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:
 Uncurry Processor:
  minus2(x,0()) -> x
  minus2(s1(x),s1(y)) -> minus2(x,y)
  f1(0()) -> s1(0())
  f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
  g1(0()) -> 0()
  g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
  map2(fun,nil()) -> nil()
  map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
  filter2(fun,nil()) -> nil()
  filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
  filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
  filter24(false(),fun,x,xs) -> filter2(fun,xs)
  app(minus1(x6),x7) -> minus2(x6,x7)
  app(minus(),x7) -> minus1(x7)
  app(s(),x7) -> s1(x7)
  app(f(),x7) -> f1(x7)
  app(g(),x7) -> g1(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:
    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,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
    filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
    app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
    app#(f(),x7) -> f{1,#}(x7)
    app#(g(),x7) -> g{1,#}(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:
    minus2(x,0()) -> x
    minus2(s1(x),s1(y)) -> minus2(x,y)
    f1(0()) -> s1(0())
    f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
    g1(0()) -> 0()
    g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
    map2(fun,nil()) -> nil()
    map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
    filter2(fun,nil()) -> nil()
    filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
    filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
    filter24(false(),fun,x,xs) -> filter2(fun,xs)
    app(minus1(x6),x7) -> minus2(x6,x7)
    app(minus(),x7) -> minus1(x7)
    app(s(),x7) -> s1(x7)
    app(f(),x7) -> f1(x7)
    app(g(),x7) -> g1(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:
     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,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     app#(f(),x7) -> f{1,#}(x7)
     app#(g(),x7) -> g{1,#}(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:
     minus2(x,0()) -> x
     minus2(s1(x),s1(y)) -> minus2(x,y)
     f1(0()) -> s1(0())
     f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
     g1(0()) -> 0()
     g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
     map2(fun,nil()) -> nil()
     map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
     filter2(fun,nil()) -> nil()
     filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
     filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
     filter24(false(),fun,x,xs) -> filter2(fun,xs)
     app(minus1(x6),x7) -> minus2(x6,x7)
     app(minus(),x7) -> minus1(x7)
     app(s(),x7) -> s1(x7)
     app(f(),x7) -> f1(x7)
     app(g(),x7) -> g1(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(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true(),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,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x6),x7) -> map{2,#}(x6,x7)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(g(),x7) -> g{1,#}(x7)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f(),x7) -> f{1,#}(x7)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7) ->
     filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7) ->
     filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x6),x7) -> map{2,#}(x6,x7) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x6),x7) -> map{2,#}(x6,x7) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(g(),x7) -> g{1,#}(x7) ->
     g{1,#}(s1(x)) -> minus{2,#}(s1(x),f1(g1(x)))
     app#(g(),x7) -> g{1,#}(x7) -> g{1,#}(s1(x)) -> f{1,#}(g1(x))
     app#(g(),x7) -> g{1,#}(x7) -> g{1,#}(s1(x)) -> g{1,#}(x)
     app#(f(),x7) -> f{1,#}(x7) ->
     f{1,#}(s1(x)) -> minus{2,#}(s1(x),g1(f1(x)))
     app#(f(),x7) -> f{1,#}(x7) -> f{1,#}(s1(x)) -> g{1,#}(f1(x))
     app#(f(),x7) -> f{1,#}(x7) ->
     f{1,#}(s1(x)) -> f{1,#}(x)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x6),x7) -> map{2,#}(x6,x7)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(g(),x7) -> g{1,#}(x7)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f(),x7) -> f{1,#}(x7)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     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,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
      filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(map1(x6),x7) -> map{2,#}(x6,x7)
      map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
      map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
      filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
      filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
      app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      f1(0()) -> s1(0())
      f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
      g1(0()) -> 0()
      g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
      map2(fun,nil()) -> nil()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil()) -> nil()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(f(),x7) -> f1(x7)
      app(g(),x7) -> g1(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(),fun,x,xs) -> filter{2,#}(fun,xs)
        app#(map1(x6),x7) -> map{2,#}(x6,x7)
        app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
        filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
        app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
       TRS:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        f1(0()) -> s1(0())
        f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
        g1(0()) -> 0()
        g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
        map2(fun,nil()) -> nil()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil()) -> nil()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x6),x7) -> minus2(x6,x7)
        app(minus(),x7) -> minus1(x7)
        app(s(),x7) -> s1(x7)
        app(f(),x7) -> f1(x7)
        app(g(),x7) -> g1(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:
      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,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      f1(0()) -> s1(0())
      f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
      g1(0()) -> 0()
      g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
      map2(fun,nil()) -> nil()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil()) -> nil()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(f(),x7) -> f1(x7)
      app(g(),x7) -> g1(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)
     Usable Rule Processor:
      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:
       f1(0()) -> s1(0())
       f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
       minus2(x,0()) -> x
       minus2(s1(x),s1(y)) -> minus2(x,y)
       g1(0()) -> 0()
       g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
      Arctic Interpretation Processor:
       dimension: 1
       usable rules:
        f1(0()) -> s1(0())
        f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        g1(0()) -> 0()
        g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
       interpretation:
        [g{1,#}](x0) = x0 + 0,
        
