Definition continuous_at(f : Rat -> Rat -> Prop)(c : Rat) :=
  forall v, f c v ->
    rat_limit f c v.

Lemma rat_inf_limit_comp : forall (f : nat -> Rat -> Prop)(g : Rat -> Rat -> Prop) a v, rat_inf_limit f a -> g a v -> continuous_at g a -> left_total f -> rat_inf_limit (fun n r => forall v', (f n v' -> g v' r)) v.