/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem:
 app(pred(),app(s(),x)) -> x
 app(app(minus(),x),0()) -> x
 app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
 app(app(quot(),0()),app(s(),y)) -> 0()
 app(app(quot(),app(s(),x)),app(s(),y)) ->
 app(s(),app(app(quot(),app(app(minus(),x),y)),app(s(),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
   quot ==> quot0/0 quot1/1 quot2/2
   0 ==> 00/0
   minus ==> minus0/0 minus1/1 minus2/2
   s ==> s0/0 s1/1
   pred ==> pred0/0 pred1/1
   
  uncurry-rules:
   app(pred0(),x4) -> pred1(x4)
   app(s0(),x6) -> s1(x6)
   app(minus1(x8),x9) -> minus2(x8,x9)
   app(minus0(),x8) -> minus1(x8)
   app(quot1(x12),x13) -> quot2(x12,x13)
   app(quot0(),x12) -> quot1(x12)
   app(map1(x15),x16) -> map2(x15,x16)
   app(map0(),x15) -> map1(x15)
   app(cons1(x19),x20) -> cons2(x19,x20)
   app(cons0(),x19) -> cons1(x19)
   app(filter1(x22),x23) -> filter2(x22,x23)
   app(filter0(),x22) -> filter1(x22)
   app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
   app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
   app(filter21(x25),x26) -> filter22(x25,x26)
   app(filter20(),x25) -> filter21(x25)
  eta-rules:
   
  problem:
   pred1(s1(x)) -> x
   minus2(x,00()) -> x
   minus2(x,s1(y)) -> pred1(minus2(x,y))
   quot2(00(),s1(y)) -> 00()
   quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
   app(s0(),x6) -> s1(x6)
   app(minus1(x8),x9) -> minus2(x8,x9)
   app(minus0(),x8) -> minus1(x8)
   app(quot1(x12),x13) -> quot2(x12,x13)
   app(quot0(),x12) -> quot1(x12)
   app(map1(x15),x16) -> map2(x15,x16)
   app(map0(),x15) -> map1(x15)
   app(cons1(x19),x20) -> cons2(x19,x20)
   app(cons0(),x19) -> cons1(x19)
   app(filter1(x22),x23) -> filter2(x22,x23)
   app(filter0(),x22) -> filter1(x22)
   app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
   app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
   app(filter21(x25),x26) -> filter22(x25,x26)
   app(filter20(),x25) -> filter21(x25)
  DP Processor:
   DPs:
    minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
    minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
    quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(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#(pred0(),x4) -> pred{1,#}(x4)
    app#(minus1(x8),x9) -> minus{2,#}(x8,x9)
    app#(quot1(x12),x13) -> quot{2,#}(x12,x13)
    app#(map1(x15),x16) -> map{2,#}(x15,x16)
    app#(filter1(x22),x23) -> filter{2,#}(x22,x23)
    app#(filter23(x25,x26,x27),x28) -> filter2{4,#}(x25,x26,x27,x28)
   TRS:
    pred1(s1(x)) -> x
    minus2(x,00()) -> x
    minus2(x,s1(y)) -> pred1(minus2(x,y))
    quot2(00(),s1(y)) -> 00()
    quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
    app(s0(),x6) -> s1(x6)
    app(minus1(x8),x9) -> minus2(x8,x9)
    app(minus0(),x8) -> minus1(x8)
    app(quot1(x12),x13) -> quot2(x12,x13)
    app(quot0(),x12) -> quot1(x12)
    app(map1(x15),x16) -> map2(x15,x16)
    app(map0(),x15) -> map1(x15)
    app(cons1(x19),x20) -> cons2(x19,x20)
    app(cons0(),x19) -> cons1(x19)
    app(filter1(x22),x23) -> filter2(x22,x23)
    app(filter0(),x22) -> filter1(x22)
    app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
    app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
    app(filter21(x25),x26) -> filter22(x25,x26)
    app(filter20(),x25) -> filter21(x25)
   TDG Processor:
    DPs:
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(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#(pred0(),x4) -> pred{1,#}(x4)
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9)
     app#(quot1(x12),x13) -> quot{2,#}(x12,x13)
     app#(map1(x15),x16) -> map{2,#}(x15,x16)
     app#(filter1(x22),x23) -> filter{2,#}(x22,x23)
     app#(filter23(x25,x26,x27),x28) -> filter2{4,#}(x25,x26,x27,x28)
    TRS:
     pred1(s1(x)) -> x
     minus2(x,00()) -> x
     minus2(x,s1(y)) -> pred1(minus2(x,y))
     quot2(00(),s1(y)) -> 00()
     quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
     app(s0(),x6) -> s1(x6)
     app(minus1(x8),x9) -> minus2(x8,x9)
     app(minus0(),x8) -> minus1(x8)
