/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(app(minus(),x),0()) -> x
 app(app(minus(),app(s(),x)),app(s(),y)) -> app(app(minus(),x),y)
 app(f(),0()) -> app(s(),0())
 app(f(),app(s(),x)) -> app(app(minus(),app(s(),x)),app(g(),app(f(),x)))
 app(g(),0()) -> 0()
 app(g(),app(s(),x)) -> app(app(minus(),app(s(),x)),app(f(),app(g(),x)))
 app(app(map(),fun),nil()) -> nil()
 app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
 app(app(filter(),fun),nil()) -> nil()
 app(app(filter(),fun),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
 app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
 app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),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
   g ==> g0/0 g1/1
   f ==> f0/0 f1/1
   s ==> s0/0 s1/1
   0 ==> 00/0
   minus ==> minus0/0 minus1/1 minus2/2
   
  uncurry-rules:
   app(minus1(x4),x5) -> minus2(x4,x5)
   app(minus0(),x4) -> minus1(x4)
   app(s0(),x8) -> s1(x8)
   app(f0(),x10) -> f1(x10)
   app(g0(),x12) -> g1(x12)
   app(map1(x14),x15) -> map2(x14,x15)
   app(map0(),x14) -> map1(x14)
   app(cons1(x18),x19) -> cons2(x18,x19)
   app(cons0(),x18) -> cons1(x18)
   app(filter1(x21),x22) -> filter2(x21,x22)
   app(filter0(),x21) -> filter1(x21)
   app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
   app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
   app(filter21(x24),x25) -> filter22(x24,x25)
   app(filter20(),x24) -> filter21(x24)
  eta-rules:
   
  problem:
   minus2(x,00()) -> x
   minus2(s1(x),s1(y)) -> minus2(x,y)
   f1(00()) -> s1(00())
   f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
   g1(00()) -> 00()
   g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
   map2(fun,nil0()) -> nil0()
   map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
   filter2(fun,nil0()) -> nil0()
   filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
   filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
   filter24(false0(),fun,x,xs) -> filter2(fun,xs)
   app(minus1(x4),x5) -> minus2(x4,x5)
   app(minus0(),x4) -> minus1(x4)
   app(s0(),x8) -> s1(x8)
   app(f0(),x10) -> f1(x10)
   app(g0(),x12) -> g1(x12)
   app(map1(x14),x15) -> map2(x14,x15)
   app(map0(),x14) -> map1(x14)
   app(cons1(x18),x19) -> cons2(x18,x19)
   app(cons0(),x18) -> cons1(x18)
   app(filter1(x21),x22) -> filter2(x21,x22)
   app(filter0(),x21) -> filter1(x21)
   app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
   app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
   app(filter21(x24),x25) -> filter22(x24,x25)
   app(filter20(),x24) -> filter21(x24)
  DP Processor:
   DPs:
    minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    f{1,#}(s1(x)) -> f{1,#}(x)
    f{1,#}(s1(x)) -> g{1,#}(f1(x))
    f{1,#}(s1(x)) -> minus{2,#}(s1(x),g1(f1(x)))
    g{1,#}(s1(x)) -> g{1,#}(x)
    g{1,#}(s1(x)) -> f{1,#}(g1(x))
    g{1,#}(s1(x)) -> minus{2,#}(s1(x),f1(g1(x)))
    map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
    map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
    filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
    filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
    filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
    filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
    app#(minus1(x4),x5) -> minus{2,#}(x4,x5)
    app#(f0(),x10) -> f{1,#}(x10)
    app#(g0(),x12) -> g{1,#}(x12)
    app#(map1(x14),x15) -> map{2,#}(x14,x15)
    app#(filter1(x21),x22) -> filter{2,#}(x21,x22)
    app#(filter23(x24,x25,x26),x27) -> filter2{4,#}(x24,x25,x26,x27)
   TRS:
    minus2(x,00()) -> x
    minus2(s1(x),s1(y)) -> minus2(x,y)
    f1(00()) -> s1(00())
    f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
    g1(00()) -> 00()
    g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
    map2(fun,nil0()) -> nil0()
    map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
    filter2(fun,nil0()) -> nil0()
    filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
    filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
