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:
 Uncurry Processor:
  if3(true(),xs,ys) -> xs
  if3(false(),xs,ys) -> ys
  lt2(s1(x),s1(y)) -> lt2(x,y)
  lt2(0(),s1(y)) -> true()
  lt2(y,0()) -> false()
  eq2(x,x) -> true()
  eq2(s1(x),0()) -> false()
  eq2(0(),s1(x)) -> false()
  merge2(xs,nil()) -> xs
  merge2(nil(),ys) -> ys
  merge2(cons2(x,xs),cons2(y,ys)) ->
  if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                   cons2(y,merge2(cons2(x,xs),ys))))
  map2(f,nil()) -> nil()
  map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
  mult2(0(),x) -> 0()
  mult2(s1(x),y) -> plus2(y,mult2(x,y))
  plus2(0(),x) -> 0()
  plus2(s1(x),y) -> s1(plus2(x,y))
  list1() -> map2(mult1(s1(s1(0()))),hamming())
  list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
  list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
  hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
  app(if2(x6,x5),x7) -> if3(x6,x5,x7)
  app(if1(x6),x7) -> if2(x6,x7)
  app(if(),x7) -> if1(x7)
  app(lt1(x6),x7) -> lt2(x6,x7)
  app(lt(),x7) -> lt1(x7)
  app(s(),x7) -> s1(x7)
  app(eq1(x6),x7) -> eq2(x6,x7)
  app(eq(),x7) -> eq1(x7)
  app(merge1(x6),x7) -> merge2(x6,x7)
  app(merge(),x7) -> merge1(x7)
  app(cons1(x6),x7) -> cons2(x6,x7)
  app(cons(),x7) -> cons1(x7)
  app(map1(x6),x7) -> map2(x6,x7)
  app(map(),x7) -> map1(x7)
  app(mult1(x6),x7) -> mult2(x6,x7)
  app(mult(),x7) -> mult1(x7)
  app(plus1(x6),x7) -> plus2(x6,x7)
  app(plus(),x7) -> plus1(x7)
  DP Processor:
   DPs:
    lt{2,#}(s1(x),s1(y)) -> lt{2,#}(x,y)
    merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
    merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
    merge{2,#}(cons2(x,xs),cons2(y,ys)) -> eq{2,#}(x,y)
    merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
    if{3,#}(eq2(x,y),cons2(x,merge2(xs,ys)),cons2(y,merge2(cons2(x,xs),ys)))
    merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
    merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y)
    merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
    if{3,#}(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                         cons2(y,merge2(cons2(x,xs),ys))))
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    mult{2,#}(s1(x),y) -> mult{2,#}(x,y)
    mult{2,#}(s1(x),y) -> plus{2,#}(y,mult2(x,y))
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    list1#() -> hamming#()
    list1#() -> map{2,#}(mult1(s1(s1(0()))),hamming())
    list2#() -> hamming#()
    list2#() -> map{2,#}(mult1(s1(s1(s1(0())))),hamming())
    list3#() -> hamming#()
    list3#() -> map{2,#}(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
    hamming#() -> list3#()
    hamming#() -> list2#()
    hamming#() -> merge{2,#}(list2(),list3())
    hamming#() -> list1#()
    hamming#() -> merge{2,#}(list1(),merge2(list2(),list3()))
    app#(if2(x6,x5),x7) -> if{3,#}(x6,x5,x7)
    app#(lt1(x6),x7) -> lt{2,#}(x6,x7)
    app#(eq1(x6),x7) -> eq{2,#}(x6,x7)
    app#(merge1(x6),x7) -> merge{2,#}(x6,x7)
    app#(map1(x6),x7) -> map{2,#}(x6,x7)
    app#(mult1(x6),x7) -> mult{2,#}(x6,x7)
    app#(plus1(x6),x7) -> plus{2,#}(x6,x7)
   TRS:
    if3(true(),xs,ys) -> xs
    if3(false(),xs,ys) -> ys
    lt2(s1(x),s1(y)) -> lt2(x,y)
    lt2(0(),s1(y)) -> true()
    lt2(y,0()) -> false()
    eq2(x,x) -> true()
    eq2(s1(x),0()) -> false()
    eq2(0(),s1(x)) -> false()
    merge2(xs,nil()) -> xs
    merge2(nil(),ys) -> ys
    merge2(cons2(x,xs),cons2(y,ys)) ->
    if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                     cons2(y,merge2(cons2(x,xs),ys))))
    map2(f,nil()) -> nil()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    mult2(0(),x) -> 0()
    mult2(s1(x),y) -> plus2(y,mult2(x,y))
    plus2(0(),x) -> 0()
    plus2(s1(x),y) -> s1(plus2(x,y))
    list1() -> map2(mult1(s1(s1(0()))),hamming())
    list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
    list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
    hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
    app(if2(x6,x5),x7) -> if3(x6,x5,x7)
    app(if1(x6),x7) -> if2(x6,x7)
    app(if(),x7) -> if1(x7)
    app(lt1(x6),x7) -> lt2(x6,x7)
    app(lt(),x7) -> lt1(x7)
    app(s(),x7) -> s1(x7)
    app(eq1(x6),x7) -> eq2(x6,x7)
    app(eq(),x7) -> eq1(x7)
    app(merge1(x6),x7) -> merge2(x6,x7)
    app(merge(),x7) -> merge1(x7)
