Inductive NoDup_gen : list A -> Prop :=
  | NoDup_gen_nil : NoDup_gen nil
  | NoDup_gen_cons : 
      forall (a : A)(ls : list A),
        (~In_gen ea a ls) -> 
        NoDup_gen ls ->
        NoDup_gen (a :: ls).Inductive NoDup_gen : list A -> Prop :=
  | NoDup_gen_nil : NoDup_gen nil
  | NoDup_gen_cons : 
      forall (a : A)(ls : list A),
        (~In_gen ea a ls) -> 
        NoDup_gen ls ->
        NoDup_gen (a :: ls).Inductive NoDup_gen : list A -> Prop :=
  | NoDup_gen_nil : NoDup_gen nil
  | NoDup_gen_cons : 
      forall (a : A)(ls : list A),
        (~In_gen ea a ls) -> 
        NoDup_gen ls ->
        NoDup_gen (a :: ls).

Lemma flatten_NoDup_gen : forall (A : Set)(ls : list (list A)), NoDup_gen (fun a b => a <> nil /\ a = b) ls -> (forall x, In x ls -> NoDup x) -> (forall x1 x2, In x1 ls -> In x2 ls -> x1 <> x2 -> NoDup (x1 ++ x2)) -> NoDup (flatten ls).