     app(quot1(x12),x13) -> quot2(x12,x13)
     app(quot0(),x12) -> quot1(x12)
     app(map1(x15),x16) -> map2(x15,x16)
     app(map0(),x15) -> map1(x15)
     app(cons1(x19),x20) -> cons2(x19,x20)
     app(cons0(),x19) -> cons1(x19)
     app(filter1(x22),x23) -> filter2(x22,x23)
     app(filter0(),x22) -> filter1(x22)
     app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
     app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
     app(filter21(x25),x26) -> filter22(x25,x26)
     app(filter20(),x25) -> filter21(x25)
    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(x25,x26,x27),x28) -> filter2{4,#}(x25,x26,x27,x28)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x22),x23) -> filter{2,#}(x22,x23)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x15),x16) -> map{2,#}(x15,x16)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(quot1(x12),x13) -> quot{2,#}(x12,x13)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(pred0(),x4) -> pred{1,#}(x4)
     app#(filter23(x25,x26,x27),x28) -> filter2{4,#}(x25,x26,x27,x28) ->
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x25,x26,x27),x28) -> filter2{4,#}(x25,x26,x27,x28) ->
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x22),x23) -> filter{2,#}(x22,x23) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x22),x23) -> filter{2,#}(x22,x23) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x15),x16) -> map{2,#}(x15,x16) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x15),x16) -> map{2,#}(x15,x16) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(quot1(x12),x13) -> quot{2,#}(x12,x13) ->
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
     app#(quot1(x12),x13) -> quot{2,#}(x12,x13) ->
     quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9) ->
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9) ->
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x25,x26,x27),x28) -> filter2{4,#}(x25,x26,x27,x28)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x22),x23) -> filter{2,#}(x22,x23)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x15),x16) -> map{2,#}(x15,x16)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(quot1(x12),x13) -> quot{2,#}(x12,x13)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(pred0(),x4) -> pred{1,#}(x4)
     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)
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y)) ->
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y)) ->
     quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y) -> minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
    SCC Processor:
     #sccs: 3
     #rules: 11
     #arcs: 36/256
     DPs:
      filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x15),x16) -> map{2,#}(x15,x16)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x22),x23) -> filter{2,#}(x22,x23)
      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(x25,x26,x27),x28) -> filter2{4,#}(x25,x26,x27,x28)
     TRS:
      pred1(s1(x)) -> x
      minus2(x,00()) -> x
      minus2(x,s1(y)) -> pred1(minus2(x,y))
      quot2(00(),s1(y)) -> 00()
      quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
      app(s0(),x6) -> s1(x6)
      app(minus1(x8),x9) -> minus2(x8,x9)
      app(minus0(),x8) -> minus1(x8)
      app(quot1(x12),x13) -> quot2(x12,x13)
      app(quot0(),x12) -> quot1(x12)
      app(map1(x15),x16) -> map2(x15,x16)
      app(map0(),x15) -> map1(x15)
      app(cons1(x19),x20) -> cons2(x19,x20)
      app(cons0(),x19) -> cons1(x19)
      app(filter1(x22),x23) -> filter2(x22,x23)
      app(filter0(),x22) -> filter1(x22)
      app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
      app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
      app(filter21(x25),x26) -> filter22(x25,x26)
      app(filter20(),x25) -> filter21(x25)
     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:
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        quot2(00(),s1(y)) -> 00()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
