MAYBE

Problem:
 app(f(),app(s(),x)) -> app(f(),x)
 app(g(),app(app(cons(),0()),y)) -> app(g(),y)
 app(g(),app(app(cons(),app(s(),x)),y)) -> app(s(),x)
 app(h(),app(app(cons(),x),y)) -> app(h(),app(g(),app(app(cons(),x),y)))
 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(s(),x)) -> app#(f(),x)
   app#(g(),app(app(cons(),0()),y)) -> app#(g(),y)
   app#(h(),app(app(cons(),x),y)) -> app#(g(),app(app(cons(),x),y))
   app#(h(),app(app(cons(),x),y)) -> app#(h(),app(g(),app(app(cons(),x),y)))
   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(s(),x)) -> app(f(),x)
   app(g(),app(app(cons(),0()),y)) -> app(g(),y)
   app(g(),app(app(cons(),app(s(),x)),y)) -> app(s(),x)
   app(h(),app(app(cons(),x),y)) -> app(h(),app(g(),app(app(cons(),x),y)))
   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(s(),x)) -> app#(f(),x)
    app#(g(),app(app(cons(),0()),y)) -> app#(g(),y)
    app#(h(),app(app(cons(),x),y)) -> app#(g(),app(app(cons(),x),y))
    app#(h(),app(app(cons(),x),y)) -> app#(h(),app(g(),app(app(cons(),x),y)))
    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(s(),x)) -> app(f(),x)
    app(g(),app(app(cons(),0()),y)) -> app(g(),y)
    app(g(),app(app(cons(),app(s(),x)),y)) -> app(s(),x)
    app(h(),app(app(cons(),x),y)) -> app(h(),app(g(),app(app(cons(),x),y)))
    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#(h(),app(app(cons(),x),y)) -> app#(h(),app(g(),app(app(cons(),x),y))) ->
    app#(h(),app(app(cons(),x),y)) -> app#(g(),app(app(cons(),x),y))
    app#(h(),app(app(cons(),x),y)) -> app#(h(),app(g(),app(app(cons(),x),y))) ->
    app#(h(),app(app(cons(),x),y)) -> app#(h(),app(g(),app(app(cons(),x),y)))
    app#(h(),app(app(cons(),x),y)) -> app#(g(),app(app(cons(),x),y)) ->
    app#(g(),app(app(cons(),0()),y)) -> app#(g(),y)
    app#(g(),app(app(cons(),0()),y)) -> app#(g(),y) ->
    app#(g(),app(app(cons(),0()),y)) -> app#(g(),y)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(f(),app(s(),x)) -> app#(f(),x)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(g(),app(app(cons(),0()),y)) -> app#(g(),y)
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(h(),app(app(cons(),x),y)) -> app#(g(),app(app(cons(),x),y))
    app#(app(filter(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(h(),app(app(cons(),x),y)) -> app#(h(),app(g(),app(app(cons(),x),y)))
    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(s(),x)) -> app#(f(),x)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(g(),app(app(cons(),0()),y)) -> app#(g(),y)
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(h(),app(app(cons(),x),y)) -> app#(g(),app(app(cons(),x),y))
    app#(app(map(),fun),app(app(cons(),x),xs)) -> app#(fun,x) ->
    app#(h(),app(app(cons(),x),y)) -> app#(h(),app(g(),app(app(cons(),x),y)))
    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(s(),x)) -> app#(f(),x) -> app#(f(),app(s(),x)) -> app#(f(),x)
   SCC Processor:
    #sccs: 4
    #rules: 8
    #arcs: 57/361
    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(s(),x)) -> app(f(),x)
     app(g(),app(app(cons(),0()),y)) -> app(g(),y)
     app(g(),app(app(cons(),app(s(),x)),y)) -> app(s(),x)
     app(h(),app(app(cons(),x),y)) -> app(h(),app(g(),app(app(cons(),x),y)))
     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)
    Open
    
    DPs:
     app#(f(),app(s(),x)) -> app#(f(),x)
    TRS:
     app(f(),app(s(),x)) -> app(f(),x)
     app(g(),app(app(cons(),0()),y)) -> app(g(),y)
     app(g(),app(app(cons(),app(s(),x)),y)) -> app(s(),x)
     app(h(),app(app(cons(),x),y)) -> app(h(),app(g(),app(app(cons(),x),y)))
     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)
    Open
    
    DPs:
     app#(h(),app(app(cons(),x),y)) -> app#(h(),app(g(),app(app(cons(),x),y)))
    TRS:
     app(f(),app(s(),x)) -> app(f(),x)
     app(g(),app(app(cons(),0()),y)) -> app(g(),y)
     app(g(),app(app(cons(),app(s(),x)),y)) -> app(s(),x)
     app(h(),app(app(cons(),x),y)) -> app(h(),app(g(),app(app(cons(),x),y)))
     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)
    Open
    
    DPs:
     app#(g(),app(app(cons(),0()),y)) -> app#(g(),y)
    TRS:
     app(f(),app(s(),x)) -> app(f(),x)
     app(g(),app(app(cons(),0()),y)) -> app(g(),y)
     app(g(),app(app(cons(),app(s(),x)),y)) -> app(s(),x)
     app(h(),app(app(cons(),x),y)) -> app(h(),app(g(),app(app(cons(),x),y)))
     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)
    Open