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)
     Usable Rule Processor:
      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:
       double1(0()) -> 0()
       double1(s1(x)) -> s1(s1(double1(x)))
       minus2(x,0()) -> x
       minus2(s1(x),s1(y)) -> minus2(x,y)
      Arctic Interpretation Processor:
       dimension: 1
       usable rules:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
       interpretation:
        [plus{2,#}](x0, x1) = x0,
        
        [double1](x0) = 1x0,
        
        [s1](x0) = 2x0 + 0,
        
        [minus2](x0, x1) = 2x0 + 0,
        
        [0] = 0
       orientation:
        plus{2,#}(s1(x),y) = 2x + 0 >= x = plus{2,#}(x,y)
        
        plus{2,#}(s1(x),y) = 2x + 0 >= x = plus{2,#}(x,s1(y))
        
        plus{2,#}(s1(x),y) = 2x + 0 >= 2x + 0 = plus{2,#}(minus2(x,y),double1(y))
        
        double1(0()) = 1 >= 0 = 0()
        
        double1(s1(x)) = 3x + 1 >= 5x + 2 = s1(s1(double1(x)))
        
        minus2(x,0()) = 2x + 0 >= x = x
        
        minus2(s1(x),s1(y)) = 4x + 2 >= 2x + 0 = minus2(x,y)
       problem:
        DPs:
         plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
         plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y))
        TRS:
         double1(0()) -> 0()
         double1(s1(x)) -> s1(s1(double1(x)))
         minus2(x,0()) -> x
         minus2(s1(x),s1(y)) -> minus2(x,y)
       Restore Modifier:
        DPs:
         plus{2,#}(s1(x),y) -> plus{2,#}(x,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)
        Usable Rule Processor:
         DPs:
          plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
          plus{2,#}(s1(x),y) -> plus{2,#}(minus2(x,y),double1(y))
         TRS:
          double1(0()) -> 0()
          double1(s1(x)) -> s1(s1(double1(x)))
          minus2(x,0()) -> x
          minus2(s1(x),s1(y)) -> minus2(x,y)
         Arctic Interpretation Processor:
          dimension: 1
          usable rules:
           minus2(x,0()) -> x
           minus2(s1(x),s1(y)) -> minus2(x,y)
          interpretation:
           [plus{2,#}](x0, x1) = 4x0,
           
           [double1](x0) = x0,
           
           [s1](x0) = 3x0 + 1,
           
           [minus2](x0, x1) = x0,
           
           [0] = 0
          orientation:
           plus{2,#}(s1(x),y) = 7x + 5 >= 4x = plus{2,#}(x,y)
           
           plus{2,#}(s1(x),y) = 7x + 5 >= 4x = plus{2,#}(minus2(x,y),double1(y))
           
           double1(0()) = 0 >= 0 = 0()
           
           double1(s1(x)) = 3x + 1 >= 6x + 4 = s1(s1(double1(x)))
           
           minus2(x,0()) = x >= x = x
           
           minus2(s1(x),s1(y)) = 3x + 1 >= x = minus2(x,y)
          problem:
           DPs:
            plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
           TRS:
            double1(0()) -> 0()
            double1(s1(x)) -> s1(s1(double1(x)))
            minus2(x,0()) -> x
            minus2(s1(x),s1(y)) -> minus2(x,y)
          Restore Modifier:
           DPs:
            plus{2,#}(s1(x),y) -> plus{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(plus{2,#}) = 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:
      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