YES

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(log(),app(s(),0())) -> 0()
 app(log(),app(s(),app(s(),x))) ->
 app(s(),app(log(),app(s(),app(app(quot(),x),app(s(),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)))
  log1(s1(0())) -> 0()
  log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
  app(minus(),x7) -> minus1(x7)
  app(s(),x7) -> s1(x7)
  app(quot1(x6),x7) -> quot2(x6,x7)
  app(quot(),x7) -> quot1(x7)
  app(log(),x7) -> log1(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)
    quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
    log{1,#}(s1(s1(x))) -> quot{2,#}(x,s1(s1(0())))
    log{1,#}(s1(s1(x))) -> log{1,#}(s1(quot2(x,s1(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(x6),x7) -> minus{2,#}(x6,x7)
    app#(quot1(x6),x7) -> quot{2,#}(x6,x7)
    app#(log(),x7) -> log{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)
    quot2(0(),s1(y)) -> 0()
    quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
    log1(s1(0())) -> 0()
    log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
    app(minus(),x7) -> minus1(x7)
    app(s(),x7) -> s1(x7)
    app(quot1(x6),x7) -> quot2(x6,x7)
    app(quot(),x7) -> quot1(x7)
    app(log(),x7) -> log1(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)
     quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
     log{1,#}(s1(s1(x))) -> quot{2,#}(x,s1(s1(0())))
     log{1,#}(s1(s1(x))) -> log{1,#}(s1(quot2(x,s1(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(x6),x7) -> minus{2,#}(x6,x7)
     app#(quot1(x6),x7) -> quot{2,#}(x6,x7)
     app#(log(),x7) -> log{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)
     quot2(0(),s1(y)) -> 0()
     quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
     log1(s1(0())) -> 0()
     log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
     app(minus(),x7) -> minus1(x7)
     app(s(),x7) -> s1(x7)
     app(quot1(x6),x7) -> quot2(x6,x7)
     app(quot(),x7) -> quot1(x7)
     app(log(),x7) -> log1(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#(log(),x7) -> log{1,#}(x7)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(quot1(x6),x7) -> quot{2,#}(x6,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#(quot1(x6),x7) -> quot{2,#}(x6,x7) ->
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
     app#(quot1(x6),x7) -> quot{2,#}(x6,x7) ->
     quot{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(log(),x7) -> log{1,#}(x7) ->
     log{1,#}(s1(s1(x))) -> log{1,#}(s1(quot2(x,s1(s1(0())))))
     app#(log(),x7) -> log{1,#}(x7) ->
     log{1,#}(s1(s1(x))) -> quot{2,#}(x,s1(s1(0())))
     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#(log(),x7) -> log{1,#}(x7)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(quot1(x6),x7) -> quot{2,#}(x6,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)
     log{1,#}(s1(s1(x))) -> log{1,#}(s1(quot2(x,s1(s1(0()))))) ->
     log{1,#}(s1(s1(x))) -> log{1,#}(s1(quot2(x,s1(s1(0())))))
     log{1,#}(s1(s1(x))) -> log{1,#}(s1(quot2(x,s1(s1(0()))))) ->
     log{1,#}(s1(s1(x))) -> quot{2,#}(x,s1(s1(0())))
     log{1,#}(s1(s1(x))) -> quot{2,#}(x,s1(s1(0()))) ->
     quot{2,#}(s1(x),s1(y)) -> quot{2,#}(minus2(x,y),s1(y))
     log{1,#}(s1(s1(x))) -> quot{2,#}(x,s1(s1(0()))) ->
     quot{2,#}(s1(x),s1(y)) -> minus{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: 12
     #arcs: 39/289
     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)
      quot2(0(),s1(y)) -> 0()
      quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
      log1(s1(0())) -> 0()
      log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(quot1(x6),x7) -> quot2(x6,x7)
      app(quot(),x7) -> quot1(x7)
      app(log(),x7) -> log1(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)
        quot2(0(),s1(y)) -> 0()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
        log1(s1(0())) -> 0()
        log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
        app(minus(),x7) -> minus1(x7)
        app(s(),x7) -> s1(x7)
        app(quot1(x6),x7) -> quot2(x6,x7)
        app(quot(),x7) -> quot1(x7)
        app(log(),x7) -> log1(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:
      log{1,#}(s1(s1(x))) -> log{1,#}(s1(quot2(x,s1(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)))
      log1(s1(0())) -> 0()
      log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(quot1(x6),x7) -> quot2(x6,x7)
      app(quot(),x7) -> quot1(x7)
      app(log(),x7) -> log1(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:
       log{1,#}(s1(s1(x))) -> log{1,#}(s1(quot2(x,s1(s1(0())))))
      TRS:
       quot2(0(),s1(y)) -> 0()
       quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
       minus2(x,0()) -> x
       minus2(s1(x),s1(y)) -> minus2(x,y)
      Arctic Interpretation Processor:
       dimension: 1
       usable rules:
        quot2(0(),s1(y)) -> 0()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
       interpretation:
        [log{1,#}](x0) = x0,
        
