MAYBE

Problem:
 app(app(minus(),x),0()) -> x
 app(app(minus(),app(s(),x)),app(s(),y)) -> 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(plus(),0()),y) -> y
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
 app(app(plus(),app(app(minus(),x),app(s(),0()))),app(app(minus(),y),app(s(),app(s(),z)))) ->
 app(app(plus(),app(app(minus(),y),app(s(),app(s(),z)))),app(app(minus(),x),app(s(),0())))
 app(app(plus(),app(app(plus(),x),app(s(),0()))),app(app(plus(),y),app(s(),app(s(),z)))) ->
 app(app(plus(),app(app(plus(),y),app(s(),app(s(),z)))),app(app(plus(),x),app(s(),0())))
 app(app(map(),f),nil()) -> nil()
 app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
 app(app(filter(),f),nil()) -> nil()
 app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
 app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
 app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

Proof:
 Uncurry Processor:
  minus2(x,0()) -> x
  minus2(s1(x),s1(y)) -> minus2(x,y)
  quot2(0(),s1(y)) -> 0()
  quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
  plus2(0(),y) -> y
  plus2(s1(x),y) -> s1(plus2(x,y))
  plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) -> plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
  plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
  map2(f,nil()) -> nil()
  map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
  filter2(f,nil()) -> nil()
  filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
  filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
  filter24(false(),f,x,xs) -> filter2(f,xs)
  app(minus1(x7),x8) -> minus2(x7,x8)
  app(minus(),x8) -> minus1(x8)
  app(s(),x8) -> s1(x8)
  app(quot1(x7),x8) -> quot2(x7,x8)
  app(quot(),x8) -> quot1(x8)
  app(plus1(x7),x8) -> plus2(x7,x8)
  app(plus(),x8) -> plus1(x8)
  app(map1(x7),x8) -> map2(x7,x8)
  app(map(),x8) -> map1(x8)
  app(cons1(x7),x8) -> cons2(x7,x8)
  app(cons(),x8) -> cons1(x8)
  app(filter1(x7),x8) -> filter2(x7,x8)
  app(filter(),x8) -> filter1(x8)
  app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
  app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
  app(filter21(x7),x8) -> filter22(x7,x8)
  app(filter2(),x8) -> filter21(x8)
  DP Processor:
   DPs:
    minus{2,#}(s1(x),s1(y)) -> minus{2,#}(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))
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
    plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
    plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
    plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
    filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
    filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
    app#(minus1(x7),x8) -> minus{2,#}(x7,x8)
    app#(quot1(x7),x8) -> quot{2,#}(x7,x8)
    app#(plus1(x7),x8) -> plus{2,#}(x7,x8)
    app#(map1(x7),x8) -> map{2,#}(x7,x8)
    app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
    app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
   TRS:
    minus2(x,0()) -> x
    minus2(s1(x),s1(y)) -> minus2(x,y)
    quot2(0(),s1(y)) -> 0()
    quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
    plus2(0(),y) -> y
    plus2(s1(x),y) -> s1(plus2(x,y))
    plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) -> plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
    plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
    map2(f,nil()) -> nil()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    filter2(f,nil()) -> nil()
    filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
    filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
    filter24(false(),f,x,xs) -> filter2(f,xs)
    app(minus1(x7),x8) -> minus2(x7,x8)
    app(minus(),x8) -> minus1(x8)
    app(s(),x8) -> s1(x8)
    app(quot1(x7),x8) -> quot2(x7,x8)
    app(quot(),x8) -> quot1(x8)
    app(plus1(x7),x8) -> plus2(x7,x8)
    app(plus(),x8) -> plus1(x8)
    app(map1(x7),x8) -> map2(x7,x8)
    app(map(),x8) -> map1(x8)
    app(cons1(x7),x8) -> cons2(x7,x8)
    app(cons(),x8) -> cons1(x8)
    app(filter1(x7),x8) -> filter2(x7,x8)
    app(filter(),x8) -> filter1(x8)
    app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
    app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
    app(filter21(x7),x8) -> filter22(x7,x8)
    app(filter2(),x8) -> filter21(x8)
   TDG Processor:
    DPs:
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(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))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
     plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
     plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
     plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(minus1(x7),x8) -> minus{2,#}(x7,x8)
     app#(quot1(x7),x8) -> quot{2,#}(x7,x8)
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8)
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
    TRS:
     minus2(x,0()) -> x
     minus2(s1(x),s1(y)) -> minus2(x,y)
     quot2(0(),s1(y)) -> 0()
     quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
     plus2(0(),y) -> y
     plus2(s1(x),y) -> s1(plus2(x,y))
     plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) -> plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
     plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
     map2(f,nil()) -> nil()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     filter2(f,nil()) -> nil()
     filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
     filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
     filter24(false(),f,x,xs) -> filter2(f,xs)
     app(minus1(x7),x8) -> minus2(x7,x8)
     app(minus(),x8) -> minus1(x8)
     app(s(),x8) -> s1(x8)
     app(quot1(x7),x8) -> quot2(x7,x8)
     app(quot(),x8) -> quot1(x8)
     app(plus1(x7),x8) -> plus2(x7,x8)
     app(plus(),x8) -> plus1(x8)
