MAYBE

Problem:
 app(app(app(if(),true()),xs),ys) -> xs
 app(app(app(if(),false()),xs),ys) -> ys
 app(app(lt(),app(s(),x)),app(s(),y)) -> app(app(lt(),x),y)
 app(app(lt(),0()),app(s(),y)) -> true()
 app(app(lt(),y),0()) -> false()
 app(app(eq(),x),x) -> true()
 app(app(eq(),app(s(),x)),0()) -> false()
 app(app(eq(),0()),app(s(),x)) -> false()
 app(app(merge(),xs),nil()) -> xs
 app(app(merge(),nil()),ys) -> ys
 app(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
 app(app(app(if(),app(app(lt(),x),y)),app(app(cons(),x),app(app(merge(),xs),app(app(cons(),y),ys)))),
     app(app(app(if(),app(app(eq(),x),y)),app(app(cons(),x),app(app(merge(),xs),ys))),
         app(app(cons(),y),app(app(merge(),app(app(cons(),x),xs)),ys))))
 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(mult(),0()),x) -> 0()
 app(app(mult(),app(s(),x)),y) -> app(app(plus(),y),app(app(mult(),x),y))
 app(app(plus(),0()),x) -> 0()
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
 list1() -> app(app(map(),app(mult(),app(s(),app(s(),0())))),hamming())
 list2() -> app(app(map(),app(mult(),app(s(),app(s(),app(s(),0()))))),hamming())
 list3() -> app(app(map(),app(mult(),app(s(),app(s(),app(s(),app(s(),app(s(),0()))))))),hamming())
 hamming() ->
 app(app(cons(),app(s(),0())),app(app(merge(),list1()),app(app(merge(),list2()),list3())))

Proof:
 DP Processor:
  DPs:
   app#(app(lt(),app(s(),x)),app(s(),y)) -> app#(lt(),x)
   app#(app(lt(),app(s(),x)),app(s(),y)) -> app#(app(lt(),x),y)
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(merge(),app(app(cons(),x),xs)),ys)
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(cons(),y),app(app(merge(),app(app(cons(),x),xs)),ys))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) -> app#(app(merge(),xs),ys)
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(cons(),x),app(app(merge(),xs),ys))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) -> app#(eq(),x)
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) -> app#(app(eq(),x),y)
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) -> app#(if(),app(app(eq(),x),y))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(if(),app(app(eq(),x),y)),app(app(cons(),x),app(app(merge(),xs),ys)))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(app(if(),app(app(eq(),x),y)),app(app(cons(),x),app(app(merge(),xs),ys))),
        app(app(cons(),y),app(app(merge(),app(app(cons(),x),xs)),ys)))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) -> app#(merge(),xs)
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(merge(),xs),app(app(cons(),y),ys))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(cons(),x),app(app(merge(),xs),app(app(cons(),y),ys)))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) -> app#(lt(),x)
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) -> app#(app(lt(),x),y)
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) -> app#(if(),app(app(lt(),x),y))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(if(),app(app(lt(),x),y)),app(app(cons(),x),app(app(merge(),xs),app(app(cons(),y),ys))))
   app#(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app#(app(app(if(),app(app(lt(),x),y)),app(app(cons(),x),app(app(merge(),xs),
                                                               app(app(cons(),y),ys)))),
        app(app(app(if(),app(app(eq(),x),y)),app(app(cons(),x),app(app(merge(),xs),ys))),
            app(app(cons(),y),app(app(merge(),app(app(cons(),x),xs)),ys))))
