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).

Theorem In_gen_weaken : forall (A : Type)(e1 e2 : A -> A -> Prop) ls a, In_gen e1 a ls -> (forall a', e1 a a' -> e2 a a') -> In_gen e2 a ls.