MAYBE

Problem:
 app(app(app(f(),x),app(c(),x)),app(c(),y)) -> app(app(app(f(),y),y),app(app(app(f(),y),x),y))
 app(app(app(f(),app(s(),x)),y),z) -> app(app(app(f(),x),app(s(),app(c(),y))),app(c(),z))
 app(app(app(f(),app(c(),x)),x),y) -> app(c(),y)
 app(app(g(),x),y) -> x
 app(app(g(),x),y) -> 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#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(app(f(),y),x)
   app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(app(app(f(),y),x),y)
   app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(f(),y)
   app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(app(f(),y),y)
   app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(app(app(f(),y),y),app(app(app(f(),y),x),y))
   app#(app(app(f(),app(s(),x)),y),z) -> app#(c(),z)
   app#(app(app(f(),app(s(),x)),y),z) -> app#(c(),y)
   app#(app(app(f(),app(s(),x)),y),z) -> app#(s(),app(c(),y))
   app#(app(app(f(),app(s(),x)),y),z) -> app#(f(),x)
   app#(app(app(f(),app(s(),x)),y),z) -> app#(app(f(),x),app(s(),app(c(),y)))
   app#(app(app(f(),app(s(),x)),y),z) -> app#(app(app(f(),x),app(s(),app(c(),y))),app(c(),z))
   app#(app(app(f(),app(c(),x)),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(app(app(f(),x),app(c(),x)),app(c(),y)) -> app(app(app(f(),y),y),app(app(app(f(),y),x),y))
   app(app(app(f(),app(s(),x)),y),z) -> app(app(app(f(),x),app(s(),app(c(),y))),app(c(),z))
   app(app(app(f(),app(c(),x)),x),y) -> app(c(),y)
   app(app(g(),x),y) -> x
   app(app(g(),x),y) -> 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)
  SCC Processor:
   #sccs: 1
   #rules: 27
   #arcs: 729/729
   DPs:
    app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(app(f(),y),x)
    app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(app(app(f(),y),x),y)
    app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(f(),y)
    app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(app(f(),y),y)
    app#(app(app(f(),x),app(c(),x)),app(c(),y)) -> app#(app(app(f(),y),y),app(app(app(f(),y),x),y))
    app#(app(app(f(),app(s(),x)),y),z) -> app#(c(),z)
    app#(app(app(f(),app(s(),x)),y),z) -> app#(c(),y)
    app#(app(app(f(),app(s(),x)),y),z) -> app#(s(),app(c(),y))
    app#(app(app(f(),app(s(),x)),y),z) -> app#(f(),x)
    app#(app(app(f(),app(s(),x)),y),z) -> app#(app(f(),x),app(s(),app(c(),y)))
    app#(app(app(f(),app(s(),x)),y),z) -> app#(app(app(f(),x),app(s(),app(c(),y))),app(c(),z))
    app#(app(app(f(),app(c(),x)),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(app(app(f(),x),app(c(),x)),app(c(),y)) -> app(app(app(f(),y),y),app(app(app(f(),y),x),y))
    app(app(app(f(),app(s(),x)),y),z) -> app(app(app(f(),x),app(s(),app(c(),y))),app(c(),z))
    app(app(app(f(),app(c(),x)),x),y) -> app(c(),y)
    app(app(g(),x),y) -> x
    app(app(g(),x),y) -> 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