Fixpoint AbsList (l:list R) (x:R) : list R :=
  match l with
    | nil => nil
    | a :: l' => (Rabs (a - x) / 2) :: (AbsList l' x)
  end.

(* Specifications: *)
Lemma MinRlist_P1 : forall (l:list R) (x:R), In x l -> MinRlist l <= x.
Lemma AbsList_P1 : forall (l:list R) (x y:R), In y l -> In (Rabs (y - x) / 2) (AbsList l x).
Lemma MinRlist_P2 : forall l:list R, (forall y:R, In y l -> 0 < y) -> 0 < MinRlist l.
Lemma AbsList_P2 : forall (l:list R) (x y:R), In y (AbsList l x) -> exists z : R, In z l /\ y = Rabs (z - x) / 2.
Lemma MaxRlist_P2 : forall l:list R, (exists y : R, In y l) -> In (MaxRlist l) l.