(** ** Complete Normed Modules *)
Module CompleteNormedModule.
Section ClassDef.
Variable K : AbsRing.
Record class_of (T : Type) := Class {
base : NormedModule.class_of K T ;
mixin : CompleteSpace.mixin_of (UniformSpace.Pack T base T)
}.
Local Coercion base : class_of >-> NormedModule.class_of.
Definition base2 T (cT : class_of T) : CompleteSpace.class_of T :=
CompleteSpace.Class _ (base T cT) (mixin T cT).
Local Coercion base2 : class_of >-> CompleteSpace.class_of.
Structure type := Pack { sort; _ : class_of sort ; _ : Type }.
Local Coercion sort : type >-> Sortclass.
Variable cT : type.
Definition class := let: Pack _ c _ := cT return class_of cT in c.
Let xT := let: Pack T _ _ := cT in T.
Notation xclass := (class : class_of xT).
Definition AbelianGroup := AbelianGroup.Pack cT xclass xT.
Definition ModuleSpace := ModuleSpace.Pack _ cT xclass xT.
Definition NormedModuleAux := NormedModuleAux.Pack _ cT xclass xT.
Definition NormedModule := NormedModule.Pack _ cT xclass xT.
Definition UniformSpace := UniformSpace.Pack cT xclass xT.
Definition CompleteSpace := CompleteSpace.Pack cT xclass xT.
End ClassDef.
Module Exports.
Coercion base : class_of >-> NormedModule.class_of.
Coercion mixin : class_of >-> CompleteSpace.mixin_of.
Coercion base2 : class_of >-> CompleteSpace.class_of.
Coercion sort : type >-> Sortclass.
Coercion AbelianGroup : type >-> AbelianGroup.type.
Canonical AbelianGroup.
Coercion ModuleSpace : type >-> ModuleSpace.type.
Canonical ModuleSpace.
Coercion NormedModuleAux : type >-> NormedModuleAux.type.
Canonical NormedModuleAux.
Coercion NormedModule : type >-> NormedModule.type.
Canonical NormedModule.
Coercion UniformSpace : type >-> UniformSpace.type.
Canonical UniformSpace.
Coercion CompleteSpace : type >-> CompleteSpace.type.
Canonical CompleteSpace.
Notation CompleteNormedModule := type.
End Exports.
End CompleteNormedModule.
Export CompleteNormedModule.Exports.
Section CompleteNormedModule1.
Context {K : AbsRing} {V : CompleteNormedModule K}.

(* Specifications: *)
Lemma iota_unique :
forall (P : V -> Prop) (x : V),
(forall y, P y -> y = x) ->
P x ->
iota P = x.
Lemma iota_correct :
forall P : V -> Prop,
(exists! x : V, P x) ->
P (iota P).
Lemma iota_is_filter_lim {F} {FF : ProperFilter' F} (l : V) :
is_filter_lim F l ->
iota (is_filter_lim F) = l.