MAYBE

Problem:
 app(half(),0()) -> 0()
 app(half(),app(s(),0())) -> 0()
 app(half(),app(s(),app(s(),x))) -> app(s(),app(half(),x))
 app(bits(),0()) -> 0()
 app(bits(),app(s(),x)) -> app(s(),app(bits(),app(half(),app(s(),x))))
 app(app(map(),f),nil()) -> nil()
 app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
 app(app(filter(),f),nil()) -> nil()
 app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
 app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
 app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

Proof:
 DP Processor:
  DPs:
   app#(half(),app(s(),app(s(),x))) -> app#(half(),x)
   app#(half(),app(s(),app(s(),x))) -> app#(s(),app(half(),x))
   app#(bits(),app(s(),x)) -> app#(half(),app(s(),x))
   app#(bits(),app(s(),x)) -> app#(bits(),app(half(),app(s(),x)))
   app#(bits(),app(s(),x)) -> app#(s(),app(bits(),app(half(),app(s(),x))))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
   app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
   app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
  TRS:
   app(half(),0()) -> 0()
   app(half(),app(s(),0())) -> 0()
   app(half(),app(s(),app(s(),x))) -> app(s(),app(half(),x))
   app(bits(),0()) -> 0()
   app(bits(),app(s(),x)) -> app(s(),app(bits(),app(half(),app(s(),x))))
   app(app(map(),f),nil()) -> nil()
   app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
   app(app(filter(),f),nil()) -> nil()
   app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
   app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
   app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)
  SCC Processor:
   #sccs: 1
   #rules: 20
   #arcs: 400/400
   DPs:
    app#(half(),app(s(),app(s(),x))) -> app#(half(),x)
    app#(half(),app(s(),app(s(),x))) -> app#(s(),app(half(),x))
    app#(bits(),app(s(),x)) -> app#(half(),app(s(),x))
    app#(bits(),app(s(),x)) -> app#(bits(),app(half(),app(s(),x)))
    app#(bits(),app(s(),x)) -> app#(s(),app(bits(),app(half(),app(s(),x))))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
   TRS:
    app(half(),0()) -> 0()
    app(half(),app(s(),0())) -> 0()
    app(half(),app(s(),app(s(),x))) -> app(s(),app(half(),x))
    app(bits(),0()) -> 0()
    app(bits(),app(s(),x)) -> app(s(),app(bits(),app(half(),app(s(),x))))
    app(app(map(),f),nil()) -> nil()
    app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
    app(app(filter(),f),nil()) -> nil()
    app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
    app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
    app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)
   Open