        [f{1,#}](x0) = x0,
        
        [g1](x0) = 1x0 + 0,
        
        [f1](x0) = 1x0 + 0,
        
        [s1](x0) = 1x0 + 0,
        
        [minus2](x0, x1) = 1x0 + x1 + 0,
        
        [0] = 4
       orientation:
        f{1,#}(s1(x)) = 1x + 0 >= x = f{1,#}(x)
        
        f{1,#}(s1(x)) = 1x + 0 >= 1x + 0 = g{1,#}(f1(x))
        
        g{1,#}(s1(x)) = 1x + 0 >= x + 0 = g{1,#}(x)
        
        g{1,#}(s1(x)) = 1x + 0 >= 1x + 0 = f{1,#}(g1(x))
        
        f1(0()) = 5 >= 5 = s1(0())
        
        f1(s1(x)) = 2x + 1 >= 2x + 1 = minus2(s1(x),g1(f1(x)))
        
        minus2(x,0()) = 1x + 4 >= x = x
        
        minus2(s1(x),s1(y)) = 2x + 1y + 1 >= 1x + y + 0 = minus2(x,y)
        
        g1(0()) = 5 >= 4 = 0()
        
        g1(s1(x)) = 2x + 1 >= 2x + 1 = minus2(s1(x),f1(g1(x)))
       problem:
        DPs:
         f{1,#}(s1(x)) -> g{1,#}(f1(x))
         g{1,#}(s1(x)) -> g{1,#}(x)
         g{1,#}(s1(x)) -> f{1,#}(g1(x))
        TRS:
         f1(0()) -> s1(0())
         f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
         minus2(x,0()) -> x
         minus2(s1(x),s1(y)) -> minus2(x,y)
         g1(0()) -> 0()
         g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
       Restore Modifier:
        DPs:
         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,0()) -> x
         minus2(s1(x),s1(y)) -> minus2(x,y)
         f1(0()) -> s1(0())
         f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
         g1(0()) -> 0()
         g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
         map2(fun,nil()) -> nil()
         map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
         filter2(fun,nil()) -> nil()
         filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
         filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
         filter24(false(),fun,x,xs) -> filter2(fun,xs)
         app(minus1(x6),x7) -> minus2(x6,x7)
         app(minus(),x7) -> minus1(x7)
         app(s(),x7) -> s1(x7)
         app(f(),x7) -> f1(x7)
         app(g(),x7) -> g1(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)
        Usable Rule Processor:
         DPs:
          f{1,#}(s1(x)) -> g{1,#}(f1(x))
          g{1,#}(s1(x)) -> g{1,#}(x)
          g{1,#}(s1(x)) -> f{1,#}(g1(x))
         TRS:
          f1(0()) -> s1(0())
          f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
          minus2(x,0()) -> x
          minus2(s1(x),s1(y)) -> minus2(x,y)
          g1(0()) -> 0()
          g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
         Arctic Interpretation Processor:
          dimension: 1
          usable rules:
           f1(0()) -> s1(0())
           f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
           minus2(x,0()) -> x
           minus2(s1(x),s1(y)) -> minus2(x,y)
           g1(0()) -> 0()
           g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
          interpretation:
           [g{1,#}](x0) = x0 + -16,
           
           [f{1,#}](x0) = x0,
           
           [g1](x0) = 2x0 + -8,
           
           [f1](x0) = 2x0 + 0,
           
           [s1](x0) = 2x0 + 0,
           
           [minus2](x0, x1) = x0 + -6x1 + -8,
           
           [0] = 5
          orientation:
           f{1,#}(s1(x)) = 2x + 0 >= 2x + 0 = g{1,#}(f1(x))
           
           g{1,#}(s1(x)) = 2x + 0 >= x + -16 = g{1,#}(x)
           
           g{1,#}(s1(x)) = 2x + 0 >= 2x + -8 = f{1,#}(g1(x))
           
           f1(0()) = 7 >= 7 = s1(0())
           
           f1(s1(x)) = 4x + 2 >= 2x + 0 = minus2(s1(x),g1(f1(x)))
           
           minus2(x,0()) = x + -1 >= x = x
           
           minus2(s1(x),s1(y)) = 2x + -4y + 0 >= x + -6y + -8 = minus2(x,y)
           
           g1(0()) = 7 >= 5 = 0()
           