    app(cons1(x6),x7) -> cons2(x6,x7)
    app(cons(),x7) -> cons1(x7)
    app(map1(x6),x7) -> map2(x6,x7)
    app(map(),x7) -> map1(x7)
    app(mult1(x6),x7) -> mult2(x6,x7)
    app(mult(),x7) -> mult1(x7)
    app(plus1(x6),x7) -> plus2(x6,x7)
    app(plus(),x7) -> plus1(x7)
   TDG Processor:
    DPs:
     lt{2,#}(s1(x),s1(y)) -> lt{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> eq{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(eq2(x,y),cons2(x,merge2(xs,ys)),cons2(y,merge2(cons2(x,xs),ys)))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                          cons2(y,merge2(cons2(x,xs),ys))))
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     mult{2,#}(s1(x),y) -> mult{2,#}(x,y)
     mult{2,#}(s1(x),y) -> plus{2,#}(y,mult2(x,y))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     list1#() -> hamming#()
     list1#() -> map{2,#}(mult1(s1(s1(0()))),hamming())
     list2#() -> hamming#()
     list2#() -> map{2,#}(mult1(s1(s1(s1(0())))),hamming())
     list3#() -> hamming#()
     list3#() -> map{2,#}(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
     hamming#() -> list3#()
     hamming#() -> list2#()
     hamming#() -> merge{2,#}(list2(),list3())
     hamming#() -> list1#()
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3()))
     app#(if2(x6,x5),x7) -> if{3,#}(x6,x5,x7)
     app#(lt1(x6),x7) -> lt{2,#}(x6,x7)
     app#(eq1(x6),x7) -> eq{2,#}(x6,x7)
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7)
     app#(map1(x6),x7) -> map{2,#}(x6,x7)
     app#(mult1(x6),x7) -> mult{2,#}(x6,x7)
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7)
    TRS:
     if3(true(),xs,ys) -> xs
     if3(false(),xs,ys) -> ys
     lt2(s1(x),s1(y)) -> lt2(x,y)
     lt2(0(),s1(y)) -> true()
     lt2(y,0()) -> false()
     eq2(x,x) -> true()
     eq2(s1(x),0()) -> false()
     eq2(0(),s1(x)) -> false()
     merge2(xs,nil()) -> xs
     merge2(nil(),ys) -> ys
     merge2(cons2(x,xs),cons2(y,ys)) ->
     if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                      cons2(y,merge2(cons2(x,xs),ys))))
     map2(f,nil()) -> nil()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     mult2(0(),x) -> 0()
     mult2(s1(x),y) -> plus2(y,mult2(x,y))
     plus2(0(),x) -> 0()
     plus2(s1(x),y) -> s1(plus2(x,y))
     list1() -> map2(mult1(s1(s1(0()))),hamming())
     list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
     list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
     hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
     app(if2(x6,x5),x7) -> if3(x6,x5,x7)
     app(if1(x6),x7) -> if2(x6,x7)
     app(if(),x7) -> if1(x7)
     app(lt1(x6),x7) -> lt2(x6,x7)
     app(lt(),x7) -> lt1(x7)
     app(s(),x7) -> s1(x7)
     app(eq1(x6),x7) -> eq2(x6,x7)
     app(eq(),x7) -> eq1(x7)
     app(merge1(x6),x7) -> merge2(x6,x7)
     app(merge(),x7) -> merge1(x7)
     app(cons1(x6),x7) -> cons2(x6,x7)
     app(cons(),x7) -> cons1(x7)
     app(map1(x6),x7) -> map2(x6,x7)
     app(map(),x7) -> map1(x7)
     app(mult1(x6),x7) -> mult2(x6,x7)
     app(mult(),x7) -> mult1(x7)
     app(plus1(x6),x7) -> plus2(x6,x7)
     app(plus(),x7) -> plus1(x7)
    graph:
     list3#() -> hamming#() ->
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3()))
     list3#() -> hamming#() -> hamming#() -> list1#()
     list3#() -> hamming#() -> hamming#() -> merge{2,#}(list2(),list3())
     list3#() -> hamming#() -> hamming#() -> list2#()
     list3#() -> hamming#() ->
     hamming#() -> list3#()
     list3#() -> map{2,#}(mult1(s1(s1(s1(s1(s1(0())))))),hamming()) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     list3#() -> map{2,#}(mult1(s1(s1(s1(s1(s1(0())))))),hamming()) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     list2#() -> hamming#() ->
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3()))
     list2#() -> hamming#() -> hamming#() -> list1#()
     list2#() -> hamming#() -> hamming#() -> merge{2,#}(list2(),list3())
     list2#() -> hamming#() -> hamming#() -> list2#()
     list2#() -> hamming#() ->
     hamming#() -> list3#()
     list2#() -> map{2,#}(mult1(s1(s1(s1(0())))),hamming()) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     list2#() -> map{2,#}(mult1(s1(s1(s1(0())))),hamming()) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     hamming#() -> list3#() ->