     app(map1(x7),x8) -> map2(x7,x8)
     app(map(),x8) -> map1(x8)
     app(cons1(x7),x8) -> cons2(x7,x8)
     app(cons(),x8) -> cons1(x8)
     app(filter1(x7),x8) -> filter2(x7,x8)
     app(filter(),x8) -> filter1(x8)
     app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
     app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
     app(filter21(x7),x8) -> filter22(x7,x8)
     app(filter2(),x8) -> filter21(x8)
    graph:
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(quot1(x7),x8) -> quot{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x7),x8) -> minus{2,#}(x7,x8)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8) ->
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8) ->
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x7),x8) -> map{2,#}(x7,x8) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x7),x8) -> map{2,#}(x7,x8) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8) ->
     plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
     plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8) ->
     plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
     plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     app#(quot1(x7),x8) -> quot{2,#}(x7,x8) ->
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
     app#(quot1(x7),x8) -> quot{2,#}(x7,x8) ->
     quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(minus1(x7),x8) -> minus{2,#}(x7,x8) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x7),x8) -> plus{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(quot1(x7),x8) -> quot{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x7),x8) -> minus{2,#}(x7,x8)
     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)
     plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
     plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0()))) ->
     plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
     plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
     plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
     plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0()))) ->
     plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
     plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
     plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
     plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0()))) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
     plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
     plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
     plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0()))) ->
     plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
     plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
     plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
     plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0()))) ->
     plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
     plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
     plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
     plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0()))) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,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)) -> 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,#}(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: 4
     #rules: 14
     #arcs: 45/324
     DPs:
      filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x7),x8) -> map{2,#}(x7,x8)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
      filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
      filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
      app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      quot2(0(),s1(y)) -> 0()
      quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) -> plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
      plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(minus1(x7),x8) -> minus2(x7,x8)
      app(minus(),x8) -> minus1(x8)
      app(s(),x8) -> s1(x8)
      app(quot1(x7),x8) -> quot2(x7,x8)
      app(quot(),x8) -> quot1(x8)
      app(plus1(x7),x8) -> plus2(x7,x8)
      app(plus(),x8) -> plus1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 1
       pi(app#) = 1
       pi(filter{2,#}) = 1
       pi(filter2{4,#}) = 3
      problem:
       DPs:
        filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
        app#(map1(x7),x8) -> map{2,#}(x7,x8)
        app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
        filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
        app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
       TRS:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        quot2(0(),s1(y)) -> 0()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
        plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
        plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil()) -> nil()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false(),f,x,xs) -> filter2(f,xs)
        app(minus1(x7),x8) -> minus2(x7,x8)
        app(minus(),x8) -> minus1(x8)
        app(s(),x8) -> s1(x8)
        app(quot1(x7),x8) -> quot2(x7,x8)
        app(quot(),x8) -> quot1(x8)
        app(plus1(x7),x8) -> plus2(x7,x8)
        app(plus(),x8) -> plus1(x8)
        app(map1(x7),x8) -> map2(x7,x8)
        app(map(),x8) -> map1(x8)
        app(cons1(x7),x8) -> cons2(x7,x8)
        app(cons(),x8) -> cons1(x8)
        app(filter1(x7),x8) -> filter2(x7,x8)
        app(filter(),x8) -> filter1(x8)
        app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
        app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
        app(filter21(x7),x8) -> filter22(x7,x8)
        app(filter2(),x8) -> filter21(x8)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 20/25
       
