Section One_predicate.
Variable P:A->Prop.
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).
Hint Constructors Exists : core.

Lemma Exists_exists (l:list A) :
Exists l <-> (exists x, In x l /\ P x).