    filter24(false0(),fun,x,xs) -> filter2(fun,xs)
    app(minus1(x4),x5) -> minus2(x4,x5)
    app(minus0(),x4) -> minus1(x4)
    app(s0(),x8) -> s1(x8)
    app(f0(),x10) -> f1(x10)
    app(g0(),x12) -> g1(x12)
    app(map1(x14),x15) -> map2(x14,x15)
    app(map0(),x14) -> map1(x14)
    app(cons1(x18),x19) -> cons2(x18,x19)
    app(cons0(),x18) -> cons1(x18)
    app(filter1(x21),x22) -> filter2(x21,x22)
    app(filter0(),x21) -> filter1(x21)
    app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
    app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
    app(filter21(x24),x25) -> filter22(x24,x25)
    app(filter20(),x24) -> filter21(x24)
   TDG Processor:
    DPs:
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     f{1,#}(s1(x)) -> f{1,#}(x)
     f{1,#}(s1(x)) -> g{1,#}(f1(x))
     f{1,#}(s1(x)) -> minus{2,#}(s1(x),g1(f1(x)))
     g{1,#}(s1(x)) -> g{1,#}(x)
     g{1,#}(s1(x)) -> f{1,#}(g1(x))
     g{1,#}(s1(x)) -> minus{2,#}(s1(x),f1(g1(x)))
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(minus1(x4),x5) -> minus{2,#}(x4,x5)
     app#(f0(),x10) -> f{1,#}(x10)
     app#(g0(),x12) -> g{1,#}(x12)
     app#(map1(x14),x15) -> map{2,#}(x14,x15)
     app#(filter1(x21),x22) -> filter{2,#}(x21,x22)
     app#(filter23(x24,x25,x26),x27) -> filter2{4,#}(x24,x25,x26,x27)
    TRS:
     minus2(x,00()) -> x
     minus2(s1(x),s1(y)) -> minus2(x,y)
     f1(00()) -> s1(00())
     f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
     g1(00()) -> 00()
     g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
     map2(fun,nil0()) -> nil0()
     map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
     filter2(fun,nil0()) -> nil0()
     filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
     filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
     filter24(false0(),fun,x,xs) -> filter2(fun,xs)
     app(minus1(x4),x5) -> minus2(x4,x5)
     app(minus0(),x4) -> minus1(x4)
     app(s0(),x8) -> s1(x8)
     app(f0(),x10) -> f1(x10)
     app(g0(),x12) -> g1(x12)
     app(map1(x14),x15) -> map2(x14,x15)
     app(map0(),x14) -> map1(x14)
     app(cons1(x18),x19) -> cons2(x18,x19)
     app(cons0(),x18) -> cons1(x18)
     app(filter1(x21),x22) -> filter2(x21,x22)
     app(filter0(),x21) -> filter1(x21)
     app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
     app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
     app(filter21(x24),x25) -> filter22(x24,x25)
     app(filter20(),x24) -> filter21(x24)
    graph:
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x24,x25,x26),x27) -> filter2{4,#}(x24,x25,x26,x27)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x21),x22) -> filter{2,#}(x21,x22)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x14),x15) -> map{2,#}(x14,x15)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(g0(),x12) -> g{1,#}(x12)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f0(),x10) -> f{1,#}(x10)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(minus1(x4),x5) -> minus{2,#}(x4,x5)
     app#(filter23(x24,x25,x26),x27) -> filter2{4,#}(x24,x25,x26,x27) ->
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter23(x24,x25,x26),x27) -> filter2{4,#}(x24,x25,x26,x27) ->
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter1(x21),x22) -> filter{2,#}(x21,x22) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     app#(filter1(x21),x22) -> filter{2,#}(x21,x22) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x14),x15) -> map{2,#}(x14,x15) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x14),x15) -> map{2,#}(x14,x15) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     app#(g0(),x12) -> g{1,#}(x12) ->
     g{1,#}(s1(x)) -> minus{2,#}(s1(x),f1(g1(x)))
     app#(g0(),x12) -> g{1,#}(x12) -> g{1,#}(s1(x)) -> f{1,#}(g1(x))
     app#(g0(),x12) -> g{1,#}(x12) -> g{1,#}(s1(x)) -> g{1,#}(x)
     app#(f0(),x10) -> f{1,#}(x10) ->
     f{1,#}(s1(x)) -> minus{2,#}(s1(x),g1(f1(x)))
     app#(f0(),x10) -> f{1,#}(x10) -> f{1,#}(s1(x)) -> g{1,#}(f1(x))
     app#(f0(),x10) -> f{1,#}(x10) ->
     f{1,#}(s1(x)) -> f{1,#}(x)
