Inductive GoodR (A : Set) (R : Rel A) : list A -> Set :=
  | FoundG :
      forall (a : A) (l : list A),
      ExistsL A (fun x : A => R x a) l -> GoodR A R (a :: l)
  | SearchG : forall (a : A) (l : list A), GoodR A R l -> GoodR A R (a :: l).Inductive GoodR (A : Set) (R : Rel A) : list A -> Set :=
  | FoundG :
      forall (a : A) (l : list A),
      ExistsL A (fun x : A => R x a) l -> GoodR A R (a :: l)
  | SearchG : forall (a : A) (l : list A), GoodR A R l -> GoodR A R (a :: l).Inductive GoodR (A : Set) (R : Rel A) : list A -> Set :=
  | FoundG :
      forall (a : A) (l : list A),
      ExistsL A (fun x : A => R x a) l -> GoodR A R (a :: l)
  | SearchG : forall (a : A) (l : list A), GoodR A R l -> GoodR A R (a :: l).

Lemma subPredExistsL : forall (A B : Set) (P : Pred A) (S : Pred B) (f : A -> B), (forall a : A, P a -> S (f a)) -> forall l : list A, ExistsL A P l -> ExistsL B S (map f l).