YES

Problem:
 app(app(times(),x),app(app(plus(),y),app(s(),z))) ->
 app(app(plus(),app(app(times(),x),app(app(plus(),y),app(app(times(),app(s(),z)),0())))),
     app(app(times(),x),app(s(),z)))
 app(app(times(),x),0()) -> 0()
 app(app(times(),x),app(s(),y)) -> app(app(plus(),app(app(times(),x),y)),x)
 app(app(plus(),x),0()) -> x
 app(app(plus(),x),app(s(),y)) -> app(s(),app(app(plus(),x),y))
 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:
 Uncurry Processor:
  times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
  times2(x,0()) -> 0()
  times2(x,s1(y)) -> plus2(times2(x,y),x)
  plus2(x,0()) -> x
  plus2(x,s1(y)) -> s1(plus2(x,y))
  map2(f,nil()) -> nil()
  map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
  filter2(f,nil()) -> nil()
  filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
  filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
  filter24(false(),f,x,xs) -> filter2(f,xs)
  app(times1(x7),x8) -> times2(x7,x8)
  app(times(),x8) -> times1(x8)
  app(plus1(x7),x8) -> plus2(x7,x8)
  app(plus(),x8) -> plus1(x8)
  app(s(),x8) -> s1(x8)
  app(map1(x7),x8) -> map2(x7,x8)
  app(map(),x8) -> map1(x8)
  app(cons1(x7),x8) -> cons2(x7,x8)
  app(cons(),x8) -> cons1(x8)
  app(filter1(x7),x8) -> filter2(x7,x8)
  app(filter(),x8) -> filter1(x8)
  app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
  app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
  app(filter21(x7),x8) -> filter22(x7,x8)
  app(filter2(),x8) -> filter21(x8)
  DP Processor:
   DPs:
    times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
    times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
    times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(y,times2(s1(z),0()))
    times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
    times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
    times{2,#}(x,s1(y)) -> times{2,#}(x,y)
    times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
    plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
    filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
    filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
    app#(times1(x7),x8) -> times{2,#}(x7,x8)
    app#(plus1(x7),x8) -> plus{2,#}(x7,x8)
    app#(map1(x7),x8) -> map{2,#}(x7,x8)
    app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
    app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
   TRS:
    times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
    times2(x,0()) -> 0()
    times2(x,s1(y)) -> plus2(times2(x,y),x)
    plus2(x,0()) -> x
    plus2(x,s1(y)) -> s1(plus2(x,y))
    map2(f,nil()) -> nil()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    filter2(f,nil()) -> nil()
    filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
    filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
    filter24(false(),f,x,xs) -> filter2(f,xs)
    app(times1(x7),x8) -> times2(x7,x8)
    app(times(),x8) -> times1(x8)
    app(plus1(x7),x8) -> plus2(x7,x8)
    app(plus(),x8) -> plus1(x8)
    app(s(),x8) -> s1(x8)
    app(map1(x7),x8) -> map2(x7,x8)
    app(map(),x8) -> map1(x8)
    app(cons1(x7),x8) -> cons2(x7,x8)
    app(cons(),x8) -> cons1(x8)
    app(filter1(x7),x8) -> filter2(x7,x8)
    app(filter(),x8) -> filter1(x8)
    app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
    app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
    app(filter21(x7),x8) -> filter22(x7,x8)
    app(filter2(),x8) -> filter21(x8)
   TDG Processor:
    DPs:
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
     times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(y,times2(s1(z),0()))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
     times{2,#}(x,plus2(y,s1(z))) ->
     plus{2,#}(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
     times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(times1(x7),x8) -> times{2,#}(x7,x8)
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8)
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
    TRS:
     times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
     times2(x,0()) -> 0()
     times2(x,s1(y)) -> plus2(times2(x,y),x)
     plus2(x,0()) -> x
     plus2(x,s1(y)) -> s1(plus2(x,y))
     map2(f,nil()) -> nil()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     filter2(f,nil()) -> nil()
     filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
     filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
     filter24(false(),f,x,xs) -> filter2(f,xs)
     app(times1(x7),x8) -> times2(x7,x8)
     app(times(),x8) -> times1(x8)
     app(plus1(x7),x8) -> plus2(x7,x8)
     app(plus(),x8) -> plus1(x8)
     app(s(),x8) -> s1(x8)
     app(map1(x7),x8) -> map2(x7,x8)
     app(map(),x8) -> map1(x8)
     app(cons1(x7),x8) -> cons2(x7,x8)
     app(cons(),x8) -> cons1(x8)
     app(filter1(x7),x8) -> filter2(x7,x8)
     app(filter(),x8) -> filter1(x8)
     app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
     app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
     app(filter21(x7),x8) -> filter22(x7,x8)
     app(filter2(),x8) -> filter21(x8)
    graph:
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(times1(x7),x8) -> times{2,#}(x7,x8)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8) ->
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8) ->
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x7),x8) -> map{2,#}(x7,x8) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x7),x8) -> map{2,#}(x7,x8) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     app#(times1(x7),x8) -> times{2,#}(x7,x8) ->
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     app#(times1(x7),x8) -> times{2,#}(x7,x8) ->
     times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     app#(times1(x7),x8) -> times{2,#}(x7,x8) ->
     times{2,#}(x,plus2(y,s1(z))) ->
     plus{2,#}(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
     app#(times1(x7),x8) -> times{2,#}(x7,x8) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
     app#(times1(x7),x8) -> times{2,#}(x7,x8) ->
     times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(y,times2(s1(z),0()))
     app#(times1(x7),x8) -> times{2,#}(x7,x8) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
     app#(times1(x7),x8) -> times{2,#}(x7,x8) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(times1(x7),x8) -> times{2,#}(x7,x8)
     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)
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
     times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
     times{2,#}(x,plus2(y,s1(z))) ->
     plus{2,#}(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
     times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(y,times2(s1(z),0()))
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
     times{2,#}(x,plus2(y,s1(z))) ->
     plus{2,#}(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z))) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(y,times2(s1(z),0())) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0()) ->
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0()) ->
     times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0()) ->
     times{2,#}(x,plus2(y,s1(z))) ->
     plus{2,#}(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0()) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0()) ->
     times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(y,times2(s1(z),0()))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0()) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0()) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z)) ->
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z)) ->
     times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z)) ->
     times{2,#}(x,plus2(y,s1(z))) ->
     plus{2,#}(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z)) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z)) ->
     times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(y,times2(s1(z),0()))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z)) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z)) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
     times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
     times{2,#}(x,plus2(y,s1(z))) ->
     plus{2,#}(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
     times{2,#}(x,plus2(y,s1(z))) -> plus{2,#}(y,times2(s1(z),0()))
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
     times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
    SCC Processor:
     #sccs: 3
     #rules: 14
     #arcs: 64/361
     DPs:
      filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x7),x8) -> map{2,#}(x7,x8)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
      filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
      filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
      app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     TRS:
      times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
      times2(x,0()) -> 0()
      times2(x,s1(y)) -> plus2(times2(x,y),x)
      plus2(x,0()) -> x
      plus2(x,s1(y)) -> s1(plus2(x,y))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(times1(x7),x8) -> times2(x7,x8)
      app(times(),x8) -> times1(x8)
      app(plus1(x7),x8) -> plus2(x7,x8)
      app(plus(),x8) -> plus1(x8)
      app(s(),x8) -> s1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 1
       pi(app#) = 1
       pi(filter{2,#}) = 1
       pi(filter2{4,#}) = 3
      problem:
       DPs:
        filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
        app#(map1(x7),x8) -> map{2,#}(x7,x8)
        app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
        filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
        app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
       TRS:
        times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
        times2(x,0()) -> 0()
        times2(x,s1(y)) -> plus2(times2(x,y),x)
        plus2(x,0()) -> x
        plus2(x,s1(y)) -> s1(plus2(x,y))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil()) -> nil()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false(),f,x,xs) -> filter2(f,xs)
        app(times1(x7),x8) -> times2(x7,x8)
        app(times(),x8) -> times1(x8)
        app(plus1(x7),x8) -> plus2(x7,x8)
        app(plus(),x8) -> plus1(x8)
        app(s(),x8) -> s1(x8)
        app(map1(x7),x8) -> map2(x7,x8)
        app(map(),x8) -> map1(x8)
        app(cons1(x7),x8) -> cons2(x7,x8)
        app(cons(),x8) -> cons1(x8)
        app(filter1(x7),x8) -> filter2(x7,x8)
        app(filter(),x8) -> filter1(x8)
        app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
        app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
        app(filter21(x7),x8) -> filter22(x7,x8)
        app(filter2(),x8) -> filter21(x8)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 20/25
       