     app#(minus1(x4),x5) -> minus{2,#}(x4,x5) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x24,x25,x26),x27) -> filter2{4,#}(x24,x25,x26,x27)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x21),x22) -> filter{2,#}(x21,x22)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x14),x15) -> map{2,#}(x14,x15)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(g0(),x12) -> g{1,#}(x12)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f0(),x10) -> f{1,#}(x10)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(minus1(x4),x5) -> minus{2,#}(x4,x5)
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     g{1,#}(s1(x)) -> g{1,#}(x) ->
     g{1,#}(s1(x)) -> minus{2,#}(s1(x),f1(g1(x)))
     g{1,#}(s1(x)) -> g{1,#}(x) -> g{1,#}(s1(x)) -> f{1,#}(g1(x))
     g{1,#}(s1(x)) -> g{1,#}(x) -> g{1,#}(s1(x)) -> g{1,#}(x)
     g{1,#}(s1(x)) -> f{1,#}(g1(x)) ->
     f{1,#}(s1(x)) -> minus{2,#}(s1(x),g1(f1(x)))
     g{1,#}(s1(x)) -> f{1,#}(g1(x)) -> f{1,#}(s1(x)) -> g{1,#}(f1(x))
     g{1,#}(s1(x)) -> f{1,#}(g1(x)) ->
     f{1,#}(s1(x)) -> f{1,#}(x)
     g{1,#}(s1(x)) -> minus{2,#}(s1(x),f1(g1(x))) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     f{1,#}(s1(x)) -> g{1,#}(f1(x)) ->
     g{1,#}(s1(x)) -> minus{2,#}(s1(x),f1(g1(x)))
     f{1,#}(s1(x)) -> g{1,#}(f1(x)) -> g{1,#}(s1(x)) -> f{1,#}(g1(x))
     f{1,#}(s1(x)) -> g{1,#}(f1(x)) -> g{1,#}(s1(x)) -> g{1,#}(x)
     f{1,#}(s1(x)) -> f{1,#}(x) ->
     f{1,#}(s1(x)) -> minus{2,#}(s1(x),g1(f1(x)))
     f{1,#}(s1(x)) -> f{1,#}(x) -> f{1,#}(s1(x)) -> g{1,#}(f1(x))
     f{1,#}(s1(x)) -> f{1,#}(x) ->
     f{1,#}(s1(x)) -> f{1,#}(x)
     f{1,#}(s1(x)) -> minus{2,#}(s1(x),g1(f1(x))) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y) -> minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    SCC Processor:
     #sccs: 3
     #rules: 14
     #arcs: 48/361
     DPs:
      filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
      filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(map1(x14),x15) -> map{2,#}(x14,x15)
      map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
      map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(filter1(x21),x22) -> filter{2,#}(x21,x22)
      filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
      filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
      app#(filter23(x24,x25,x26),x27) -> filter2{4,#}(x24,x25,x26,x27)
     TRS:
      minus2(x,00()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      f1(00()) -> s1(00())
      f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
      g1(00()) -> 00()
      g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
      map2(fun,nil0()) -> nil0()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil0()) -> nil0()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false0(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x4),x5) -> minus2(x4,x5)
      app(minus0(),x4) -> minus1(x4)
      app(s0(),x8) -> s1(x8)
      app(f0(),x10) -> f1(x10)
      app(g0(),x12) -> g1(x12)
      app(map1(x14),x15) -> map2(x14,x15)
      app(map0(),x14) -> map1(x14)
      app(cons1(x18),x19) -> cons2(x18,x19)
      app(cons0(),x18) -> cons1(x18)
      app(filter1(x21),x22) -> filter2(x21,x22)
      app(filter0(),x21) -> filter1(x21)
      app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
      app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
      app(filter21(x24),x25) -> filter22(x24,x25)
      app(filter20(),x24) -> filter21(x24)
     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(),fun,x,xs) -> filter{2,#}(fun,xs)
        filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
        map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
        map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
        filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
        filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
       TRS:
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        f1(00()) -> s1(00())
        f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
        g1(00()) -> 00()