           g1(s1(x)) = 4x + 2 >= 2x + 0 = minus2(s1(x),f1(g1(x)))
          problem:
           DPs:
            f{1,#}(s1(x)) -> g{1,#}(f1(x))
            g{1,#}(s1(x)) -> f{1,#}(g1(x))
           TRS:
            f1(0()) -> s1(0())
            f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
            minus2(x,0()) -> x
            minus2(s1(x),s1(y)) -> minus2(x,y)
            g1(0()) -> 0()
            g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
          Restore Modifier:
           DPs:
            f{1,#}(s1(x)) -> g{1,#}(f1(x))
            g{1,#}(s1(x)) -> f{1,#}(g1(x))
           TRS:
            minus2(x,0()) -> x
            minus2(s1(x),s1(y)) -> minus2(x,y)
            f1(0()) -> s1(0())
            f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
            g1(0()) -> 0()
            g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
            map2(fun,nil()) -> nil()
            map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
            filter2(fun,nil()) -> nil()
            filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
            filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
            filter24(false(),fun,x,xs) -> filter2(fun,xs)
            app(minus1(x6),x7) -> minus2(x6,x7)
            app(minus(),x7) -> minus1(x7)
            app(s(),x7) -> s1(x7)
            app(f(),x7) -> f1(x7)
            app(g(),x7) -> g1(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)
           Usable Rule Processor:
            DPs:
             f{1,#}(s1(x)) -> g{1,#}(f1(x))
             g{1,#}(s1(x)) -> f{1,#}(g1(x))
            TRS:
             f1(0()) -> s1(0())
             f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
             minus2(x,0()) -> x
             minus2(s1(x),s1(y)) -> minus2(x,y)
             g1(0()) -> 0()
             g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
            Arctic Interpretation Processor:
             dimension: 1
             usable rules:
              f1(0()) -> s1(0())
              f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
              minus2(x,0()) -> x
              minus2(s1(x),s1(y)) -> minus2(x,y)
              g1(0()) -> 0()
              g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
             interpretation:
              [g{1,#}](x0) = x0,
              
              [f{1,#}](x0) = x0,
              
              [g1](x0) = x0,
              
              [f1](x0) = 1x0,
              
              [s1](x0) = 1x0 + 0,
              
              [minus2](x0, x1) = x0 + x1 + 0,
              
              [0] = 0
             orientation:
              f{1,#}(s1(x)) = 1x + 0 >= 1x = g{1,#}(f1(x))
              
              g{1,#}(s1(x)) = 1x + 0 >= x = f{1,#}(g1(x))
              
              f1(0()) = 1 >= 1 = s1(0())
              
              f1(s1(x)) = 2x + 1 >= 1x + 0 = minus2(s1(x),g1(f1(x)))
              
              minus2(x,0()) = x + 0 >= x = x
              
              minus2(s1(x),s1(y)) = 1x + 1y + 0 >= x + y + 0 = minus2(x,y)
              
              g1(0()) = 0 >= 0 = 0()
              
              g1(s1(x)) = 1x + 0 >= 1x + 0 = minus2(s1(x),f1(g1(x)))
             problem:
              DPs:
               f{1,#}(s1(x)) -> g{1,#}(f1(x))
              TRS:
               f1(0()) -> s1(0())
               f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
               minus2(x,0()) -> x
               minus2(s1(x),s1(y)) -> minus2(x,y)
               g1(0()) -> 0()
               g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
             Restore Modifier:
              DPs:
               f{1,#}(s1(x)) -> g{1,#}(f1(x))
              TRS:
               minus2(x,0()) -> x
               minus2(s1(x),s1(y)) -> minus2(x,y)
               f1(0()) -> s1(0())
               f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
               g1(0()) -> 0()
               g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
               map2(fun,nil()) -> nil()
               map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
               filter2(fun,nil()) -> nil()
               filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
               filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
               filter24(false(),fun,x,xs) -> filter2(fun,xs)
               app(minus1(x6),x7) -> minus2(x6,x7)
               app(minus(),x7) -> minus1(x7)
               app(s(),x7) -> s1(x7)
               app(f(),x7) -> f1(x7)
               app(g(),x7) -> g1(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: 8/1
               
     
     DPs:
      minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      f1(0()) -> s1(0())
      f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
      g1(0()) -> 0()
      g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
      map2(fun,nil()) -> nil()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil()) -> nil()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(f(),x7) -> f1(x7)
      app(g(),x7) -> g1(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(minus{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        f1(0()) -> s1(0())
        f1(s1(x)) -> minus2(s1(x),g1(f1(x)))
        g1(0()) -> 0()
        g1(s1(x)) -> minus2(s1(x),f1(g1(x)))
        map2(fun,nil()) -> nil()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil()) -> nil()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x6),x7) -> minus2(x6,x7)
        app(minus(),x7) -> minus1(x7)
        app(s(),x7) -> s1(x7)
        app(f(),x7) -> f1(x7)
        app(g(),x7) -> g1(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