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
      plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
      plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
      plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
      plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      quot2(0(),s1(y)) -> 0()
      quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) -> plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
      plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(minus1(x7),x8) -> minus2(x7,x8)
      app(minus(),x8) -> minus1(x8)
      app(s(),x8) -> s1(x8)
      app(quot1(x7),x8) -> quot2(x7,x8)
      app(quot(),x8) -> quot1(x8)
      app(plus1(x7),x8) -> plus2(x7,x8)
      app(plus(),x8) -> plus1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     CDG Processor:
      DPs:
       plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
       plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
       plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
       plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
       plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
      TRS:
       minus2(x,0()) -> x
       minus2(s1(x),s1(y)) -> minus2(x,y)
       quot2(0(),s1(y)) -> 0()
       quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
       plus2(0(),y) -> y
       plus2(s1(x),y) -> s1(plus2(x,y))
       plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) -> plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
       plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
       map2(f,nil()) -> nil()
       map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
       filter2(f,nil()) -> nil()
       filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
       filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
       filter24(false(),f,x,xs) -> filter2(f,xs)
       app(minus1(x7),x8) -> minus2(x7,x8)
       app(minus(),x8) -> minus1(x8)
       app(s(),x8) -> s1(x8)
       app(quot1(x7),x8) -> quot2(x7,x8)
       app(quot(),x8) -> quot1(x8)
       app(plus1(x7),x8) -> plus2(x7,x8)
       app(plus(),x8) -> plus1(x8)
       app(map1(x7),x8) -> map2(x7,x8)
       app(map(),x8) -> map1(x8)
       app(cons1(x7),x8) -> cons2(x7,x8)
       app(cons(),x8) -> cons1(x8)
       app(filter1(x7),x8) -> filter2(x7,x8)
       app(filter(),x8) -> filter1(x8)
       app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
       app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
       app(filter21(x7),x8) -> filter22(x7,x8)
       app(filter2(),x8) -> filter21(x8)
      graph:
       plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
       plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0()))) ->
       plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
       plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
       plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
       plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
       plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
       plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
       plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
       plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
       plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
       plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
       plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0()))) ->
       plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
       plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
       plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0()))) ->
       plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
       plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
       plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
       plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0()))) ->
       plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
       plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [plus{2,#}](x0, x1) = x0 + x1,
        