        g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x4),x5) -> minus2(x4,x5)
        app(minus0(),x4) -> minus1(x4)
        app(s0(),x8) -> s1(x8)
        app(f0(),x10) -> f1(x10)
        app(g0(),x12) -> g1(x12)
        app(map1(x14),x15) -> map2(x14,x15)
        app(map0(),x14) -> map1(x14)
        app(cons1(x18),x19) -> cons2(x18,x19)
        app(cons0(),x18) -> cons1(x18)
        app(filter1(x21),x22) -> filter2(x21,x22)
        app(filter0(),x21) -> filter1(x21)
        app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
        app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
        app(filter21(x24),x25) -> filter22(x24,x25)
        app(filter20(),x24) -> filter21(x24)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
       TRS:
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        f1(00()) -> s1(00())
        f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
        g1(00()) -> 00()
        g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x4),x5) -> minus2(x4,x5)
        app(minus0(),x4) -> minus1(x4)
        app(s0(),x8) -> s1(x8)
        app(f0(),x10) -> f1(x10)
        app(g0(),x12) -> g1(x12)
        app(map1(x14),x15) -> map2(x14,x15)
        app(map0(),x14) -> map1(x14)
        app(cons1(x18),x19) -> cons2(x18,x19)
        app(cons0(),x18) -> cons1(x18)
        app(filter1(x21),x22) -> filter2(x21,x22)
        app(filter0(),x21) -> filter1(x21)
        app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
        app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
        app(filter21(x24),x25) -> filter22(x24,x25)
        app(filter20(),x24) -> filter21(x24)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          minus2(x,00()) -> x
          minus2(s1(x),s1(y)) -> minus2(x,y)
          f1(00()) -> s1(00())
          f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
          g1(00()) -> 00()
          g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
          map2(fun,nil0()) -> nil0()
          map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
          filter2(fun,nil0()) -> nil0()
          filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
          filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
          filter24(false0(),fun,x,xs) -> filter2(fun,xs)
          app(minus1(x4),x5) -> minus2(x4,x5)
          app(minus0(),x4) -> minus1(x4)
          app(s0(),x8) -> s1(x8)
          app(f0(),x10) -> f1(x10)
          app(g0(),x12) -> g1(x12)
          app(map1(x14),x15) -> map2(x14,x15)
          app(map0(),x14) -> map1(x14)
          app(cons1(x18),x19) -> cons2(x18,x19)
          app(cons0(),x18) -> cons1(x18)
          app(filter1(x21),x22) -> filter2(x21,x22)
          app(filter0(),x21) -> filter1(x21)
          app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
          app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
          app(filter21(x24),x25) -> filter22(x24,x25)
          app(filter20(),x24) -> filter21(x24)
        Qed
       
       DPs:
        filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
        filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
        filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
       TRS:
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        f1(00()) -> s1(00())
        f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
        g1(00()) -> 00()
        g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x4),x5) -> minus2(x4,x5)
        app(minus0(),x4) -> minus1(x4)
        app(s0(),x8) -> s1(x8)
        app(f0(),x10) -> f1(x10)
        app(g0(),x12) -> g1(x12)
        app(map1(x14),x15) -> map2(x14,x15)
        app(map0(),x14) -> map1(x14)
        app(cons1(x18),x19) -> cons2(x18,x19)
        app(cons0(),x18) -> cons1(x18)
        app(filter1(x21),x22) -> filter2(x21,x22)
        app(filter0(),x21) -> filter1(x21)
        app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
        app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
        app(filter21(x24),x25) -> filter22(x24,x25)
