Section No_Duplicate.
Variable A : Set.
Hypothesis Aeq_dec : forall a b : A, {a = b} + {a <> b}.
Definition no_dup (l : list A) : list A :=
list_rec (fun _ => list A) nil
(fun (a : A) _ (rec : list A) =>
match In_dec Aeq_dec a rec with
| left _ => rec
| right _ => a :: rec
end) l.

Lemma no_dup3 : forall (k l : list A) (a : A), no_dup k = a :: l -> ~ In a l.