     list3#() -> map{2,#}(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
     hamming#() -> list3#() -> list3#() -> hamming#()
     hamming#() -> list2#() ->
     list2#() -> map{2,#}(mult1(s1(s1(s1(0())))),hamming())
     hamming#() -> list2#() -> list2#() -> hamming#()
     hamming#() -> list1#() ->
     list1#() -> map{2,#}(mult1(s1(s1(0()))),hamming())
     hamming#() -> list1#() -> list1#() -> hamming#()
     hamming#() -> merge{2,#}(list2(),list3()) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                          cons2(y,merge2(cons2(x,xs),ys))))
     hamming#() -> merge{2,#}(list2(),list3()) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y)
     hamming#() -> merge{2,#}(list2(),list3()) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
     hamming#() -> merge{2,#}(list2(),list3()) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(eq2(x,y),cons2(x,merge2(xs,ys)),cons2(y,merge2(cons2(x,xs),ys)))
     hamming#() -> merge{2,#}(list2(),list3()) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> eq{2,#}(x,y)
     hamming#() -> merge{2,#}(list2(),list3()) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
     hamming#() -> merge{2,#}(list2(),list3()) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3())) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                          cons2(y,merge2(cons2(x,xs),ys))))
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3())) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y)
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3())) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3())) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(eq2(x,y),cons2(x,merge2(xs,ys)),cons2(y,merge2(cons2(x,xs),ys)))
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3())) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> eq{2,#}(x,y)
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3())) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3())) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
     list1#() -> hamming#() ->
     hamming#() -> merge{2,#}(list1(),merge2(list2(),list3()))
     list1#() -> hamming#() -> hamming#() -> list1#()
     list1#() -> hamming#() -> hamming#() -> merge{2,#}(list2(),list3())
     list1#() -> hamming#() -> hamming#() -> list2#()
     list1#() -> hamming#() ->
     hamming#() -> list3#()
     list1#() -> map{2,#}(mult1(s1(s1(0()))),hamming()) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     list1#() -> map{2,#}(mult1(s1(s1(0()))),hamming()) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     mult{2,#}(s1(x),y) -> plus{2,#}(y,mult2(x,y)) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     mult{2,#}(s1(x),y) -> mult{2,#}(x,y) ->
     mult{2,#}(s1(x),y) -> plus{2,#}(y,mult2(x,y))
     mult{2,#}(s1(x),y) -> mult{2,#}(x,y) ->
     mult{2,#}(s1(x),y) -> mult{2,#}(x,y)
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     app#(mult1(x6),x7) -> mult{2,#}(x6,x7) ->
     mult{2,#}(s1(x),y) -> plus{2,#}(y,mult2(x,y))
     app#(mult1(x6),x7) -> mult{2,#}(x6,x7) ->
     mult{2,#}(s1(x),y) -> mult{2,#}(x,y)
     app#(map1(x6),x7) -> map{2,#}(x6,x7) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x6),x7) -> map{2,#}(x6,x7) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                          cons2(y,merge2(cons2(x,xs),ys))))
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y)
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(eq2(x,y),cons2(x,merge2(xs,ys)),cons2(y,merge2(cons2(x,xs),ys)))
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> eq{2,#}(x,y)
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
     app#(lt1(x6),x7) -> lt{2,#}(x6,x7) ->
     lt{2,#}(s1(x),s1(y)) -> lt{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(mult1(x6),x7) -> mult{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x6),x7) -> map{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(merge1(x6),x7) -> merge{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(eq1(x6),x7) -> eq{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(lt1(x6),x7) -> lt{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(if2(x6,x5),x7) -> if{3,#}(x6,x5,x7)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                          