        [filter21](x0) = 4x0 + 1,
        
        [filter22](x0, x1) = x0 + 2x1 + 3,
        
        [filter24](x0, x1, x2, x3) = 4x1,
        
        [filter23](x0, x1, x2) = 5x1 + 4x2 + 1,
        
        [filter2](x0, x1) = 4x0,
        
        [filter1](x0) = 4x0 + 6,
        
        [cons2](x0, x1) = x1,
        
        [cons1](x0) = x0 + 1,
        
        [map2](x0, x1) = 4x0 + 4x1 + 4,
        
        [map1](x0) = 4x0 + 1,
        
        [plus2](x0, x1) = x0 + x1,
        
        [plus1](x0) = 4x0 + 1,
        
        [quot2](x0, x1) = x0 + 4,
        
        [quot1](x0) = 2x0 + 1,
        
        [s1](x0) = x0 + 4,
        
        [minus2](x0, x1) = x0,
        
        [minus1](x0) = 4x0 + 5,
        
        [false] = 0,
        
        [true] = 0,
        
        [filter2] = 4,
        
        [filter] = 2,
        
        [cons] = 2,
        
        [nil] = 0,
        
        [map] = 4,
        
        [plus] = 4,
        
        [quot] = 7,
        
        [s] = 3,
        
        [0] = 1,
        
        [app](x0, x1) = 4x0 + 4x1 + 2,
        
        [minus] = 2
       orientation:
        plus{2,#}(s1(x),y) = x + y + 4 >= x + y = plus{2,#}(x,y)
        
        plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) = x + y >= x + y = plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
        
        plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) = x + y + z + 13 >= x + y + z + 13 = plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
        
        minus2(x,0()) = x >= x = x
        
        minus2(s1(x),s1(y)) = x + 4 >= x = minus2(x,y)
        
        quot2(0(),s1(y)) = 5 >= 1 = 0()
        
        quot2(s1(x),s1(y)) = x + 8 >= x + 8 = s1(quot2(minus2(x,y),s1(y)))
        
        plus2(0(),y) = y + 1 >= y = y
        
        plus2(s1(x),y) = x + y + 4 >= x + y + 4 = s1(plus2(x,y))
        
        plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) = x + y >= x + y = plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
        
        plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) = x + y + z + 13 >= x + y + z + 13 = plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
        
        map2(f,nil()) = 4f + 4 >= 0 = nil()
        
        map2(f,cons2(x,xs)) = 4f + 4xs + 4 >= 4f + 4xs + 4 = cons2(app(f,x),map2(f,xs))
        
        filter2(f,nil()) = 4f >= 0 = nil()
        
        filter2(f,cons2(x,xs)) = 4f >= 4f = filter24(app(f,x),f,x,xs)
        
        filter24(true(),f,x,xs) = 4f >= 4f = cons2(x,filter2(f,xs))
        
        filter24(false(),f,x,xs) = 4f >= 4f = filter2(f,xs)
        
        app(minus1(x7),x8) = 16x7 + 4x8 + 22 >= x7 = minus2(x7,x8)
        
        app(minus(),x8) = 4x8 + 10 >= 4x8 + 5 = minus1(x8)
        
        app(s(),x8) = 4x8 + 14 >= x8 + 4 = s1(x8)
        
        app(quot1(x7),x8) = 8x7 + 4x8 + 6 >= x7 + 4 = quot2(x7,x8)
        
        app(quot(),x8) = 4x8 + 30 >= 2x8 + 1 = quot1(x8)
        
        app(plus1(x7),x8) = 16x7 + 4x8 + 6 >= x7 + x8 = plus2(x7,x8)
        
        app(plus(),x8) = 4x8 + 18 >= 4x8 + 1 = plus1(x8)
        
        app(map1(x7),x8) = 16x7 + 4x8 + 6 >= 4x7 + 4x8 + 4 = map2(x7,x8)
        
        app(map(),x8) = 4x8 + 18 >= 4x8 + 1 = map1(x8)
        
        app(cons1(x7),x8) = 4x7 + 4x8 + 6 >= x8 = cons2(x7,x8)
        
        app(cons(),x8) = 4x8 + 10 >= x8 + 1 = cons1(x8)
        
        app(filter1(x7),x8) = 16x7 + 4x8 + 26 >= 4x7 = filter2(x7,x8)
        
        app(filter(),x8) = 4x8 + 10 >= 4x8 + 6 = filter1(x8)
        
        app(filter23(x7,x6,x5),x8) = 16x5 + 20x6 + 4x8 + 6 >= 4x6 = filter24(x7,x6,x5,x8)
        
        app(filter22(x7,x6),x8) = 8x6 + 4x7 + 4x8 + 14 >= 5x6 + 4x8 + 1 = filter23(x7,x6,x8)
        
