End Removal.
Hint Resolve rem_not_in rem_incl rem_mono rem_cons rem_cons' rem_app rem_in rem_neq.
(* * Duplicate-free lists *)
Inductive dupfree (X : Type) : list X -> Prop :=
| dupfreeN : dupfree nil
| dupfreeC x A : ~ x el A -> dupfree A -> dupfree (x::A).
Section Dupfree.
Variable X : Type.
Implicit Types A B : list X.

(* Specifications: *)
Lemma dupfree_inv x A :
dupfree (x::A) <-> ~ x el A /\ dupfree A.
Lemma dupfree_map Y A (f : X -> Y) :
(forall x y, x el A -> y el A -> f x = f y -> x=y) ->
dupfree A -> dupfree (map f A).
Lemma dupfree_filter p (p_dec : forall x, dec (p x)) A :
dupfree A -> dupfree (filter p A).
Lemma dupfree_dec A :
eq_dec X -> dec (dupfree A).