        app(s0(),x6) -> s1(x6)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus0(),x8) -> minus1(x8)
        app(quot1(x12),x13) -> quot2(x12,x13)
        app(quot0(),x12) -> quot1(x12)
        app(map1(x15),x16) -> map2(x15,x16)
        app(map0(),x15) -> map1(x15)
        app(cons1(x19),x20) -> cons2(x19,x20)
        app(cons0(),x19) -> cons1(x19)
        app(filter1(x22),x23) -> filter2(x22,x23)
        app(filter0(),x22) -> filter1(x22)
        app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
        app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
        app(filter21(x25),x26) -> filter22(x25,x26)
        app(filter20(),x25) -> filter21(x25)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
       TRS:
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        quot2(00(),s1(y)) -> 00()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
        app(s0(),x6) -> s1(x6)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus0(),x8) -> minus1(x8)
        app(quot1(x12),x13) -> quot2(x12,x13)
        app(quot0(),x12) -> quot1(x12)
        app(map1(x15),x16) -> map2(x15,x16)
        app(map0(),x15) -> map1(x15)
        app(cons1(x19),x20) -> cons2(x19,x20)
        app(cons0(),x19) -> cons1(x19)
        app(filter1(x22),x23) -> filter2(x22,x23)
        app(filter0(),x22) -> filter1(x22)
        app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
        app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
        app(filter21(x25),x26) -> filter22(x25,x26)
        app(filter20(),x25) -> filter21(x25)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          pred1(s1(x)) -> x
          minus2(x,00()) -> x
          minus2(x,s1(y)) -> pred1(minus2(x,y))
          quot2(00(),s1(y)) -> 00()
          quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
          app(s0(),x6) -> s1(x6)
          app(minus1(x8),x9) -> minus2(x8,x9)
          app(minus0(),x8) -> minus1(x8)
          app(quot1(x12),x13) -> quot2(x12,x13)
          app(quot0(),x12) -> quot1(x12)
          app(map1(x15),x16) -> map2(x15,x16)
          app(map0(),x15) -> map1(x15)
          app(cons1(x19),x20) -> cons2(x19,x20)
          app(cons0(),x19) -> cons1(x19)
          app(filter1(x22),x23) -> filter2(x22,x23)
          app(filter0(),x22) -> filter1(x22)
          app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
          app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
          app(filter21(x25),x26) -> filter22(x25,x26)
          app(filter20(),x25) -> filter21(x25)
        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:
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        quot2(00(),s1(y)) -> 00()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
        app(s0(),x6) -> s1(x6)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus0(),x8) -> minus1(x8)
        app(quot1(x12),x13) -> quot2(x12,x13)
        app(quot0(),x12) -> quot1(x12)
        app(map1(x15),x16) -> map2(x15,x16)
        app(map0(),x15) -> map1(x15)
        app(cons1(x19),x20) -> cons2(x19,x20)
        app(cons0(),x19) -> cons1(x19)
        app(filter1(x22),x23) -> filter2(x22,x23)
        app(filter0(),x22) -> filter1(x22)
        app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
        app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
        app(filter21(x25),x26) -> filter22(x25,x26)
        app(filter20(),x25) -> filter21(x25)
       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:
          pred1(s1(x)) -> x
          minus2(x,00()) -> x
          minus2(x,s1(y)) -> pred1(minus2(x,y))
          quot2(00(),s1(y)) -> 00()
          quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
          app(s0(),x6) -> s1(x6)
          app(minus1(x8),x9) -> minus2(x8,x9)
          app(minus0(),x8) -> minus1(x8)
          app(quot1(x12),x13) -> quot2(x12,x13)
          app(quot0(),x12) -> quot1(x12)
          app(map1(x15),x16) -> map2(x15,x16)
          app(map0(),x15) -> map1(x15)
          app(cons1(x19),x20) -> cons2(x19,x20)
          app(cons0(),x19) -> cons1(x19)
          app(filter1(x22),x23) -> filter2(x22,x23)
