MAYBE

Problem:
 app(app(app(if(),true()),x),y) -> x
 app(app(app(if(),true()),x),y) -> y
 app(app(takeWhile(),p),nil()) -> nil()
 app(app(takeWhile(),p),app(app(cons(),x),xs)) ->
 app(app(app(if(),app(p,x)),app(app(cons(),x),app(app(takeWhile(),p),xs))),nil())
 app(app(dropWhile(),p),nil()) -> nil()
 app(app(dropWhile(),p),app(app(cons(),x),xs)) ->
 app(app(app(if(),app(p,x)),app(app(dropWhile(),p),xs)),app(app(cons(),x),xs))

Proof:
 DP Processor:
  DPs:
   app#(app(takeWhile(),p),app(app(cons(),x),xs)) -> app#(app(takeWhile(),p),xs)
   app#(app(takeWhile(),p),app(app(cons(),x),xs)) -> app#(app(cons(),x),app(app(takeWhile(),p),xs))
   app#(app(takeWhile(),p),app(app(cons(),x),xs)) -> app#(p,x)
   app#(app(takeWhile(),p),app(app(cons(),x),xs)) -> app#(if(),app(p,x))
   app#(app(takeWhile(),p),app(app(cons(),x),xs)) ->
   app#(app(if(),app(p,x)),app(app(cons(),x),app(app(takeWhile(),p),xs)))
   app#(app(takeWhile(),p),app(app(cons(),x),xs)) ->
   app#(app(app(if(),app(p,x)),app(app(cons(),x),app(app(takeWhile(),p),xs))),nil())
   app#(app(dropWhile(),p),app(app(cons(),x),xs)) -> app#(app(dropWhile(),p),xs)
   app#(app(dropWhile(),p),app(app(cons(),x),xs)) -> app#(p,x)
   app#(app(dropWhile(),p),app(app(cons(),x),xs)) -> app#(if(),app(p,x))
   app#(app(dropWhile(),p),app(app(cons(),x),xs)) ->
   app#(app(if(),app(p,x)),app(app(dropWhile(),p),xs))
   app#(app(dropWhile(),p),app(app(cons(),x),xs)) ->
   app#(app(app(if(),app(p,x)),app(app(dropWhile(),p),xs)),app(app(cons(),x),xs))
  TRS:
   app(app(app(if(),true()),x),y) -> x
   app(app(app(if(),true()),x),y) -> y
   app(app(takeWhile(),p),nil()) -> nil()
   app(app(takeWhile(),p),app(app(cons(),x),xs)) ->
   app(app(app(if(),app(p,x)),app(app(cons(),x),app(app(takeWhile(),p),xs))),nil())
   app(app(dropWhile(),p),nil()) -> nil()
   app(app(dropWhile(),p),app(app(cons(),x),xs)) ->
   app(app(app(if(),app(p,x)),app(app(dropWhile(),p),xs)),app(app(cons(),x),xs))
  Open