YES

Problem:
 app(app(ack(),0()),y) -> app(succ(),y)
 app(app(ack(),app(succ(),x)),y) -> app(app(ack(),x),app(succ(),0()))
 app(app(ack(),app(succ(),x)),app(succ(),y)) -> app(app(ack(),x),app(app(ack(),app(succ(),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:
  ack2(0(),y) -> succ1(y)
  ack2(succ1(x),y) -> ack2(x,succ1(0()))
  ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(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(ack1(x6),x7) -> ack2(x6,x7)
  app(ack(),x7) -> ack1(x7)
  app(succ(),x7) -> succ1(x7)
  app(map1(x6),x7) -> map2(x6,x7)
  app(map(),x7) -> map1(x7)
  app(cons1(x6),x7) -> cons2(x6,x7)
  app(cons(),x7) -> cons1(x7)
  app(filter1(x6),x7) -> filter2(x6,x7)
  app(filter(),x7) -> filter1(x7)
  app(filter23(x6,x5,x4),x7) -> filter24(x6,x5,x4,x7)
  app(filter22(x6,x5),x7) -> filter23(x6,x5,x7)
  app(filter21(x6),x7) -> filter22(x6,x7)
  app(filter2(),x7) -> filter21(x7)
  DP Processor:
   DPs:
    ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0()))
    ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
    ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(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#(ack1(x6),x7) -> ack{2,#}(x6,x7)
    app#(map1(x6),x7) -> map{2,#}(x6,x7)
    app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
    app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
   TRS:
    ack2(0(),y) -> succ1(y)
    ack2(succ1(x),y) -> ack2(x,succ1(0()))
    ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(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(ack1(x6),x7) -> ack2(x6,x7)
    app(ack(),x7) -> ack1(x7)
    app(succ(),x7) -> succ1(x7)
    app(map1(x6),x7) -> map2(x6,x7)
    app(map(),x7) -> map1(x7)
    app(cons1(x6),x7) -> cons2(x6,x7)
    app(cons(),x7) -> cons1(x7)
    app(filter1(x6),x7) -> filter2(x6,x7)
    app(filter(),x7) -> filter1(x7)
    app(filter23(x6,x5,x4),x7) -> filter24(x6,x5,x4,x7)
    app(filter22(x6,x5),x7) -> filter23(x6,x5,x7)
    app(filter21(x6),x7) -> filter22(x6,x7)
    app(filter2(),x7) -> filter21(x7)
   TDG Processor:
    DPs:
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0()))
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(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#(ack1(x6),x7) -> ack{2,#}(x6,x7)
     app#(map1(x6),x7) -> map{2,#}(x6,x7)
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
    TRS:
     ack2(0(),y) -> succ1(y)
     ack2(succ1(x),y) -> ack2(x,succ1(0()))
     ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(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(ack1(x6),x7) -> ack2(x6,x7)
     app(ack(),x7) -> ack1(x7)
     app(succ(),x7) -> succ1(x7)
     app(map1(x6),x7) -> map2(x6,x7)
     app(map(),x7) -> map1(x7)
     app(cons1(x6),x7) -> cons2(x6,x7)
     app(cons(),x7) -> cons1(x7)
     app(filter1(x6),x7) -> filter2(x6,x7)
     app(filter(),x7) -> filter1(x7)
     app(filter23(x6,x5,x4),x7) -> filter24(x6,x5,x4,x7)
     app(filter22(x6,x5),x7) -> filter23(x6,x5,x7)
     app(filter21(x6),x7) -> filter22(x6,x7)
     app(filter2(),x7) -> filter21(x7)
    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(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x6),x7) -> map{2,#}(x6,x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(ack1(x6),x7) -> ack{2,#}(x6,x7)
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7) ->
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7) ->
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7) ->
     filter{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#(map1(x6),x7) -> map{2,#}(x6,x7) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(ack1(x6),x7) -> ack{2,#}(x6,x7) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y))
     app#(ack1(x6),x7) -> ack{2,#}(x6,x7) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
     app#(ack1(x6),x7) -> ack{2,#}(x6,x7) ->
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0()))
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x6),x7) -> filter{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#(ack1(x6),x7) -> ack{2,#}(x6,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)
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y))
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y) ->
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0()))
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y)) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y))
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y)) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y)) ->
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0()))
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0())) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y))
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0())) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0())) ->
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0()))
    SCC Processor:
     #sccs: 2
     #rules: 12
     #arcs: 34/169
     DPs:
      filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
      filter{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)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
      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(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     TRS:
      ack2(0(),y) -> succ1(y)
      ack2(succ1(x),y) -> ack2(x,succ1(0()))
      ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(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(ack1(x6),x7) -> ack2(x6,x7)
      app(ack(),x7) -> ack1(x7)
      app(succ(),x7) -> succ1(x7)
      app(map1(x6),x7) -> map2(x6,x7)
      app(map(),x7) -> map1(x7)
      app(cons1(x6),x7) -> cons2(x6,x7)
      app(cons(),x7) -> cons1(x7)
      app(filter1(x6),x7) -> filter2(x6,x7)
      app(filter(),x7) -> filter1(x7)
      app(filter23(x6,x5,x4),x7) -> filter24(x6,x5,x4,x7)
      app(filter22(x6,x5),x7) -> filter23(x6,x5,x7)
      app(filter21(x6),x7) -> filter22(x6,x7)
      app(filter2(),x7) -> filter21(x7)
     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(x6),x7) -> map{2,#}(x6,x7)
        app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
        filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
        app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
       TRS:
        ack2(0(),y) -> succ1(y)
        ack2(succ1(x),y) -> ack2(x,succ1(0()))
        ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(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(ack1(x6),x7) -> ack2(x6,x7)
        app(ack(),x7) -> ack1(x7)
        app(succ(),x7) -> succ1(x7)
        app(map1(x6),x7) -> map2(x6,x7)
        app(map(),x7) -> map1(x7)
        app(cons1(x6),x7) -> cons2(x6,x7)
        app(cons(),x7) -> cons1(x7)
        app(filter1(x6),x7) -> filter2(x6,x7)
        app(filter(),x7) -> filter1(x7)
        app(filter23(x6,x5,x4),x7) -> filter24(x6,x5,x4,x7)
        app(filter22(x6,x5),x7) -> filter23(x6,x5,x7)
        app(filter21(x6),x7) -> filter22(x6,x7)
        app(filter2(),x7) -> filter21(x7)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 20/25
       