cons2(y,merge2(cons2(x,xs),ys))))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(eq2(x,y),cons2(x,merge2(xs,ys)),cons2(y,merge2(cons2(x,xs),ys)))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> eq{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys)) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                          cons2(y,merge2(cons2(x,xs),ys))))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys)) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys)) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys)) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(eq2(x,y),cons2(x,merge2(xs,ys)),cons2(y,merge2(cons2(x,xs),ys)))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys)) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> eq{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys)) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys)) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                          cons2(y,merge2(cons2(x,xs),ys))))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) ->
     if{3,#}(eq2(x,y),cons2(x,merge2(xs,ys)),cons2(y,merge2(cons2(x,xs),ys)))
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> eq{2,#}(x,y)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys) ->
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
     merge{2,#}(cons2(x,xs),cons2(y,ys)) -> lt{2,#}(x,y) ->
     lt{2,#}(s1(x),s1(y)) -> lt{2,#}(x,y)
     lt{2,#}(s1(x),s1(y)) -> lt{2,#}(x,y) -> lt{2,#}(s1(x),s1(y)) -> lt{2,#}(x,y)
    SCC Processor:
     #sccs: 6
     #rules: 15
     #arcs: 90/961
     DPs:
      list3#() -> hamming#()
      hamming#() -> list3#()
      hamming#() -> list2#()
      list2#() -> hamming#()
      hamming#() -> list1#()
      list1#() -> hamming#()
     TRS:
      if3(true(),xs,ys) -> xs
      if3(false(),xs,ys) -> ys
      lt2(s1(x),s1(y)) -> lt2(x,y)
      lt2(0(),s1(y)) -> true()
      lt2(y,0()) -> false()
      eq2(x,x) -> true()
      eq2(s1(x),0()) -> false()
      eq2(0(),s1(x)) -> false()
      merge2(xs,nil()) -> xs
      merge2(nil(),ys) -> ys
      merge2(cons2(x,xs),cons2(y,ys)) ->
      if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                       cons2(y,merge2(cons2(x,xs),ys))))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      mult2(0(),x) -> 0()
      mult2(s1(x),y) -> plus2(y,mult2(x,y))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      list1() -> map2(mult1(s1(s1(0()))),hamming())
      list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
      list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
      hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
      app(if2(x6,x5),x7) -> if3(x6,x5,x7)
      app(if1(x6),x7) -> if2(x6,x7)
      app(if(),x7) -> if1(x7)
      app(lt1(x6),x7) -> lt2(x6,x7)
      app(lt(),x7) -> lt1(x7)
      app(s(),x7) -> s1(x7)
      app(eq1(x6),x7) -> eq2(x6,x7)
      app(eq(),x7) -> eq1(x7)
      app(merge1(x6),x7) -> merge2(x6,x7)
      app(merge(),x7) -> merge1(x7)
      app(cons1(x6),x7) -> cons2(x6,x7)
      app(cons(),x7) -> cons1(x7)
      app(map1(x6),x7) -> map2(x6,x7)
      app(map(),x7) -> map1(x7)
      app(mult1(x6),x7) -> mult2(x6,x7)
      app(mult(),x7) -> mult1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(x7)
     Open
     
     DPs:
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x6),x7) -> map{2,#}(x6,x7)
     TRS:
      if3(true(),xs,ys) -> xs
      if3(false(),xs,ys) -> ys
      lt2(s1(x),s1(y)) -> lt2(x,y)
      lt2(0(),s1(y)) -> true()
      lt2(y,0()) -> false()
      eq2(x,x) -> true()
      eq2(s1(x),0()) -> false()
      eq2(0(),s1(x)) -> false()
      merge2(xs,nil()) -> xs
      merge2(nil(),ys) -> ys
      merge2(cons2(x,xs),cons2(y,ys)) ->
      if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                       cons2(y,merge2(cons2(x,xs),ys))))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      mult2(0(),x) -> 0()
      mult2(s1(x),y) -> plus2(y,mult2(x,y))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      list1() -> map2(mult1(s1(s1(0()))),hamming())
      list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
      list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
      hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
      app(if2(x6,x5),x7) -> if3(x6,x5,x7)
      app(if1(x6),x7) -> if2(x6,x7)
      app(if(),x7) -> if1(x7)
      app(lt1(x6),x7) -> lt2(x6,x7)
      app(lt(),x7) -> lt1(x7)
      app(s(),x7) -> s1(x7)
      app(eq1(x6),x7) -> eq2(x6,x7)
