/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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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:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   false ==> false0/0
   true ==> true0/0
   filter2 ==> filter20/0 filter21/1 filter22/2 filter23/3 filter24/4
   filter ==> filter0/0 filter1/1 filter2/2
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   succ ==> succ0/0 succ1/1
   0 ==> 00/0
   ack ==> ack0/0 ack1/1 ack2/2
   
  uncurry-rules:
   app(ack1(x4),x5) -> ack2(x4,x5)
   app(ack0(),x4) -> ack1(x4)
   app(succ0(),x8) -> succ1(x8)
   app(map1(x10),x11) -> map2(x10,x11)
   app(map0(),x10) -> map1(x10)
   app(cons1(x14),x15) -> cons2(x14,x15)
   app(cons0(),x14) -> cons1(x14)
   app(filter1(x17),x18) -> filter2(x17,x18)
   app(filter0(),x17) -> filter1(x17)
   app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
   app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
   app(filter21(x20),x21) -> filter22(x20,x21)
   app(filter20(),x20) -> filter21(x20)
  eta-rules:
   
  problem:
   ack2(00(),y) -> succ1(y)
   ack2(succ1(x),y) -> ack2(x,succ1(00()))
   ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   filter2(f,nil0()) -> nil0()
   filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
   filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
   filter24(false0(),f,x,xs) -> filter2(f,xs)
   app(ack1(x4),x5) -> ack2(x4,x5)
   app(ack0(),x4) -> ack1(x4)
   app(succ0(),x8) -> succ1(x8)
   app(map1(x10),x11) -> map2(x10,x11)
   app(map0(),x10) -> map1(x10)
   app(cons1(x14),x15) -> cons2(x14,x15)
   app(cons0(),x14) -> cons1(x14)
   app(filter1(x17),x18) -> filter2(x17,x18)
   app(filter0(),x17) -> filter1(x17)
   app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
   app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
   app(filter21(x20),x21) -> filter22(x20,x21)
   app(filter20(),x20) -> filter21(x20)
  DP Processor:
   DPs:
    ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(00()))
    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,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
    filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
    app#(ack1(x4),x5) -> ack{2,#}(x4,x5)
    app#(map1(x10),x11) -> map{2,#}(x10,x11)
    app#(filter1(x17),x18) -> filter{2,#}(x17,x18)
    app#(filter23(x20,x21,x22),x23) -> filter2{4,#}(x20,x21,x22,x23)
   TRS:
    ack2(00(),y) -> succ1(y)
    ack2(succ1(x),y) -> ack2(x,succ1(00()))
    ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    filter2(f,nil0()) -> nil0()
    filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
    filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
    filter24(false0(),f,x,xs) -> filter2(f,xs)
    app(ack1(x4),x5) -> ack2(x4,x5)
    app(ack0(),x4) -> ack1(x4)
    app(succ0(),x8) -> succ1(x8)
    app(map1(x10),x11) -> map2(x10,x11)
    app(map0(),x10) -> map1(x10)
    app(cons1(x14),x15) -> cons2(x14,x15)
    app(cons0(),x14) -> cons1(x14)
    app(filter1(x17),x18) -> filter2(x17,x18)
    app(filter0(),x17) -> filter1(x17)
    app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
    app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
    app(filter21(x20),x21) -> filter22(x20,x21)
    app(filter20(),x20) -> filter21(x20)
   TDG Processor:
    DPs:
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(00()))
     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,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(ack1(x4),x5) -> ack{2,#}(x4,x5)
     app#(map1(x10),x11) -> map{2,#}(x10,x11)
     app#(filter1(x17),x18) -> filter{2,#}(x17,x18)
     app#(filter23(x20,x21,x22),x23) -> filter2{4,#}(x20,x21,x22,x23)
    TRS:
     ack2(00(),y) -> succ1(y)
     ack2(succ1(x),y) -> ack2(x,succ1(00()))
     ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     filter2(f,nil0()) -> nil0()
     filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
     filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
     filter24(false0(),f,x,xs) -> filter2(f,xs)
     app(ack1(x4),x5) -> ack2(x4,x5)
     app(ack0(),x4) -> ack1(x4)
     app(succ0(),x8) -> succ1(x8)
     app(map1(x10),x11) -> map2(x10,x11)
     app(map0(),x10) -> map1(x10)
     app(cons1(x14),x15) -> cons2(x14,x15)
     app(cons0(),x14) -> cons1(x14)
     app(filter1(x17),x18) -> filter2(x17,x18)
     app(filter0(),x17) -> filter1(x17)
     app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
     app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
     app(filter21(x20),x21) -> filter22(x20,x21)
     app(filter20(),x20) -> filter21(x20)
    graph:
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true0(),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,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x20,x21,x22),x23) -> filter2{4,#}(x20,x21,x22,x23)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x17),x18) -> filter{2,#}(x17,x18)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x10),x11) -> map{2,#}(x10,x11)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(ack1(x4),x5) -> ack{2,#}(x4,x5)
     app#(filter23(x20,x21,x22),x23) -> filter2{4,#}(x20,x21,x22,x23) ->
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x20,x21,x22),x23) -> filter2{4,#}(x20,x21,x22,x23) ->
