Inductive Exists : list A -> Prop :=
      | Exists_cons_hd : forall x l, P x -> Exists (x::l)
      | Exists_cons_tl : forall x l, Exists l -> Exists (x::l).Inductive Exists : list A -> Prop :=
      | Exists_cons_hd : forall x l, P x -> Exists (x::l)
      | Exists_cons_tl : forall x l, Exists l -> Exists (x::l).Inductive Exists : list A -> Prop :=
      | Exists_cons_hd : forall x l, P x -> Exists (x::l)
      | Exists_cons_tl : forall x l, Exists l -> Exists (x::l).

Lemma Exists_dec l: (forall x:A, {P x} + { ~ P x }) -> {Exists l} + {~ Exists l}.