Definition Pred (A : Set) := A -> Prop.Definition Rel (A : Set) := A -> A -> Prop.Inductive ExistsL (A : Set) (P : Pred A) : list A -> Set :=
  | FoundE : forall (a : A) (l : list A), P a -> ExistsL A P (a :: l)
  | SearchE :
      forall (a : A) (l : list A), ExistsL A P l -> ExistsL A P (a :: l).Inductive ExistsL (A : Set) (P : Pred A) : list A -> Set :=
  | FoundE : forall (a : A) (l : list A), P a -> ExistsL A P (a :: l)
  | SearchE :
      forall (a : A) (l : list A), ExistsL A P l -> ExistsL A P (a :: l).Inductive ExistsL (A : Set) (P : Pred A) : list A -> Set :=
  | FoundE : forall (a : A) (l : list A), P a -> ExistsL A P (a :: l)
  | SearchE :
      forall (a : A) (l : list A), ExistsL A P l -> ExistsL A P (a :: l).

(* Specifications: *)
Lemma monExistsL1 : forall (A : Set) (P : Pred A) (xs bs : list A), ExistsL A P bs -> ExistsL A P (xs ++ bs).
Lemma monExistsL : forall (A : Set) (P : Pred A) (xs bs cs : list A), ExistsL A P (xs ++ bs) -> ExistsL A P (xs ++ cs ++ bs).