End Lemmas.
End Compare.
Notation list_compare := list_compare.
Section Exists_Forall.
(** * Existential and universal predicates over lists *)
Variable A:Type.
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).
#[local]
Hint Constructors Exists : core.

Lemma Exists_nth l :
Exists l <-> exists i d, i < length l /\ P (nth i l d).