     
     DPs:
      times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
      times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
      times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
      times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     TRS:
      times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
      times2(x,0()) -> 0()
      times2(x,s1(y)) -> plus2(times2(x,y),x)
      plus2(x,0()) -> x
      plus2(x,s1(y)) -> s1(plus2(x,y))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(times1(x7),x8) -> times2(x7,x8)
      app(times(),x8) -> times1(x8)
      app(plus1(x7),x8) -> plus2(x7,x8)
      app(plus(),x8) -> plus1(x8)
      app(s(),x8) -> s1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     EDG Processor:
      DPs:
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
       times{2,#}(x,s1(y)) -> times{2,#}(x,y)
      TRS:
       times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
       times2(x,0()) -> 0()
       times2(x,s1(y)) -> plus2(times2(x,y),x)
       plus2(x,0()) -> x
       plus2(x,s1(y)) -> s1(plus2(x,y))
       map2(f,nil()) -> nil()
       map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
       filter2(f,nil()) -> nil()
       filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
       filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
       filter24(false(),f,x,xs) -> filter2(f,xs)
       app(times1(x7),x8) -> times2(x7,x8)
       app(times(),x8) -> times1(x8)
       app(plus1(x7),x8) -> plus2(x7,x8)
       app(plus(),x8) -> plus1(x8)
       app(s(),x8) -> s1(x8)
       app(map1(x7),x8) -> map2(x7,x8)
       app(map(),x8) -> map1(x8)
       app(cons1(x7),x8) -> cons2(x7,x8)
       app(cons(),x8) -> cons1(x8)
       app(filter1(x7),x8) -> filter2(x7,x8)
       app(filter(),x8) -> filter1(x8)
       app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
       app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
       app(filter21(x7),x8) -> filter22(x7,x8)
       app(filter2(),x8) -> filter21(x8)
      graph:
       times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
       times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
       times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
       times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
       times{2,#}(x,s1(y)) -> times{2,#}(x,y)
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z)) ->
       times{2,#}(x,s1(y)) -> times{2,#}(x,y)
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(s1(z),0())
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
       times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0()))) ->
       times{2,#}(x,s1(y)) -> times{2,#}(x,y)
      SCC Processor:
       #sccs: 1
       #rules: 3
       #arcs: 9/16
       DPs:
        times{2,#}(x,s1(y)) -> times{2,#}(x,y)
        times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
        times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
       TRS:
        times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
        times2(x,0()) -> 0()
        times2(x,s1(y)) -> plus2(times2(x,y),x)
        plus2(x,0()) -> x
        plus2(x,s1(y)) -> s1(plus2(x,y))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil()) -> nil()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false(),f,x,xs) -> filter2(f,xs)
        app(times1(x7),x8) -> times2(x7,x8)
        app(times(),x8) -> times1(x8)
        app(plus1(x7),x8) -> plus2(x7,x8)
        app(plus(),x8) -> plus1(x8)
        app(s(),x8) -> s1(x8)
        app(map1(x7),x8) -> map2(x7,x8)
        app(map(),x8) -> map1(x8)
        app(cons1(x7),x8) -> cons2(x7,x8)
        app(cons(),x8) -> cons1(x8)
        app(filter1(x7),x8) -> filter2(x7,x8)
        app(filter(),x8) -> filter1(x8)
        app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
        app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
        app(filter21(x7),x8) -> filter22(x7,x8)
        app(filter2(),x8) -> filter21(x8)
       Usable Rule Processor:
        DPs:
         times{2,#}(x,s1(y)) -> times{2,#}(x,y)
         times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,plus2(y,times2(s1(z),0())))
         times{2,#}(x,plus2(y,s1(z))) -> times{2,#}(x,s1(z))
        TRS:
         times2(x,0()) -> 0()
         plus2(x,0()) -> x
         plus2(x,s1(y)) -> s1(plus2(x,y))
        Semantic Labeling Processor:
         dimension: 1
         usable rules:
          times2(x,0()) -> 0()
          plus2(x,0()) -> x
          plus2(x,s1(y)) -> s1(plus2(x,y))
         interpretation:
          [s1](x0) = x0,
          
