YES

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

Proof:
 Uncurry Processor:
  minus2(x,0()) -> x
  minus2(s1(x),s1(y)) -> minus2(x,y)
  double1(0()) -> 0()
  double1(s1(x)) -> s1(s1(double1(x)))
  plus2(0(),y) -> y
  plus2(s1(x),y) -> s1(plus2(x,y))
  plus2(s1(x),y) -> plus2(x,s1(y))
  plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
  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(x6),x7) -> minus2(x6,x7)
  app(minus(),x7) -> minus1(x7)
  app(s(),x7) -> s1(x7)
  app(double(),x7) -> double1(x7)
  app(plus1(x6),x7) -> plus2(x6,x7)
  app(plus(),x7) -> plus1(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)
    double{1,#}(s1(x)) -> double{1,#}(x)
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y))
    plus{2,#}(s1(x),y) -> double{1,#}(y)
    plus{2,#}(s1(x),y) -> minus{2,#}(x,y)
    plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y))
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
    filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
    filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
    app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
    app#(double(),x7) -> double{1,#}(x7)
    app#(plus1(x6),x7) -> plus{2,#}(x6,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)
    double1(0()) -> 0()
    double1(s1(x)) -> s1(s1(double1(x)))
    plus2(0(),y) -> y
    plus2(s1(x),y) -> s1(plus2(x,y))
    plus2(s1(x),y) -> plus2(x,s1(y))
    plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
    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(x6),x7) -> minus2(x6,x7)
    app(minus(),x7) -> minus1(x7)
    app(s(),x7) -> s1(x7)
    app(double(),x7) -> double1(x7)
    app(plus1(x6),x7) -> plus2(x6,x7)
    app(plus(),x7) -> plus1(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)
     double{1,#}(s1(x)) -> double{1,#}(x)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y))
     plus{2,#}(s1(x),y) -> double{1,#}(y)
     plus{2,#}(s1(x),y) -> minus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y))
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     app#(double(),x7) -> double{1,#}(x7)
     app#(plus1(x6),x7) -> plus{2,#}(x6,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)
     double1(0()) -> 0()
     double1(s1(x)) -> s1(s1(double1(x)))
     plus2(0(),y) -> y
     plus2(s1(x),y) -> s1(plus2(x,y))
     plus2(s1(x),y) -> plus2(x,s1(y))
     plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
     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(x6),x7) -> minus2(x6,x7)
     app(minus(),x7) -> minus1(x7)
     app(s(),x7) -> s1(x7)
     app(double(),x7) -> double1(x7)
     app(plus1(x6),x7) -> plus2(x6,x7)
     app(plus(),x7) -> plus1(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(),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(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x6),x7) -> map{2,#}(x6,x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(double(),x7) -> double{1,#}(x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7) ->
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7) ->
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x6),x7) -> map{2,#}(x6,x7) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x6),x7) -> map{2,#}(x6,x7) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y))
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7) ->
     plus{2,#}(s1(x),y) -> minus{2,#}(x,y)
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7) ->
     plus{2,#}(s1(x),y) -> double{1,#}(y)
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y))
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(double(),x7) -> double{1,#}(x7) ->
     double{1,#}(s1(x)) -> double{1,#}(x)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x6),x7) -> map{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x6),x7) -> plus{2,#}(x6,x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(double(),x7) -> double{1,#}(x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     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,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y)) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y))
     plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y)) ->
     plus{2,#}(s1(x),y) -> minus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y)) ->
     plus{2,#}(s1(x),y) -> double{1,#}(y)
     plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y)) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y))
     plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y)) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y)) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y)) ->
     plus{2,#}(s1(x),y) -> minus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y)) ->
     plus{2,#}(s1(x),y) -> double{1,#}(y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y)) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(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,#}(minus2(x,y),double1(y))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> minus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> double{1,#}(y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> double{1,#}(y) ->
     double{1,#}(s1(x)) -> double{1,#}(x)
     plus{2,#}(s1(x),y) -> minus{2,#}(x,y) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     double{1,#}(s1(x)) -> double{1,#}(x) ->
     double{1,#}(s1(x)) -> double{1,#}(x)
     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: 52/361
     DPs:
      filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x6),x7) -> map{2,#}(x6,x7)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
      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(x6,x5,x4),x7) -> filter2{4,#}(x6,x5,x4,x7)
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      double1(0()) -> 0()
      double1(s1(x)) -> s1(s1(double1(x)))
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      plus2(s1(x),y) -> plus2(x,s1(y))
      plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
      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(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(double(),x7) -> double1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(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(),f,x,xs) -> filter{2,#}(f,xs)
        app#(map1(x6),x7) -> map{2,#}(x6,x7)
        app#(filter1(x6),x7) -> filter{2,#}(x6,x7)
        filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,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)
        double1(0()) -> 0()
        double1(s1(x)) -> s1(s1(double1(x)))
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        plus2(s1(x),y) -> plus2(x,s1(y))
        plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
        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(x6),x7) -> minus2(x6,x7)
        app(minus(),x7) -> minus1(x7)
        app(s(),x7) -> s1(x7)
        app(double(),x7) -> double1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(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:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
      plus{2,#}(s1(x),y) -> plus{2,#}(x,s1(y))
      plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y))
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      double1(0()) -> 0()
      double1(s1(x)) -> s1(s1(double1(x)))
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      plus2(s1(x),y) -> plus2(x,s1(y))
      plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
      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(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(double(),x7) -> double1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(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)
     LPO Processor:
      argument filtering:
       pi(minus) = []
       pi(app) = [0,1]
       pi(0) = []
       pi(s) = []
       pi(double) = []
       pi(plus) = []
       pi(map) = []
       pi(nil) = []
       pi(cons) = []
       pi(filter) = []
       pi(filter2) = []
       pi(true) = []
       pi(false) = []
       pi(minus1) = 0
       pi(minus2) = 0
       pi(s1) = [0]
       pi(double1) = [0]
       pi(plus1) = [0]
       pi(plus2) = [0,1]
       pi(map1) = []
       pi(map2) = 1
       pi(cons1) = 0
       pi(cons2) = 1
       pi(filter1) = 0
       pi(filter2) = 1
       pi(filter23) = 2
       pi(filter24) = 3
       pi(filter22) = 1
       pi(filter21) = 0
       pi(plus{2,#}) = 0
      precedence:
       app > map > plus2 > double1 > plus{2,#} ~ filter21 ~ filter22 ~ 
       filter24 ~ filter23 ~ filter2 ~ filter1 ~ cons2 ~ cons1 ~ map2 ~ 
       map1 ~ plus1 ~ s1 ~ minus2 ~ minus1 ~ false ~ true ~ filter2 ~ 
       filter ~ cons ~ nil ~ plus ~ double ~ s ~ 0 ~ minus
      problem:
       DPs:
        
