProperFilter (at_left x).
(* *)

(* Specifications: *)
Lemma sum_n_Reals : forall a N, sum_n a N = sum_f_R0 a N.
Lemma sum_n_m_Reals a n m : (n <= m)%nat -> sum_n_m a n m = sum_f n m a.
Lemma sum_n_m_const (n m : nat) (a : R) :
sum_n_m (fun _ => a) n m = INR (S m - n) * a.
Lemma sum_n_const (n : nat) (a : R) :
sum_n (fun _ => a) n = INR (S n) * a.
Lemma norm_sum_n_m {K : AbsRing} {V : NormedModule K} (a : nat -> V) (n m : nat) :
norm (sum_n_m a n m) <= sum_n_m (fun n => norm (a n)) n m.
Lemma sum_n_m_le (a b : nat -> R) (n m : nat) :
(forall k, a k <= b k)
-> sum_n_m a n m <= sum_n_m b n m.
Lemma pow_n_pow :
forall (x : R) k, pow_n x k = x^k.