        app(filter20(),x24) -> filter21(x24)
       Subterm Criterion Processor:
        simple projection:
         pi(filter{2,#}) = 1
         pi(filter2{4,#}) = 3
        problem:
         DPs:
          filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
          filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
         TRS:
          minus2(x,00()) -> x
          minus2(s1(x),s1(y)) -> minus2(x,y)
          f1(00()) -> s1(00())
          f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
          g1(00()) -> 00()
          g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
          map2(fun,nil0()) -> nil0()
          map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
          filter2(fun,nil0()) -> nil0()
          filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
          filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
          filter24(false0(),fun,x,xs) -> filter2(fun,xs)
          app(minus1(x4),x5) -> minus2(x4,x5)
          app(minus0(),x4) -> minus1(x4)
          app(s0(),x8) -> s1(x8)
          app(f0(),x10) -> f1(x10)
          app(g0(),x12) -> g1(x12)
          app(map1(x14),x15) -> map2(x14,x15)
          app(map0(),x14) -> map1(x14)
          app(cons1(x18),x19) -> cons2(x18,x19)
          app(cons0(),x18) -> cons1(x18)
          app(filter1(x21),x22) -> filter2(x21,x22)
          app(filter0(),x21) -> filter1(x21)
          app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
          app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
          app(filter21(x24),x25) -> filter22(x24,x25)
          app(filter20(),x24) -> filter21(x24)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      f{1,#}(s1(x)) -> f{1,#}(x)
      f{1,#}(s1(x)) -> g{1,#}(f1(x))
      g{1,#}(s1(x)) -> g{1,#}(x)
      g{1,#}(s1(x)) -> f{1,#}(g1(x))
     TRS:
      minus2(x,00()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      f1(00()) -> s1(00())
      f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
      g1(00()) -> 00()
      g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
      map2(fun,nil0()) -> nil0()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil0()) -> nil0()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false0(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x4),x5) -> minus2(x4,x5)
      app(minus0(),x4) -> minus1(x4)
      app(s0(),x8) -> s1(x8)
      app(f0(),x10) -> f1(x10)
      app(g0(),x12) -> g1(x12)
      app(map1(x14),x15) -> map2(x14,x15)
      app(map0(),x14) -> map1(x14)
      app(cons1(x18),x19) -> cons2(x18,x19)
      app(cons0(),x18) -> cons1(x18)
      app(filter1(x21),x22) -> filter2(x21,x22)
      app(filter0(),x21) -> filter1(x21)
      app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
      app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
      app(filter21(x24),x25) -> filter22(x24,x25)
      app(filter20(),x24) -> filter21(x24)
     Subterm Criterion Processor:
      simple projection:
       pi(minus2) = [0,0]
       pi(s1) = [0,0]
       pi(f1) = [0,0]
       pi(g1) = [0,0]
       pi(cons2) = [0,0,0,1,1,1]
       pi(f{1,#}) = [0,0]
       pi(g{1,#}) = [0,0]
      problem:
       DPs:
        f{1,#}(s1(x)) -> g{1,#}(f1(x))
        g{1,#}(s1(x)) -> f{1,#}(g1(x))
       TRS:
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        f1(00()) -> s1(00())
        f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
        g1(00()) -> 00()
        g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x4),x5) -> minus2(x4,x5)
        app(minus0(),x4) -> minus1(x4)
        app(s0(),x8) -> s1(x8)
        app(f0(),x10) -> f1(x10)
        app(g0(),x12) -> g1(x12)
        app(map1(x14),x15) -> map2(x14,x15)
        app(map0(),x14) -> map1(x14)
        app(cons1(x18),x19) -> cons2(x18,x19)
        app(cons0(),x18) -> cons1(x18)
        app(filter1(x21),x22) -> filter2(x21,x22)
        app(filter0(),x21) -> filter1(x21)
        app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
        app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
