Definition rat_inf_limit(f : nat -> Rat -> Prop)(r : Rat) :=
  inf_limit eqRat ratDistance rat0 leRat f r.Definition rat_inf_limit_2(f1 f2 : nat -> Rat -> Prop) :=
  inf_limit_2 eqRat ratDistance rat0 leRat f1 f2.Definition rat_limits_eq :=
  limits_eq eq_Rat_dec ratDistance_comm ratHalf ratHalf_ne_0 ratIdentityIndiscernables eqRat_impl_leRat leRat_trans ratAdd ratAdd_leRat_compat ratTriangleInequality ratHalf_add le_ratHalf_0.

Lemma rat_inf_limit_difference : forall (f1 f2 : nat -> Rat -> Prop) c1 c2, left_total f1 -> left_total f2 -> rat_inf_limit f1 c1 -> rat_inf_limit f2 c2 -> rat_inf_limit (fun n => (ratSubtract_rel (f1 n) (f2 n))) (ratSubtract c1 c2).