        [quot2](x0, x1) = x0,
        
        [s1](x0) = 2x0,
        
        [minus2](x0, x1) = x0,
        
        [0] = 0
       orientation:
        log{1,#}(s1(s1(x))) = 4x >= 2x = log{1,#}(s1(quot2(x,s1(s1(0())))))
        
        quot2(0(),s1(y)) = 0 >= 0 = 0()
        
        quot2(s1(x),s1(y)) = 2x >= 2x = s1(quot2(minus2(x,y),s1(y)))
        
        minus2(x,0()) = x >= x = x
        
        minus2(s1(x),s1(y)) = 2x >= x = minus2(x,y)
       problem:
        DPs:
         
        TRS:
         quot2(0(),s1(y)) -> 0()
         quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
         minus2(x,0()) -> x
         minus2(s1(x),s1(y)) -> minus2(x,y)
       Qed
     
     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)))
      log1(s1(0())) -> 0()
      log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(quot1(x6),x7) -> quot2(x6,x7)
      app(quot(),x7) -> quot1(x7)
      app(log(),x7) -> log1(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:
       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)
      Arctic Interpretation Processor:
       dimension: 1
       usable rules:
        minus2(x,0()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
       interpretation:
        [quot{2,#}](x0, x1) = 3x0 + -16,
        
        [s1](x0) = 1x0 + -3,
        
        [minus2](x0, x1) = x0 + -11,
        
        [0] = 8
       orientation:
        quot{2,#}(s1(x),s1(y)) = 4x + 0 >= 3x + -8 = quot{2,#}(minus2(x,y),s1(y))
        
        minus2(x,0()) = x + -11 >= x = x
        
        minus2(s1(x),s1(y)) = 1x + -3 >= x + -11 = minus2(x,y)
       problem:
        DPs:
         
        TRS:
         minus2(x,0()) -> x
         minus2(s1(x),s1(y)) -> minus2(x,y)
       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)))
      log1(s1(0())) -> 0()
      log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
      app(minus(),x7) -> minus1(x7)
      app(s(),x7) -> s1(x7)
      app(quot1(x6),x7) -> quot2(x6,x7)
      app(quot(),x7) -> quot1(x7)
      app(log(),x7) -> log1(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)
        quot2(0(),s1(y)) -> 0()
        quot2(s1(x),s1(y)) -> s1(quot2(minus2(x,y),s1(y)))
        log1(s1(0())) -> 0()
        log1(s1(s1(x))) -> s1(log1(s1(quot2(x,s1(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(x6),x7) -> minus2(x6,x7)
        app(minus(),x7) -> minus1(x7)
        app(s(),x7) -> s1(x7)
        app(quot1(x6),x7) -> quot2(x6,x7)
        app(quot(),x7) -> quot1(x7)
        app(log(),x7) -> log1(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