      app(eq(),x7) -> eq1(x7)
      app(merge1(x6),x7) -> merge2(x6,x7)
      app(merge(),x7) -> merge1(x7)
      app(cons1(x6),x7) -> cons2(x6,x7)
      app(cons(),x7) -> cons1(x7)
      app(map1(x6),x7) -> map2(x6,x7)
      app(map(),x7) -> map1(x7)
      app(mult1(x6),x7) -> mult2(x6,x7)
      app(mult(),x7) -> mult1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(x7)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 1
       pi(app#) = 1
      problem:
       DPs:
        app#(map1(x6),x7) -> map{2,#}(x6,x7)
       TRS:
        if3(true(),xs,ys) -> xs
        if3(false(),xs,ys) -> ys
        lt2(s1(x),s1(y)) -> lt2(x,y)
        lt2(0(),s1(y)) -> true()
        lt2(y,0()) -> false()
        eq2(x,x) -> true()
        eq2(s1(x),0()) -> false()
        eq2(0(),s1(x)) -> false()
        merge2(xs,nil()) -> xs
        merge2(nil(),ys) -> ys
        merge2(cons2(x,xs),cons2(y,ys)) ->
        if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                         cons2(y,merge2(cons2(x,xs),ys))))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        mult2(0(),x) -> 0()
        mult2(s1(x),y) -> plus2(y,mult2(x,y))
        plus2(0(),x) -> 0()
        plus2(s1(x),y) -> s1(plus2(x,y))
        list1() -> map2(mult1(s1(s1(0()))),hamming())
        list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
        list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
        hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
        app(if2(x6,x5),x7) -> if3(x6,x5,x7)
        app(if1(x6),x7) -> if2(x6,x7)
        app(if(),x7) -> if1(x7)
        app(lt1(x6),x7) -> lt2(x6,x7)
        app(lt(),x7) -> lt1(x7)
        app(s(),x7) -> s1(x7)
        app(eq1(x6),x7) -> eq2(x6,x7)
        app(eq(),x7) -> eq1(x7)
        app(merge1(x6),x7) -> merge2(x6,x7)
        app(merge(),x7) -> merge1(x7)
        app(cons1(x6),x7) -> cons2(x6,x7)
        app(cons(),x7) -> cons1(x7)
        app(map1(x6),x7) -> map2(x6,x7)
        app(map(),x7) -> map1(x7)
        app(mult1(x6),x7) -> mult2(x6,x7)
        app(mult(),x7) -> mult1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(x7)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 5/1
       
     
     DPs:
      mult{2,#}(s1(x),y) -> mult{2,#}(x,y)
     TRS:
      if3(true(),xs,ys) -> xs
      if3(false(),xs,ys) -> ys
      lt2(s1(x),s1(y)) -> lt2(x,y)
      lt2(0(),s1(y)) -> true()
      lt2(y,0()) -> false()
      eq2(x,x) -> true()
      eq2(s1(x),0()) -> false()
      eq2(0(),s1(x)) -> false()
      merge2(xs,nil()) -> xs
      merge2(nil(),ys) -> ys
      merge2(cons2(x,xs),cons2(y,ys)) ->
      if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                       cons2(y,merge2(cons2(x,xs),ys))))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      mult2(0(),x) -> 0()
      mult2(s1(x),y) -> plus2(y,mult2(x,y))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      list1() -> map2(mult1(s1(s1(0()))),hamming())
      list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
      list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
      hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
      app(if2(x6,x5),x7) -> if3(x6,x5,x7)
      app(if1(x6),x7) -> if2(x6,x7)
      app(if(),x7) -> if1(x7)
      app(lt1(x6),x7) -> lt2(x6,x7)
      app(lt(),x7) -> lt1(x7)
      app(s(),x7) -> s1(x7)
      app(eq1(x6),x7) -> eq2(x6,x7)
      app(eq(),x7) -> eq1(x7)
      app(merge1(x6),x7) -> merge2(x6,x7)
      app(merge(),x7) -> merge1(x7)
      app(cons1(x6),x7) -> cons2(x6,x7)
      app(cons(),x7) -> cons1(x7)
      app(map1(x6),x7) -> map2(x6,x7)
      app(map(),x7) -> map1(x7)
      app(mult1(x6),x7) -> mult2(x6,x7)
      app(mult(),x7) -> mult1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(x7)
     Subterm Criterion Processor:
      simple projection:
       pi(mult{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        if3(true(),xs,ys) -> xs
        if3(false(),xs,ys) -> ys
        lt2(s1(x),s1(y)) -> lt2(x,y)
        lt2(0(),s1(y)) -> true()
        lt2(y,0()) -> false()
        eq2(x,x) -> true()
        eq2(s1(x),0()) -> false()
        eq2(0(),s1(x)) -> false()
        merge2(xs,nil()) -> xs
        merge2(nil(),ys) -> ys
        merge2(cons2(x,xs),cons2(y,ys)) ->
