YES

Problem:
 and(P,forall()) -> forall()
 or(P,forall()) -> forall()
 and(forall(),P) -> forall()
 or(forall(),P) -> forall()
 not(forall()) -> exists()
 and(P,exists()) -> exists()
 or(P,exists()) -> exists()
 and(exists(),P) -> exists()
 or(exists(),P) -> exists()
 not(exists()) -> forall()

Proof:
 DP Processor:
  DPs:
   
  TRS:
   and(P,forall()) -> forall()
   or(P,forall()) -> forall()
   and(forall(),P) -> forall()
   or(forall(),P) -> forall()
   not(forall()) -> exists()
   and(P,exists()) -> exists()
   or(P,exists()) -> exists()
   and(exists(),P) -> exists()
   or(exists(),P) -> exists()
   not(exists()) -> forall()
  Qed