Definition filterlimi {T U : Type} (f : T -> U -> Prop) F G :=
filter_le (filtermapi f F) G.

(* Specifications: *)
Lemma filterlimi_comp :
forall T U V (f : T -> U) (g : U -> V -> Prop) F G H,
filterlim f F G -> filterlimi g G H ->
filterlimi (fun x => g (f x)) F H.
Lemma filterlimi_ext_loc :
forall {T U F G} {FF : Filter F} (f g : T -> U -> Prop),
F (fun x => forall y, f x y <-> g x y) ->
filterlimi f F G ->
filterlimi g F G.
Lemma filterlimi_ext :
forall {T U F G} {FF : Filter F} (f g : T -> U -> Prop),
(forall x y, f x y <-> g x y) ->
filterlimi f F G ->
filterlimi g F G.
Lemma filterlimi_filter_le_1 :
forall {T U F G H} (f : T -> U -> Prop),
filter_le G F ->
filterlimi f F H ->
filterlimi f G H.
Lemma filterlimi_filter_le_2 :
forall {T U F G H} (f : T -> U -> Prop),
filter_le G H ->
filterlimi f F G ->
filterlimi f F H.