        app(filter21(x24),x25) -> filter22(x24,x25)
        app(filter20(),x24) -> filter21(x24)
      Subterm Criterion Processor:
       simple projection:
        pi(minus1) = [0,0]
        pi(minus2) = 0
        pi(s1) = [0,0]
        pi(f1) = [0,0]
        pi(g1) = 0
        pi(cons1) = [0,0]
        pi(cons2) = [0,0,0,1,1,1]
        pi(filter1) = [0,0]
        pi(filter21) = [0,0]
        pi(filter22) = [1,1]
        pi(f{1,#}) = [0,0]
        pi(g{1,#}) = [0,0]
       problem:
        DPs:
         f{1,#}(s1(x)) -> g{1,#}(f1(x))
        TRS:
         minus2(x,00()) -> x
         minus2(s1(x),s1(y)) -> minus2(x,y)
         f1(00()) -> s1(00())
         f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
         g1(00()) -> 00()
         g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
         map2(fun,nil0()) -> nil0()
         map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
         filter2(fun,nil0()) -> nil0()
         filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
         filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
         filter24(false0(),fun,x,xs) -> filter2(fun,xs)
         app(minus1(x4),x5) -> minus2(x4,x5)
         app(minus0(),x4) -> minus1(x4)
         app(s0(),x8) -> s1(x8)
         app(f0(),x10) -> f1(x10)
         app(g0(),x12) -> g1(x12)
         app(map1(x14),x15) -> map2(x14,x15)
         app(map0(),x14) -> map1(x14)
         app(cons1(x18),x19) -> cons2(x18,x19)
         app(cons0(),x18) -> cons1(x18)
         app(filter1(x21),x22) -> filter2(x21,x22)
         app(filter0(),x21) -> filter1(x21)
         app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
         app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
         app(filter21(x24),x25) -> filter22(x24,x25)
         app(filter20(),x24) -> filter21(x24)
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 8/1
        
     
     DPs:
      minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     TRS:
      minus2(x,00()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      f1(00()) -> s1(00())
      f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
      g1(00()) -> 00()
      g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
      map2(fun,nil0()) -> nil0()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil0()) -> nil0()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false0(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x4),x5) -> minus2(x4,x5)
      app(minus0(),x4) -> minus1(x4)
      app(s0(),x8) -> s1(x8)
      app(f0(),x10) -> f1(x10)
      app(g0(),x12) -> g1(x12)
      app(map1(x14),x15) -> map2(x14,x15)
      app(map0(),x14) -> map1(x14)
      app(cons1(x18),x19) -> cons2(x18,x19)
      app(cons0(),x18) -> cons1(x18)
      app(filter1(x21),x22) -> filter2(x21,x22)
      app(filter0(),x21) -> filter1(x21)
      app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
      app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
      app(filter21(x24),x25) -> filter22(x24,x25)
      app(filter20(),x24) -> filter21(x24)
     Subterm Criterion Processor:
      simple projection:
       pi(minus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        f1(00()) -> s1(00())
        f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
        g1(00()) -> 00()
        g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x4),x5) -> minus2(x4,x5)
        app(minus0(),x4) -> minus1(x4)
        app(s0(),x8) -> s1(x8)
        app(f0(),x10) -> f1(x10)
        app(g0(),x12) -> g1(x12)
        app(map1(x14),x15) -> map2(x14,x15)
        app(map0(),x14) -> map1(x14)
        app(cons1(x18),x19) -> cons2(x18,x19)
        app(cons0(),x18) -> cons1(x18)
        app(filter1(x21),x22) -> filter2(x21,x22)
        app(filter0(),x21) -> filter1(x21)
        app(filter23(x24,x25,x26),x27) -> filter24(x24,x25,x26,x27)
        app(filter22(x24,x25),x26) -> filter23(x24,x25,x26)
        app(filter21(x24),x25) -> filter22(x24,x25)
        app(filter20(),x24) -> filter21(x24)
      Qed