        app(filter21(x7),x8) = 16x7 + 4x8 + 6 >= x7 + 2x8 + 3 = filter22(x7,x8)
        
        app(filter2(),x8) = 4x8 + 18 >= 4x8 + 1 = filter21(x8)
       problem:
        DPs:
         plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
         plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
         plus{2,#}(plus2(x,s1(0())),plus2(y,s1(s1(z)))) ->
         plus{2,#}(plus2(y,s1(s1(z))),plus2(x,s1(0())))
        TRS:
         minus2(x,0()) -> x
         minus2(s1(x),s1(y)) -> minus2(x,y)
         quot2(0(),s1(y)) -> 0()
         quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
         plus2(0(),y) -> y
         plus2(s1(x),y) -> s1(plus2(x,y))
         plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
         plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
         plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
         map2(f,nil()) -> nil()
         map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
         filter2(f,nil()) -> nil()
         filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
         filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
         filter24(false(),f,x,xs) -> filter2(f,xs)
         app(minus1(x7),x8) -> minus2(x7,x8)
         app(minus(),x8) -> minus1(x8)
         app(s(),x8) -> s1(x8)
         app(quot1(x7),x8) -> quot2(x7,x8)
         app(quot(),x8) -> quot1(x8)
         app(plus1(x7),x8) -> plus2(x7,x8)
         app(plus(),x8) -> plus1(x8)
         app(map1(x7),x8) -> map2(x7,x8)
         app(map(),x8) -> map1(x8)
         app(cons1(x7),x8) -> cons2(x7,x8)
         app(cons(),x8) -> cons1(x8)
         app(filter1(x7),x8) -> filter2(x7,x8)
         app(filter(),x8) -> filter1(x8)
         app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
         app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
         app(filter21(x7),x8) -> filter22(x7,x8)
         app(filter2(),x8) -> filter21(x8)
       SCC Processor:
        #sccs: 1
        #rules: 1
        #arcs: 7/4
        DPs:
         plus{2,#}(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
         plus{2,#}(minus2(y,s1(s1(z))),minus2(x,s1(0())))
        TRS:
         minus2(x,0()) -> x
         minus2(s1(x),s1(y)) -> minus2(x,y)
         quot2(0(),s1(y)) -> 0()
         quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
         plus2(0(),y) -> y
         plus2(s1(x),y) -> s1(plus2(x,y))
         plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
         plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
         plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
         map2(f,nil()) -> nil()
         map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
         filter2(f,nil()) -> nil()
         filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
         filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
         filter24(false(),f,x,xs) -> filter2(f,xs)
         app(minus1(x7),x8) -> minus2(x7,x8)
         app(minus(),x8) -> minus1(x8)
         app(s(),x8) -> s1(x8)
         app(quot1(x7),x8) -> quot2(x7,x8)
         app(quot(),x8) -> quot1(x8)
         app(plus1(x7),x8) -> plus2(x7,x8)
         app(plus(),x8) -> plus1(x8)
         app(map1(x7),x8) -> map2(x7,x8)
         app(map(),x8) -> map1(x8)
         app(cons1(x7),x8) -> cons2(x7,x8)
         app(cons(),x8) -> cons1(x8)
         app(filter1(x7),x8) -> filter2(x7,x8)
         app(filter(),x8) -> filter1(x8)
         app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
         app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
         app(filter21(x7),x8) -> filter22(x7,x8)
         app(filter2(),x8) -> filter21(x8)
        Open
     
     DPs:
      quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      quot2(0(),s1(y)) -> 0()
      quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) -> plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
      plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(minus1(x7),x8) -> minus2(x7,x8)
      app(minus(),x8) -> minus1(x8)
      app(s(),x8) -> s1(x8)
      app(quot1(x7),x8) -> quot2(x7,x8)
      app(quot(),x8) -> quot1(x8)
      app(plus1(x7),x8) -> plus2(x7,x8)
      app(plus(),x8) -> plus1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [quot{2,#}](x0, x1) = 1/2x0 + 1,
       