     
     DPs:
      ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(0()))
      ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
      ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y))
     TRS:
      ack2(0(),y) -> succ1(y)
      ack2(succ1(x),y) -> ack2(x,succ1(0()))
      ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(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(ack1(x6),x7) -> ack2(x6,x7)
      app(ack(),x7) -> ack1(x7)
      app(succ(),x7) -> succ1(x7)
      app(map1(x6),x7) -> map2(x6,x7)
      app(map(),x7) -> map1(x7)
      app(cons1(x6),x7) -> cons2(x6,x7)
      app(cons(),x7) -> cons1(x7)
      app(filter1(x6),x7) -> filter2(x6,x7)
      app(filter(),x7) -> filter1(x7)
      app(filter23(x6,x5,x4),x7) -> filter24(x6,x5,x4,x7)
      app(filter22(x6,x5),x7) -> filter23(x6,x5,x7)
      app(filter21(x6),x7) -> filter22(x6,x7)
      app(filter2(),x7) -> filter21(x7)
     Subterm Criterion Processor:
      simple projection:
       pi(ack{2,#}) = 0
      problem:
       DPs:
        ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
       TRS:
        ack2(0(),y) -> succ1(y)
        ack2(succ1(x),y) -> ack2(x,succ1(0()))
        ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(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(ack1(x6),x7) -> ack2(x6,x7)
        app(ack(),x7) -> ack1(x7)
        app(succ(),x7) -> succ1(x7)
        app(map1(x6),x7) -> map2(x6,x7)
        app(map(),x7) -> map1(x7)
        app(cons1(x6),x7) -> cons2(x6,x7)
        app(cons(),x7) -> cons1(x7)
        app(filter1(x6),x7) -> filter2(x6,x7)
        app(filter(),x7) -> filter1(x7)
        app(filter23(x6,x5,x4),x7) -> filter24(x6,x5,x4,x7)
        app(filter22(x6,x5),x7) -> filter23(x6,x5,x7)
        app(filter21(x6),x7) -> filter22(x6,x7)
        app(filter2(),x7) -> filter21(x7)
      Subterm Criterion Processor:
       simple projection:
        pi(ack{2,#}) = 1
       problem:
        DPs:
         
        TRS:
         ack2(0(),y) -> succ1(y)
         ack2(succ1(x),y) -> ack2(x,succ1(0()))
         ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(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(ack1(x6),x7) -> ack2(x6,x7)
         app(ack(),x7) -> ack1(x7)
         app(succ(),x7) -> succ1(x7)
         app(map1(x6),x7) -> map2(x6,x7)
         app(map(),x7) -> map1(x7)
         app(cons1(x6),x7) -> cons2(x6,x7)
         app(cons(),x7) -> cons1(x7)
         app(filter1(x6),x7) -> filter2(x6,x7)
         app(filter(),x7) -> filter1(x7)
         app(filter23(x6,x5,x4),x7) -> filter24(x6,x5,x4,x7)
         app(filter22(x6,x5),x7) -> filter23(x6,x5,x7)
         app(filter21(x6),x7) -> filter22(x6,x7)
         app(filter2(),x7) -> filter21(x7)
       Qed