        if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                         cons2(y,merge2(cons2(x,xs),ys))))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        mult2(0(),x) -> 0()
        mult2(s1(x),y) -> plus2(y,mult2(x,y))
        plus2(0(),x) -> 0()
        plus2(s1(x),y) -> s1(plus2(x,y))
        list1() -> map2(mult1(s1(s1(0()))),hamming())
        list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
        list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
        hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
        app(if2(x6,x5),x7) -> if3(x6,x5,x7)
        app(if1(x6),x7) -> if2(x6,x7)
        app(if(),x7) -> if1(x7)
        app(lt1(x6),x7) -> lt2(x6,x7)
        app(lt(),x7) -> lt1(x7)
        app(s(),x7) -> s1(x7)
        app(eq1(x6),x7) -> eq2(x6,x7)
        app(eq(),x7) -> eq1(x7)
        app(merge1(x6),x7) -> merge2(x6,x7)
        app(merge(),x7) -> merge1(x7)
        app(cons1(x6),x7) -> cons2(x6,x7)
        app(cons(),x7) -> cons1(x7)
        app(map1(x6),x7) -> map2(x6,x7)
        app(map(),x7) -> map1(x7)
        app(mult1(x6),x7) -> mult2(x6,x7)
        app(mult(),x7) -> mult1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(x7)
      Qed
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      if3(true(),xs,ys) -> xs
      if3(false(),xs,ys) -> ys
      lt2(s1(x),s1(y)) -> lt2(x,y)
      lt2(0(),s1(y)) -> true()
      lt2(y,0()) -> false()
      eq2(x,x) -> true()
      eq2(s1(x),0()) -> false()
      eq2(0(),s1(x)) -> false()
      merge2(xs,nil()) -> xs
      merge2(nil(),ys) -> ys
      merge2(cons2(x,xs),cons2(y,ys)) ->
      if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                       cons2(y,merge2(cons2(x,xs),ys))))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      mult2(0(),x) -> 0()
      mult2(s1(x),y) -> plus2(y,mult2(x,y))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      list1() -> map2(mult1(s1(s1(0()))),hamming())
      list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
      list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
      hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
      app(if2(x6,x5),x7) -> if3(x6,x5,x7)
      app(if1(x6),x7) -> if2(x6,x7)
      app(if(),x7) -> if1(x7)
      app(lt1(x6),x7) -> lt2(x6,x7)
      app(lt(),x7) -> lt1(x7)
      app(s(),x7) -> s1(x7)
      app(eq1(x6),x7) -> eq2(x6,x7)
      app(eq(),x7) -> eq1(x7)
      app(merge1(x6),x7) -> merge2(x6,x7)
      app(merge(),x7) -> merge1(x7)
      app(cons1(x6),x7) -> cons2(x6,x7)
      app(cons(),x7) -> cons1(x7)
      app(map1(x6),x7) -> map2(x6,x7)
      app(map(),x7) -> map1(x7)
      app(mult1(x6),x7) -> mult2(x6,x7)
      app(mult(),x7) -> mult1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(x7)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        if3(true(),xs,ys) -> xs
        if3(false(),xs,ys) -> ys
        lt2(s1(x),s1(y)) -> lt2(x,y)
        lt2(0(),s1(y)) -> true()
        lt2(y,0()) -> false()
        eq2(x,x) -> true()
        eq2(s1(x),0()) -> false()
        eq2(0(),s1(x)) -> false()
        merge2(xs,nil()) -> xs
        merge2(nil(),ys) -> ys
        merge2(cons2(x,xs),cons2(y,ys)) ->
        if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                         cons2(y,merge2(cons2(x,xs),ys))))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        mult2(0(),x) -> 0()
        mult2(s1(x),y) -> plus2(y,mult2(x,y))
        plus2(0(),x) -> 0()
        plus2(s1(x),y) -> s1(plus2(x,y))
        list1() -> map2(mult1(s1(s1(0()))),hamming())
        list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
        list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
        hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
        app(if2(x6,x5),x7) -> if3(x6,x5,x7)
        app(if1(x6),x7) -> if2(x6,x7)
        app(if(),x7) -> if1(x7)
        app(lt1(x6),x7) -> lt2(x6,x7)
        app(lt(),x7) -> lt1(x7)
        app(s(),x7) -> s1(x7)
        app(eq1(x6),x7) -> eq2(x6,x7)
        app(eq(),x7) -> eq1(x7)
        app(merge1(x6),x7) -> merge2(x6,x7)
        app(merge(),x7) -> merge1(x7)
        app(cons1(x6),x7) -> cons2(x6,x7)
        app(cons(),x7) -> cons1(x7)
        app(map1(x6),x7) -> map2(x6,x7)
        app(map(),x7) -> map1(x7)
        app(mult1(x6),x7) -> mult2(x6,x7)
        app(mult(),x7) -> mult1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(x7)
      Qed
     