       [filter21](x0) = 2x0 + 1/2,
       
       [filter22](x0, x1) = x0 + 1/2,
       
       [filter24](x0, x1, x2, x3) = 2x3,
       
       [filter23](x0, x1, x2) = 1/2x2 + 1/2,
       
       [filter2](x0, x1) = 2x1,
       
       [filter1](x0) = 1/2,
       
       [cons2](x0, x1) = x1,
       
       [cons1](x0) = 2x0 + 2,
       
       [map2](x0, x1) = 1,
       
       [map1](x0) = 1,
       
       [plus2](x0, x1) = x0 + x1,
       
       [plus1](x0) = 2x0 + 1,
       
       [quot2](x0, x1) = x0 + 1/2,
       
       [quot1](x0) = 2x0 + 1/2,
       
       [s1](x0) = x0 + 1,
       
       [minus2](x0, x1) = x0,
       
       [minus1](x0) = 2x0 + 2,
       
       [false] = 0,
       
       [true] = 0,
       
       [filter2] = 3,
       
       [filter] = 1,
       
       [cons] = 2,
       
       [nil] = 1/2,
       
       [map] = 1,
       
       [plus] = 3/2,
       
       [quot] = 1/2,
       
       [s] = 2,
       
       [0] = 0,
       
       [app](x0, x1) = x0 + 2x1,
       
       [minus] = 3
      orientation:
       quot{2,#}(s1(x),s1(y)) = 1/2x + 3/2 >= 1/2x + 1 = quot{2,#}(minus2(x,y),s1(y))
       
       minus2(x,0()) = x >= x = x
       
       minus2(s1(x),s1(y)) = x + 1 >= x = minus2(x,y)
       
       quot2(0(),s1(y)) = 1/2 >= 0 = 0()
       
       quot2(s1(x),s1(y)) = x + 3/2 >= x + 3/2 = s1(quot2(minus2(x,y),s1(y)))
       
       plus2(0(),y) = y >= y = y
       
       plus2(s1(x),y) = x + y + 1 >= x + y + 1 = s1(plus2(x,y))
       
       plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) = x + y >= x + y = plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
       
       plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) = x + y + z + 3 >= x + y + z + 3 = plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
       
       map2(f,nil()) = 1 >= 1/2 = nil()
       
       map2(f,cons2(x,xs)) = 1 >= 1 = cons2(app(f,x),map2(f,xs))
       
       filter2(f,nil()) = 1 >= 1/2 = nil()
       
       filter2(f,cons2(x,xs)) = 2xs >= 2xs = filter24(app(f,x),f,x,xs)
       
       filter24(true(),f,x,xs) = 2xs >= 2xs = cons2(x,filter2(f,xs))
       
       filter24(false(),f,x,xs) = 2xs >= 2xs = filter2(f,xs)
       
       app(minus1(x7),x8) = 2x7 + 2x8 + 2 >= x7 = minus2(x7,x8)
       
       app(minus(),x8) = 2x8 + 3 >= 2x8 + 2 = minus1(x8)
       
       app(s(),x8) = 2x8 + 2 >= x8 + 1 = s1(x8)
       
       app(quot1(x7),x8) = 2x7 + 2x8 + 1/2 >= x7 + 1/2 = quot2(x7,x8)
       
       app(quot(),x8) = 2x8 + 1/2 >= 2x8 + 1/2 = quot1(x8)
       
       app(plus1(x7),x8) = 2x7 + 2x8 + 1 >= x7 + x8 = plus2(x7,x8)
       
       app(plus(),x8) = 2x8 + 3/2 >= 2x8 + 1 = plus1(x8)
       
       app(map1(x7),x8) = 2x8 + 1 >= 1 = map2(x7,x8)
       
       app(map(),x8) = 2x8 + 1 >= 1 = map1(x8)
       