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x17),x18) -> filter{2,#}(x17,x18) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x17),x18) -> filter{2,#}(x17,x18) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x10),x11) -> map{2,#}(x10,x11) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x10),x11) -> map{2,#}(x10,x11) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(ack1(x4),x5) -> ack{2,#}(x4,x5) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y))
     app#(ack1(x4),x5) -> ack{2,#}(x4,x5) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
     app#(ack1(x4),x5) -> ack{2,#}(x4,x5) ->
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(00()))
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x20,x21,x22),x23) -> filter2{4,#}(x20,x21,x22,x23)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x17),x18) -> filter{2,#}(x17,x18)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x10),x11) -> map{2,#}(x10,x11)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(ack1(x4),x5) -> ack{2,#}(x4,x5)
     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(00()))
     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(00()))
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(00())) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(x,ack2(succ1(x),y))
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(00())) ->
     ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(00())) ->
     ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(00()))
    SCC Processor:
     #sccs: 2
     #rules: 12
     #arcs: 34/169
     DPs:
      filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x10),x11) -> map{2,#}(x10,x11)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x17),x18) -> filter{2,#}(x17,x18)
      filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
      filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
      app#(filter23(x20,x21,x22),x23) -> filter2{4,#}(x20,x21,x22,x23)
     TRS:
      ack2(00(),y) -> succ1(y)
      ack2(succ1(x),y) -> ack2(x,succ1(00()))
      ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(ack1(x4),x5) -> ack2(x4,x5)
      app(ack0(),x4) -> ack1(x4)
      app(succ0(),x8) -> succ1(x8)
      app(map1(x10),x11) -> map2(x10,x11)
      app(map0(),x10) -> map1(x10)
      app(cons1(x14),x15) -> cons2(x14,x15)
      app(cons0(),x14) -> cons1(x14)
      app(filter1(x17),x18) -> filter2(x17,x18)
      app(filter0(),x17) -> filter1(x17)
      app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
      app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
      app(filter21(x20),x21) -> filter22(x20,x21)
      app(filter20(),x20) -> filter21(x20)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 0
       pi(app#) = 0
       pi(filter{2,#}) = 0
       pi(filter2{4,#}) = 1
      problem:
       DPs:
        filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
        filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
        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)) -> filter2{4,#}(app(f,x),f,x,xs)
        filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
       TRS:
        ack2(00(),y) -> succ1(y)
        ack2(succ1(x),y) -> ack2(x,succ1(00()))
        ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(ack1(x4),x5) -> ack2(x4,x5)
        app(ack0(),x4) -> ack1(x4)
        app(succ0(),x8) -> succ1(x8)
        app(map1(x10),x11) -> map2(x10,x11)
        app(map0(),x10) -> map1(x10)
        app(cons1(x14),x15) -> cons2(x14,x15)
        app(cons0(),x14) -> cons1(x14)
        app(filter1(x17),x18) -> filter2(x17,x18)
        app(filter0(),x17) -> filter1(x17)
        app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
        app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
        app(filter21(x20),x21) -> filter22(x20,x21)
        app(filter20(),x20) -> filter21(x20)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
       TRS:
        ack2(00(),y) -> succ1(y)
        ack2(succ1(x),y) -> ack2(x,succ1(00()))
        ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(ack1(x4),x5) -> ack2(x4,x5)
        app(ack0(),x4) -> ack1(x4)
        app(succ0(),x8) -> succ1(x8)
        app(map1(x10),x11) -> map2(x10,x11)
        app(map0(),x10) -> map1(x10)
        app(cons1(x14),x15) -> cons2(x14,x15)
        app(cons0(),x14) -> cons1(x14)
        app(filter1(x17),x18) -> filter2(x17,x18)
        app(filter0(),x17) -> filter1(x17)
        app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
        app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
        app(filter21(x20),x21) -> filter22(x20,x21)
        app(filter20(),x20) -> filter21(x20)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          ack2(00(),y) -> succ1(y)
          ack2(succ1(x),y) -> ack2(x,succ1(00()))
          ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
          map2(f,nil0()) -> nil0()
          map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
          filter2(f,nil0()) -> nil0()
          filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
          filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
          filter24(false0(),f,x,xs) -> filter2(f,xs)
          app(ack1(x4),x5) -> ack2(x4,x5)
          app(ack0(),x4) -> ack1(x4)
          app(succ0(),x8) -> succ1(x8)
          app(map1(x10),x11) -> map2(x10,x11)
          app(map0(),x10) -> map1(x10)
          app(cons1(x14),x15) -> cons2(x14,x15)
          app(cons0(),x14) -> cons1(x14)
          app(filter1(x17),x18) -> filter2(x17,x18)
          app(filter0(),x17) -> filter1(x17)
          app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
          app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