   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(mult(),app(s(),x)),y) -> app#(mult(),x)
   app#(app(mult(),app(s(),x)),y) -> app#(app(mult(),x),y)
   app#(app(mult(),app(s(),x)),y) -> app#(plus(),y)
   app#(app(mult(),app(s(),x)),y) -> app#(app(plus(),y),app(app(mult(),x),y))
   app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
   app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
   app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
   list1#() -> hamming#()
   list1#() -> app#(s(),0())
   list1#() -> app#(s(),app(s(),0()))
   list1#() -> app#(mult(),app(s(),app(s(),0())))
   list1#() -> app#(map(),app(mult(),app(s(),app(s(),0()))))
   list1#() -> app#(app(map(),app(mult(),app(s(),app(s(),0())))),hamming())
   list2#() -> hamming#()
   list2#() -> app#(s(),0())
   list2#() -> app#(s(),app(s(),0()))
   list2#() -> app#(s(),app(s(),app(s(),0())))
   list2#() -> app#(mult(),app(s(),app(s(),app(s(),0()))))
   list2#() -> app#(map(),app(mult(),app(s(),app(s(),app(s(),0())))))
   list2#() -> app#(app(map(),app(mult(),app(s(),app(s(),app(s(),0()))))),hamming())
   list3#() -> hamming#()
   list3#() -> app#(s(),0())
   list3#() -> app#(s(),app(s(),0()))
   list3#() -> app#(s(),app(s(),app(s(),0())))
   list3#() -> app#(s(),app(s(),app(s(),app(s(),0()))))
   list3#() -> app#(s(),app(s(),app(s(),app(s(),app(s(),0())))))
   list3#() -> app#(mult(),app(s(),app(s(),app(s(),app(s(),app(s(),0()))))))
   list3#() -> app#(map(),app(mult(),app(s(),app(s(),app(s(),app(s(),app(s(),0())))))))
   list3#() ->
   app#(app(map(),app(mult(),app(s(),app(s(),app(s(),app(s(),app(s(),0()))))))),hamming())
   hamming#() -> list3#()
   hamming#() -> list2#()
   hamming#() -> app#(merge(),list2())
   hamming#() -> app#(app(merge(),list2()),list3())
   hamming#() -> list1#()
   hamming#() -> app#(merge(),list1())
   hamming#() -> app#(app(merge(),list1()),app(app(merge(),list2()),list3()))
   hamming#() -> app#(s(),0())
   hamming#() -> app#(cons(),app(s(),0()))
   hamming#() ->
   app#(app(cons(),app(s(),0())),app(app(merge(),list1()),app(app(merge(),list2()),list3())))
  TRS:
   app(app(app(if(),true()),xs),ys) -> xs
   app(app(app(if(),false()),xs),ys) -> ys
   app(app(lt(),app(s(),x)),app(s(),y)) -> app(app(lt(),x),y)
   app(app(lt(),0()),app(s(),y)) -> true()
   app(app(lt(),y),0()) -> false()
   app(app(eq(),x),x) -> true()
   app(app(eq(),app(s(),x)),0()) -> false()
   app(app(eq(),0()),app(s(),x)) -> false()
   app(app(merge(),xs),nil()) -> xs
   app(app(merge(),nil()),ys) -> ys
   app(app(merge(),app(app(cons(),x),xs)),app(app(cons(),y),ys)) ->
   app(app(app(if(),app(app(lt(),x),y)),app(app(cons(),x),app(app(merge(),xs),app(app(cons(),y),ys)))),
       app(app(app(if(),app(app(eq(),x),y)),app(app(cons(),x),app(app(merge(),xs),ys))),
           app(app(cons(),y),app(app(merge(),app(app(cons(),x),xs)),ys))))
   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(mult(),0()),x) -> 0()
   app(app(mult(),app(s(),x)),y) -> app(app(plus(),y),app(app(mult(),x),y))
   app(app(plus(),0()),x) -> 0()
   app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
   list1() -> app(app(map(),app(mult(),app(s(),app(s(),0())))),hamming())
   list2() -> app(app(map(),app(mult(),app(s(),app(s(),app(s(),0()))))),hamming())
   list3() ->
   app(app(map(),app(mult(),app(s(),app(s(),app(s(),app(s(),app(s(),0()))))))),hamming())
   hamming() ->
   app(app(cons(),app(s(),0())),app(app(merge(),list1()),app(app(merge(),list2()),list3())))
  Open