     DPs:
      merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(cons2(x,xs),ys)
      merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,ys)
      merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
     TRS:
      if3(true(),xs,ys) -> xs
      if3(false(),xs,ys) -> ys
      lt2(s1(x),s1(y)) -> lt2(x,y)
      lt2(0(),s1(y)) -> true()
      lt2(y,0()) -> false()
      eq2(x,x) -> true()
      eq2(s1(x),0()) -> false()
      eq2(0(),s1(x)) -> false()
      merge2(xs,nil()) -> xs
      merge2(nil(),ys) -> ys
      merge2(cons2(x,xs),cons2(y,ys)) ->
      if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                       cons2(y,merge2(cons2(x,xs),ys))))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      mult2(0(),x) -> 0()
      mult2(s1(x),y) -> plus2(y,mult2(x,y))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      list1() -> map2(mult1(s1(s1(0()))),hamming())
      list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
      list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
      hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
      app(if2(x6,x5),x7) -> if3(x6,x5,x7)
      app(if1(x6),x7) -> if2(x6,x7)
      app(if(),x7) -> if1(x7)
      app(lt1(x6),x7) -> lt2(x6,x7)
      app(lt(),x7) -> lt1(x7)
      app(s(),x7) -> s1(x7)
      app(eq1(x6),x7) -> eq2(x6,x7)
      app(eq(),x7) -> eq1(x7)
      app(merge1(x6),x7) -> merge2(x6,x7)
      app(merge(),x7) -> merge1(x7)
      app(cons1(x6),x7) -> cons2(x6,x7)
      app(cons(),x7) -> cons1(x7)
      app(map1(x6),x7) -> map2(x6,x7)
      app(map(),x7) -> map1(x7)
      app(mult1(x6),x7) -> mult2(x6,x7)
      app(mult(),x7) -> mult1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(x7)
     Subterm Criterion Processor:
      simple projection:
       pi(merge{2,#}) = 1
      problem:
       DPs:
        merge{2,#}(cons2(x,xs),cons2(y,ys)) -> merge{2,#}(xs,cons2(y,ys))
       TRS:
        if3(true(),xs,ys) -> xs
        if3(false(),xs,ys) -> ys
        lt2(s1(x),s1(y)) -> lt2(x,y)
        lt2(0(),s1(y)) -> true()
        lt2(y,0()) -> false()
        eq2(x,x) -> true()
        eq2(s1(x),0()) -> false()
        eq2(0(),s1(x)) -> false()
        merge2(xs,nil()) -> xs
        merge2(nil(),ys) -> ys
        merge2(cons2(x,xs),cons2(y,ys)) ->
        if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                         cons2(y,merge2(cons2(x,xs),ys))))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        mult2(0(),x) -> 0()
        mult2(s1(x),y) -> plus2(y,mult2(x,y))
        plus2(0(),x) -> 0()
        plus2(s1(x),y) -> s1(plus2(x,y))
        list1() -> map2(mult1(s1(s1(0()))),hamming())
        list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
        list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
        hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
        app(if2(x6,x5),x7) -> if3(x6,x5,x7)
        app(if1(x6),x7) -> if2(x6,x7)
        app(if(),x7) -> if1(x7)
        app(lt1(x6),x7) -> lt2(x6,x7)
        app(lt(),x7) -> lt1(x7)
        app(s(),x7) -> s1(x7)
        app(eq1(x6),x7) -> eq2(x6,x7)
        app(eq(),x7) -> eq1(x7)
        app(merge1(x6),x7) -> merge2(x6,x7)
        app(merge(),x7) -> merge1(x7)
        app(cons1(x6),x7) -> cons2(x6,x7)
        app(cons(),x7) -> cons1(x7)
        app(map1(x6),x7) -> map2(x6,x7)
        app(map(),x7) -> map1(x7)
        app(mult1(x6),x7) -> mult2(x6,x7)
        app(mult(),x7) -> mult1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(x7)
      Subterm Criterion Processor:
       simple projection:
        pi(merge{2,#}) = 0
       problem:
        DPs:
         
        TRS:
         if3(true(),xs,ys) -> xs
         if3(false(),xs,ys) -> ys
         lt2(s1(x),s1(y)) -> lt2(x,y)
         lt2(0(),s1(y)) -> true()
         lt2(y,0()) -> false()
         eq2(x,x) -> true()
         eq2(s1(x),0()) -> false()
         eq2(0(),s1(x)) -> false()
         merge2(xs,nil()) -> xs
         merge2(nil(),ys) -> ys
         merge2(cons2(x,xs),cons2(y,ys)) ->
         if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                          cons2(y,merge2(cons2(x,xs),ys))))
         map2(f,nil()) -> nil()
         map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
         mult2(0(),x) -> 0()
         mult2(s1(x),y) -> plus2(y,mult2(x,y))
         plus2(0(),x) -> 0()