          [plus2](x0, x1) = x0 + x1,
          
          [times2](x0, x1) = x1,
          
          [0] = 1
          labeled:
          usable (for model):
          argument filtering:
           pi(0) = []
           pi(times2) = 1
           pi(plus2) = [0,1]
           pi(s1) = [0]
           pi(times{2,#}) = [1]
         precedence:
          plus2 > s1 > times{2,#} ~ times2 ~ 0
         problem:
          DPs:
           
          TRS:
           times2(x,0()) -> 0()
           plus2(x,0()) -> x
           plus2(x,s1(y)) -> s1(plus2(x,y))
         Qed
     
     DPs:
      plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     TRS:
      times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
      times2(x,0()) -> 0()
      times2(x,s1(y)) -> plus2(times2(x,y),x)
      plus2(x,0()) -> x
      plus2(x,s1(y)) -> s1(plus2(x,y))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(times1(x7),x8) -> times2(x7,x8)
      app(times(),x8) -> times1(x8)
      app(plus1(x7),x8) -> plus2(x7,x8)
      app(plus(),x8) -> plus1(x8)
      app(s(),x8) -> s1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        times2(x,plus2(y,s1(z))) -> plus2(times2(x,plus2(y,times2(s1(z),0()))),times2(x,s1(z)))
        times2(x,0()) -> 0()
        times2(x,s1(y)) -> plus2(times2(x,y),x)
        plus2(x,0()) -> x
        plus2(x,s1(y)) -> s1(plus2(x,y))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil()) -> nil()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false(),f,x,xs) -> filter2(f,xs)
        app(times1(x7),x8) -> times2(x7,x8)
        app(times(),x8) -> times1(x8)
        app(plus1(x7),x8) -> plus2(x7,x8)
        app(plus(),x8) -> plus1(x8)
        app(s(),x8) -> s1(x8)
        app(map1(x7),x8) -> map2(x7,x8)
        app(map(),x8) -> map1(x8)
        app(cons1(x7),x8) -> cons2(x7,x8)
        app(cons(),x8) -> cons1(x8)
        app(filter1(x7),x8) -> filter2(x7,x8)
        app(filter(),x8) -> filter1(x8)
        app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
        app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
        app(filter21(x7),x8) -> filter22(x7,x8)
        app(filter2(),x8) -> filter21(x8)
      Qed