End ForallPairs.

(* Specifications: *)
Lemma NoDup_iff_ForallOrdPairs [A] (l: list A):
NoDup l <-> ForallOrdPairs (fun a b => a <> b) l.
Lemma NoDup_map_NoDup_ForallPairs [A B] (f: A->B) (l: list A) :
ForallPairs (fun x y => f x = f y -> x = y) l -> NoDup l -> NoDup (map f l).
Lemma NoDup_concat [A] (L: list (list A)):
Forall (@NoDup A) L ->
ForallOrdPairs (fun l1 l2 => forall a, In a l1 -> ~ In a l2) L ->
NoDup (concat L).