MAYBE

Problem:
 app(f(),0()) -> true()
 app(f(),1()) -> false()
 app(f(),app(s(),x)) -> app(f(),x)
 app(app(app(if(),true()),app(s(),x)),app(s(),y)) -> app(s(),x)
 app(app(app(if(),false()),app(s(),x)),app(s(),y)) -> app(s(),y)
 app(app(g(),x),app(c(),y)) -> app(c(),app(app(g(),x),y))
 app(app(g(),x),app(c(),y)) ->
 app(app(g(),x),app(app(app(if(),app(f(),x)),app(c(),app(app(g(),app(s(),x)),y))),app(c(),y)))
 app(app(map(),fun),nil()) -> nil()
 app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
 app(app(filter(),fun),nil()) -> nil()
 app(app(filter(),fun),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
 app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
 app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)

Proof:
 DP Processor:
  DPs:
   app#(f(),app(s(),x)) -> app#(f(),x)
   app#(app(g(),x),app(c(),y)) -> app#(app(g(),x),y)
   app#(app(g(),x),app(c(),y)) -> app#(c(),app(app(g(),x),y))
   app#(app(g(),x),app(c(),y)) -> app#(s(),x)
   app#(app(g(),x),app(c(),y)) -> app#(g(),app(s(),x))
   app#(app(g(),x),app(c(),y)) -> app#(app(g(),app(s(),x)),y)
   app#(app(g(),x),app(c(),y)) -> app#(c(),app(app(g(),app(s(),x)),y))
   app#(app(g(),x),app(c(),y)) -> app#(f(),x)
   app#(app(g(),x),app(c(),y)) -> app#(if(),app(f(),x))
   app#(app(g(),x),app(c(),y)) -> app#(app(if(),app(f(),x)),app(c(),app(app(g(),app(s(),x)),y)))
   app#(app(g(),x),app(c(),y)) ->
   app#(app(app(if(),app(f(),x)),app(c(),app(app(g(),app(s(),x)),y))),app(c(),y))
   app#(app(g(),x),app(c(),y)) ->
   app#(app(g(),x),app(app(app(if(),app(f(),x)),app(c(),app(app(g(),app(s(),x)),y))),app(c(),y)))
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
   app#(app(map(),fun),app(app(cons(),x),xs)) ->
   app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
   app#(app(filter(),fun),app(app(cons(),x),xs)) ->
   app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
   app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
   app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
  TRS:
   app(f(),0()) -> true()
   app(f(),1()) -> false()
   app(f(),app(s(),x)) -> app(f(),x)
   app(app(app(if(),true()),app(s(),x)),app(s(),y)) -> app(s(),x)
   app(app(app(if(),false()),app(s(),x)),app(s(),y)) -> app(s(),y)
   app(app(g(),x),app(c(),y)) -> app(c(),app(app(g(),x),y))
   app(app(g(),x),app(c(),y)) ->
   app(app(g(),x),app(app(app(if(),app(f(),x)),app(c(),app(app(g(),app(s(),x)),y))),app(c(),y)))
   app(app(map(),fun),nil()) -> nil()
   app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
   app(app(filter(),fun),nil()) -> nil()
   app(app(filter(),fun),app(app(cons(),x),xs)) ->
   app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
   app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
   app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
  Open