YES

Problem:
 app(f(),app(g(),x)) -> app(g(),app(g(),app(f(),x)))
 app(f(),app(g(),x)) -> app(g(),app(g(),app(g(),x)))
 app(app(map(),fun),nil()) -> nil()
 app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
 app(app(filter(),fun),nil()) -> nil()
 app(app(filter(),fun),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
 app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
 app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)

Proof:
 DP Processor:
  DPs:
   app#(f(),app(g(),x)) -> app#(f(),x)
   app#(f(),app(g(),x)) -> app#(g(),app(f(),x))
   app#(f(),app(g(),x)) -> app#(g(),app(g(),app(f(),x)))
   app#(f(),app(g(),x)) -> app#(g(),app(g(),x))
   app#(f(),app(g(),x)) -> app#(g(),app(g(),app(g(),x)))
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
   app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
   app#(app(map(),fun),app(app(cons(),x),xs)) ->
   app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
   app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
   app#(app(filter(),fun),app(app(cons(),x),xs)) ->
   app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
   app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
   app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
   app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
  TRS:
   app(f(),app(g(),x)) -> app(g(),app(g(),app(f(),x)))
   app(f(),app(g(),x)) -> app(g(),app(g(),app(g(),x)))
   app(app(map(),fun),nil()) -> nil()
   app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
   app(app(filter(),fun),nil()) -> nil()
   app(app(filter(),fun),app(app(cons(),x),xs)) ->
   app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
   app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
   app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
  EDG Processor:
   DPs:
    app#(f(),app(g(),x)) -> app#(f(),x)
    app#(f(),app(g(),x)) -> app#(g(),app(f(),x))
    app#(f(),app(g(),x)) -> app#(g(),app(g(),app(f(),x)))
    app#(f(),app(g(),x)) -> app#(g(),app(g(),x))
    app#(f(),app(g(),x)) -> app#(g(),app(g(),app(g(),x)))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
    app#(app(map(),fun),app(app(cons(),x),xs)) ->
    app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
   TRS:
    app(f(),app(g(),x)) -> app(g(),app(g(),app(f(),x)))
    app(f(),app(g(),x)) -> app(g(),app(g(),app(g(),x)))
    app(app(map(),fun),nil()) -> nil()
    app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app(app(filter(),fun),nil()) -> nil()
    app(app(filter(),fun),app(app(cons(),x),xs)) ->
    app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
    app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
   graph:
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(f(),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(f(),x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(g(),app(f(),x)))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(g(),x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(g(),app(g(),x)))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) ->
    app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) ->
    app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(f(),x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(f(),x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(g(),app(f(),x)))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(g(),x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(g(),app(g(),x)))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(cons(),app(fun,x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(map(),fun),app(app(cons(),x),xs)) ->
    app#(app(cons(),app(fun,x)),app(app(map(),fun),xs))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(filter(),fun)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(cons(),x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(cons(),x),app(app(filter(),fun),xs))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(filter(),fun)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(filter2(),app(fun,x))
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(filter2(),app(fun,x)),fun)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(fun,x)),fun),x)
    app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs) ->
    app#(app(filter(),fun),app(app(cons(),x),xs)) ->
    app#(app(app(app(filter2(),app(fun,x)),fun),x),xs)
    app#(f(),app(g(),x)) -> app#(f(),x) ->
    app#(f(),app(g(),x)) -> app#(f(),x)
    app#(f(),app(g(),x)) -> app#(f(),x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(f(),x))
    app#(f(),app(g(),x)) -> app#(f(),x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(g(),app(f(),x)))
    app#(f(),app(g(),x)) -> app#(f(),x) ->
    app#(f(),app(g(),x)) -> app#(g(),app(g(),x))
    app#(f(),app(g(),x)) -> app#(f(),x) -> app#(f(),app(g(),x)) -> app#(g(),app(g(),app(g(),x)))
   SCC Processor:
    #sccs: 2
    #rules: 6
    #arcs: 59/400
    DPs:
     app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
     app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
     app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
     app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x)
     app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(app(map(),fun),xs)
    TRS:
     app(f(),app(g(),x)) -> app(g(),app(g(),app(f(),x)))
     app(f(),app(g(),x)) -> app(g(),app(g(),app(g(),x)))
     app(app(map(),fun),nil()) -> nil()
     app(app(map(),fun),app(app(cons(),x),xs)) ->
     app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
     app(app(filter(),fun),nil()) -> nil()
     app(app(filter(),fun),app(app(cons(),x),xs)) ->
     app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
     app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
     app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 1
     problem:
      DPs:
       app#(app(app(app(filter2(),false()),fun),x),xs) -> app#(app(filter(),fun),xs)
       app#(app(app(app(filter2(),true()),fun),x),xs) -> app#(app(filter(),fun),xs)
      TRS:
       app(f(),app(g(),x)) -> app(g(),app(g(),app(f(),x)))
       app(f(),app(g(),x)) -> app(g(),app(g(),app(g(),x)))
       app(app(map(),fun),nil()) -> nil()
       app(app(map(),fun),app(app(cons(),x),xs)) ->
       app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
       app(app(filter(),fun),nil()) -> nil()
       app(app(filter(),fun),app(app(cons(),x),xs)) ->
       app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
       app(app(app(app(filter2(),true()),fun),x),xs) ->
       app(app(cons(),x),app(app(filter(),fun),xs))
       app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 14/4
      
    
    DPs:
     app#(f(),app(g(),x)) -> app#(f(),x)
    TRS:
     app(f(),app(g(),x)) -> app(g(),app(g(),app(f(),x)))
     app(f(),app(g(),x)) -> app(g(),app(g(),app(g(),x)))
     app(app(map(),fun),nil()) -> nil()
     app(app(map(),fun),app(app(cons(),x),xs)) ->
     app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
     app(app(filter(),fun),nil()) -> nil()
     app(app(filter(),fun),app(app(cons(),x),xs)) ->
     app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
     app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
     app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 1
     problem:
      DPs:
       
      TRS:
       app(f(),app(g(),x)) -> app(g(),app(g(),app(f(),x)))
       app(f(),app(g(),x)) -> app(g(),app(g(),app(g(),x)))
       app(app(map(),fun),nil()) -> nil()
       app(app(map(),fun),app(app(cons(),x),xs)) ->
       app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
       app(app(filter(),fun),nil()) -> nil()
       app(app(filter(),fun),app(app(cons(),x),xs)) ->
       app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
       app(app(app(app(filter2(),true()),fun),x),xs) ->
       app(app(cons(),x),app(app(filter(),fun),xs))
       app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)
     Qed