          app(filter0(),x22) -> filter1(x22)
          app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
          app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
          app(filter21(x25),x26) -> filter22(x25,x26)
          app(filter20(),x25) -> filter21(x25)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
     TRS:
      pred1(s1(x)) -> x
      minus2(x,00()) -> x
      minus2(x,s1(y)) -> pred1(minus2(x,y))
      quot2(00(),s1(y)) -> 00()
      quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
      app(s0(),x6) -> s1(x6)
      app(minus1(x8),x9) -> minus2(x8,x9)
      app(minus0(),x8) -> minus1(x8)
      app(quot1(x12),x13) -> quot2(x12,x13)
      app(quot0(),x12) -> quot1(x12)
      app(map1(x15),x16) -> map2(x15,x16)
      app(map0(),x15) -> map1(x15)
      app(cons1(x19),x20) -> cons2(x19,x20)
      app(cons0(),x19) -> cons1(x19)
      app(filter1(x22),x23) -> filter2(x22,x23)
      app(filter0(),x22) -> filter1(x22)
      app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
      app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
      app(filter21(x25),x26) -> filter22(x25,x26)
      app(filter20(),x25) -> filter21(x25)
     Subterm Criterion Processor:
      simple projection:
       pi(pred1) = 0
       pi(minus2) = 0
       pi(quot{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        quot2(00(),s1(y)) -> 00()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
        app(s0(),x6) -> s1(x6)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus0(),x8) -> minus1(x8)
        app(quot1(x12),x13) -> quot2(x12,x13)
        app(quot0(),x12) -> quot1(x12)
        app(map1(x15),x16) -> map2(x15,x16)
        app(map0(),x15) -> map1(x15)
        app(cons1(x19),x20) -> cons2(x19,x20)
        app(cons0(),x19) -> cons1(x19)
        app(filter1(x22),x23) -> filter2(x22,x23)
        app(filter0(),x22) -> filter1(x22)
        app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
        app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
        app(filter21(x25),x26) -> filter22(x25,x26)
        app(filter20(),x25) -> filter21(x25)
      Qed
     
     DPs:
      minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     TRS:
      pred1(s1(x)) -> x
      minus2(x,00()) -> x
      minus2(x,s1(y)) -> pred1(minus2(x,y))
      quot2(00(),s1(y)) -> 00()
      quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
      app(s0(),x6) -> s1(x6)
      app(minus1(x8),x9) -> minus2(x8,x9)
      app(minus0(),x8) -> minus1(x8)
      app(quot1(x12),x13) -> quot2(x12,x13)
      app(quot0(),x12) -> quot1(x12)
      app(map1(x15),x16) -> map2(x15,x16)
      app(map0(),x15) -> map1(x15)
      app(cons1(x19),x20) -> cons2(x19,x20)
      app(cons0(),x19) -> cons1(x19)
      app(filter1(x22),x23) -> filter2(x22,x23)
      app(filter0(),x22) -> filter1(x22)
      app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
      app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
      app(filter21(x25),x26) -> filter22(x25,x26)
      app(filter20(),x25) -> filter21(x25)
     Subterm Criterion Processor:
      simple projection:
       pi(minus{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        quot2(00(),s1(y)) -> 00()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(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(pred0(),x4) -> pred1(x4)
        app(s0(),x6) -> s1(x6)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus0(),x8) -> minus1(x8)
        app(quot1(x12),x13) -> quot2(x12,x13)
        app(quot0(),x12) -> quot1(x12)
        app(map1(x15),x16) -> map2(x15,x16)
        app(map0(),x15) -> map1(x15)
        app(cons1(x19),x20) -> cons2(x19,x20)
        app(cons0(),x19) -> cons1(x19)
        app(filter1(x22),x23) -> filter2(x22,x23)
        app(filter0(),x22) -> filter1(x22)
        app(filter23(x25,x26,x27),x28) -> filter24(x25,x26,x27,x28)
        app(filter22(x25,x26),x27) -> filter23(x25,x26,x27)
        app(filter21(x25),x26) -> filter22(x25,x26)
        app(filter20(),x25) -> filter21(x25)
      Qed