          app(filter21(x20),x21) -> filter22(x20,x21)
          app(filter20(),x20) -> filter21(x20)
        Qed
       
       DPs:
        filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
        filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
        filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
       TRS:
        ack2(00(),y) -> succ1(y)
        ack2(succ1(x),y) -> ack2(x,succ1(00()))
        ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(ack1(x4),x5) -> ack2(x4,x5)
        app(ack0(),x4) -> ack1(x4)
        app(succ0(),x8) -> succ1(x8)
        app(map1(x10),x11) -> map2(x10,x11)
        app(map0(),x10) -> map1(x10)
        app(cons1(x14),x15) -> cons2(x14,x15)
        app(cons0(),x14) -> cons1(x14)
        app(filter1(x17),x18) -> filter2(x17,x18)
        app(filter0(),x17) -> filter1(x17)
        app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
        app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
        app(filter21(x20),x21) -> filter22(x20,x21)
        app(filter20(),x20) -> filter21(x20)
       Subterm Criterion Processor:
        simple projection:
         pi(filter{2,#}) = 1
         pi(filter2{4,#}) = 3
        problem:
         DPs:
          filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
          filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
         TRS:
          ack2(00(),y) -> succ1(y)
          ack2(succ1(x),y) -> ack2(x,succ1(00()))
          ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
          map2(f,nil0()) -> nil0()
          map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
          filter2(f,nil0()) -> nil0()
          filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
          filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
          filter24(false0(),f,x,xs) -> filter2(f,xs)
          app(ack1(x4),x5) -> ack2(x4,x5)
          app(ack0(),x4) -> ack1(x4)
          app(succ0(),x8) -> succ1(x8)
          app(map1(x10),x11) -> map2(x10,x11)
          app(map0(),x10) -> map1(x10)
          app(cons1(x14),x15) -> cons2(x14,x15)
          app(cons0(),x14) -> cons1(x14)
          app(filter1(x17),x18) -> filter2(x17,x18)
          app(filter0(),x17) -> filter1(x17)
          app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
          app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
          app(filter21(x20),x21) -> filter22(x20,x21)
          app(filter20(),x20) -> filter21(x20)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      ack{2,#}(succ1(x),y) -> ack{2,#}(x,succ1(00()))
      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(00(),y) -> succ1(y)
      ack2(succ1(x),y) -> ack2(x,succ1(00()))
      ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(ack1(x4),x5) -> ack2(x4,x5)
      app(ack0(),x4) -> ack1(x4)
      app(succ0(),x8) -> succ1(x8)
      app(map1(x10),x11) -> map2(x10,x11)
      app(map0(),x10) -> map1(x10)
      app(cons1(x14),x15) -> cons2(x14,x15)
      app(cons0(),x14) -> cons1(x14)
      app(filter1(x17),x18) -> filter2(x17,x18)
      app(filter0(),x17) -> filter1(x17)
      app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
      app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
      app(filter21(x20),x21) -> filter22(x20,x21)
      app(filter20(),x20) -> filter21(x20)
     Subterm Criterion Processor:
      simple projection:
       pi(ack{2,#}) = 0
      problem:
       DPs:
        ack{2,#}(succ1(x),succ1(y)) -> ack{2,#}(succ1(x),y)
       TRS:
        ack2(00(),y) -> succ1(y)
        ack2(succ1(x),y) -> ack2(x,succ1(00()))
        ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(ack1(x4),x5) -> ack2(x4,x5)
        app(ack0(),x4) -> ack1(x4)
        app(succ0(),x8) -> succ1(x8)
        app(map1(x10),x11) -> map2(x10,x11)
        app(map0(),x10) -> map1(x10)
        app(cons1(x14),x15) -> cons2(x14,x15)
        app(cons0(),x14) -> cons1(x14)
        app(filter1(x17),x18) -> filter2(x17,x18)
        app(filter0(),x17) -> filter1(x17)
        app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
        app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
        app(filter21(x20),x21) -> filter22(x20,x21)
        app(filter20(),x20) -> filter21(x20)
      Subterm Criterion Processor:
       simple projection:
        pi(ack{2,#}) = 1
       problem:
        DPs:
         
        TRS:
         ack2(00(),y) -> succ1(y)
         ack2(succ1(x),y) -> ack2(x,succ1(00()))
         ack2(succ1(x),succ1(y)) -> ack2(x,ack2(succ1(x),y))
         map2(f,nil0()) -> nil0()
         map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
         filter2(f,nil0()) -> nil0()
         filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
         filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
         filter24(false0(),f,x,xs) -> filter2(f,xs)
         app(ack1(x4),x5) -> ack2(x4,x5)
         app(ack0(),x4) -> ack1(x4)
         app(succ0(),x8) -> succ1(x8)
         app(map1(x10),x11) -> map2(x10,x11)
         app(map0(),x10) -> map1(x10)
         app(cons1(x14),x15) -> cons2(x14,x15)
         app(cons0(),x14) -> cons1(x14)
         app(filter1(x17),x18) -> filter2(x17,x18)
         app(filter0(),x17) -> filter1(x17)
         app(filter23(x20,x21,x22),x23) -> filter24(x20,x21,x22,x23)
         app(filter22(x20,x21),x22) -> filter23(x20,x21,x22)
         app(filter21(x20),x21) -> filter22(x20,x21)
         app(filter20(),x20) -> filter21(x20)
       Qed