         plus2(s1(x),y) -> s1(plus2(x,y))
         list1() -> map2(mult1(s1(s1(0()))),hamming())
         list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
         list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
         hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
         app(if2(x6,x5),x7) -> if3(x6,x5,x7)
         app(if1(x6),x7) -> if2(x6,x7)
         app(if(),x7) -> if1(x7)
         app(lt1(x6),x7) -> lt2(x6,x7)
         app(lt(),x7) -> lt1(x7)
         app(s(),x7) -> s1(x7)
         app(eq1(x6),x7) -> eq2(x6,x7)
         app(eq(),x7) -> eq1(x7)
         app(merge1(x6),x7) -> merge2(x6,x7)
         app(merge(),x7) -> merge1(x7)
         app(cons1(x6),x7) -> cons2(x6,x7)
         app(cons(),x7) -> cons1(x7)
         app(map1(x6),x7) -> map2(x6,x7)
         app(map(),x7) -> map1(x7)
         app(mult1(x6),x7) -> mult2(x6,x7)
         app(mult(),x7) -> mult1(x7)
         app(plus1(x6),x7) -> plus2(x6,x7)
         app(plus(),x7) -> plus1(x7)
       Qed
     
     DPs:
      lt{2,#}(s1(x),s1(y)) -> lt{2,#}(x,y)
     TRS:
      if3(true(),xs,ys) -> xs
      if3(false(),xs,ys) -> ys
      lt2(s1(x),s1(y)) -> lt2(x,y)
      lt2(0(),s1(y)) -> true()
      lt2(y,0()) -> false()
      eq2(x,x) -> true()
      eq2(s1(x),0()) -> false()
      eq2(0(),s1(x)) -> false()
      merge2(xs,nil()) -> xs
      merge2(nil(),ys) -> ys
      merge2(cons2(x,xs),cons2(y,ys)) ->
      if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                       cons2(y,merge2(cons2(x,xs),ys))))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      mult2(0(),x) -> 0()
      mult2(s1(x),y) -> plus2(y,mult2(x,y))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      list1() -> map2(mult1(s1(s1(0()))),hamming())
      list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
      list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
      hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
      app(if2(x6,x5),x7) -> if3(x6,x5,x7)
      app(if1(x6),x7) -> if2(x6,x7)
      app(if(),x7) -> if1(x7)
      app(lt1(x6),x7) -> lt2(x6,x7)
      app(lt(),x7) -> lt1(x7)
      app(s(),x7) -> s1(x7)
      app(eq1(x6),x7) -> eq2(x6,x7)
      app(eq(),x7) -> eq1(x7)
      app(merge1(x6),x7) -> merge2(x6,x7)
      app(merge(),x7) -> merge1(x7)
      app(cons1(x6),x7) -> cons2(x6,x7)
      app(cons(),x7) -> cons1(x7)
      app(map1(x6),x7) -> map2(x6,x7)
      app(map(),x7) -> map1(x7)
      app(mult1(x6),x7) -> mult2(x6,x7)
      app(mult(),x7) -> mult1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(x7)
     Subterm Criterion Processor:
      simple projection:
       pi(lt{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        if3(true(),xs,ys) -> xs
        if3(false(),xs,ys) -> ys
        lt2(s1(x),s1(y)) -> lt2(x,y)
        lt2(0(),s1(y)) -> true()
        lt2(y,0()) -> false()
        eq2(x,x) -> true()
        eq2(s1(x),0()) -> false()
        eq2(0(),s1(x)) -> false()
        merge2(xs,nil()) -> xs
        merge2(nil(),ys) -> ys
        merge2(cons2(x,xs),cons2(y,ys)) ->
        if3(lt2(x,y),cons2(x,merge2(xs,cons2(y,ys))),if3(eq2(x,y),cons2(x,merge2(xs,ys)),
                                                         cons2(y,merge2(cons2(x,xs),ys))))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        mult2(0(),x) -> 0()
        mult2(s1(x),y) -> plus2(y,mult2(x,y))
        plus2(0(),x) -> 0()
        plus2(s1(x),y) -> s1(plus2(x,y))
        list1() -> map2(mult1(s1(s1(0()))),hamming())
        list2() -> map2(mult1(s1(s1(s1(0())))),hamming())
        list3() -> map2(mult1(s1(s1(s1(s1(s1(0())))))),hamming())
        hamming() -> cons2(s1(0()),merge2(list1(),merge2(list2(),list3())))
        app(if2(x6,x5),x7) -> if3(x6,x5,x7)
        app(if1(x6),x7) -> if2(x6,x7)
        app(if(),x7) -> if1(x7)
        app(lt1(x6),x7) -> lt2(x6,x7)
        app(lt(),x7) -> lt1(x7)
        app(s(),x7) -> s1(x7)
        app(eq1(x6),x7) -> eq2(x6,x7)
        app(eq(),x7) -> eq1(x7)
        app(merge1(x6),x7) -> merge2(x6,x7)
        app(merge(),x7) -> merge1(x7)
        app(cons1(x6),x7) -> cons2(x6,x7)
        app(cons(),x7) -> cons1(x7)
        app(map1(x6),x7) -> map2(x6,x7)
        app(map(),x7) -> map1(x7)
        app(mult1(x6),x7) -> mult2(x6,x7)
        app(mult(),x7) -> mult1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(x7)
      Qed