       app(cons1(x7),x8) = 2x7 + 2x8 + 2 >= x8 = cons2(x7,x8)
       
       app(cons(),x8) = 2x8 + 2 >= 2x8 + 2 = cons1(x8)
       
       app(filter1(x7),x8) = 2x8 + 1/2 >= 2x8 = filter2(x7,x8)
       
       app(filter(),x8) = 2x8 + 1 >= 1/2 = filter1(x8)
       
       app(filter23(x7,x6,x5),x8) = 1/2x5 + 2x8 + 1/2 >= 2x8 = filter24(x7,x6,x5,x8)
       
       app(filter22(x7,x6),x8) = x7 + 2x8 + 1/2 >= 1/2x8 + 1/2 = filter23(x7,x6,x8)
       
       app(filter21(x7),x8) = 2x7 + 2x8 + 1/2 >= x7 + 1/2 = filter22(x7,x8)
       
       app(filter2(),x8) = 2x8 + 3 >= 2x8 + 1/2 = filter21(x8)
      problem:
       DPs:
        
       TRS:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        quot2(0(),s1(y)) -> 0()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
        plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
        plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil()) -> nil()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false(),f,x,xs) -> filter2(f,xs)
        app(minus1(x7),x8) -> minus2(x7,x8)
        app(minus(),x8) -> minus1(x8)
        app(s(),x8) -> s1(x8)
        app(quot1(x7),x8) -> quot2(x7,x8)
        app(quot(),x8) -> quot1(x8)
        app(plus1(x7),x8) -> plus2(x7,x8)
        app(plus(),x8) -> plus1(x8)
        app(map1(x7),x8) -> map2(x7,x8)
        app(map(),x8) -> map1(x8)
        app(cons1(x7),x8) -> cons2(x7,x8)
        app(cons(),x8) -> cons1(x8)
        app(filter1(x7),x8) -> filter2(x7,x8)
        app(filter(),x8) -> filter1(x8)
        app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
        app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
        app(filter21(x7),x8) -> filter22(x7,x8)
        app(filter2(),x8) -> filter21(x8)
      Qed
     
     DPs:
      minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      quot2(0(),s1(y)) -> 0()
      quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) -> plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
      plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(minus1(x7),x8) -> minus2(x7,x8)
      app(minus(),x8) -> minus1(x8)
      app(s(),x8) -> s1(x8)
      app(quot1(x7),x8) -> quot2(x7,x8)
      app(quot(),x8) -> quot1(x8)
      app(plus1(x7),x8) -> plus2(x7,x8)
      app(plus(),x8) -> plus1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     Subterm Criterion Processor:
      simple projection:
       pi(minus{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        quot2(0(),s1(y)) -> 0()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        plus2(minus2(x,s1(0())),minus2(y,s1(s1(z)))) ->
        plus2(minus2(y,s1(s1(z))),minus2(x,s1(0())))
        plus2(plus2(x,s1(0())),plus2(y,s1(s1(z)))) -> plus2(plus2(y,s1(s1(z))),plus2(x,s1(0())))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil()) -> nil()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false(),f,x,xs) -> filter2(f,xs)
        app(minus1(x7),x8) -> minus2(x7,x8)
        app(minus(),x8) -> minus1(x8)
        app(s(),x8) -> s1(x8)
        app(quot1(x7),x8) -> quot2(x7,x8)
        app(quot(),x8) -> quot1(x8)
        app(plus1(x7),x8) -> plus2(x7,x8)
        app(plus(),x8) -> plus1(x8)
        app(map1(x7),x8) -> map2(x7,x8)
        app(map(),x8) -> map1(x8)
        app(cons1(x7),x8) -> cons2(x7,x8)
        app(cons(),x8) -> cons1(x8)
        app(filter1(x7),x8) -> filter2(x7,x8)
        app(filter(),x8) -> filter1(x8)
        app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
        app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
        app(filter21(x7),x8) -> filter22(x7,x8)
        app(filter2(),x8) -> filter21(x8)
      Qed