       TRS:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        double1(0()) -> 0()
        double1(s1(x)) -> s1(s1(double1(x)))
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        plus2(s1(x),y) -> plus2(x,s1(y))
        plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
        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(x6),x7) -> minus2(x6,x7)
        app(minus(),x7) -> minus1(x7)
        app(s(),x7) -> s1(x7)
        app(double(),x7) -> double1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(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
     
     DPs:
      double{1,#}(s1(x)) -> double{1,#}(x)
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      double1(0()) -> 0()
      double1(s1(x)) -> s1(s1(double1(x)))
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      plus2(s1(x),y) -> plus2(x,s1(y))
      plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
      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(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(double(),x7) -> double1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(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(double{1,#}) = 0
      problem:
       DPs:
        
       TRS:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        double1(0()) -> 0()
        double1(s1(x)) -> s1(s1(double1(x)))
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        plus2(s1(x),y) -> plus2(x,s1(y))
        plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
        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(x6),x7) -> minus2(x6,x7)
        app(minus(),x7) -> minus1(x7)
        app(s(),x7) -> s1(x7)
        app(double(),x7) -> double1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(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
     
     DPs:
      minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     TRS:
      minus2(x,0()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      double1(0()) -> 0()
      double1(s1(x)) -> s1(s1(double1(x)))
      plus2(0(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      plus2(s1(x),y) -> plus2(x,s1(y))
      plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
      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(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(double(),x7) -> double1(x7)
      app(plus1(x6),x7) -> plus2(x6,x7)
      app(plus(),x7) -> plus1(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)
        double1(0()) -> 0()
        double1(s1(x)) -> s1(s1(double1(x)))
        plus2(0(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        plus2(s1(x),y) -> plus2(x,s1(y))
        plus2(s1(x),y) -> s1(plus2(minus2(x,y),double1(y)))
        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(x6),x7) -> minus2(x6,x7)
        app(minus(),x7) -> minus1(x7)
        app(s(),x7) -> s1(x7)
        app(double(),x7) -> double1(x7)
        app(plus1(x6),x7) -> plus2(x6,x7)
        app(plus(),x7) -> plus1(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