MAYBE

Problem:
 app(app(plus(),0()),y) -> y
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
 app(app(sumwith(),f),nil()) -> nil()
 app(app(sumwith(),f),app(app(cons(),x),xs)) -> app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))

Proof:
 DP Processor:
  DPs:
   app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
   app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
   app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
   app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs)
   app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(plus(),app(f,x))
   app#(app(sumwith(),f),app(app(cons(),x),xs)) ->
   app#(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
  TRS:
   app(app(plus(),0()),y) -> y
   app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
   app(app(sumwith(),f),nil()) -> nil()
   app(app(sumwith(),f),app(app(cons(),x),xs)) ->
   app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
  EDG Processor:
   DPs:
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(plus(),app(f,x))
    app#(app(sumwith(),f),app(app(cons(),x),xs)) ->
    app#(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
   TRS:
    app(app(plus(),0()),y) -> y
    app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
    app(app(sumwith(),f),nil()) -> nil()
    app(app(sumwith(),f),app(app(cons(),x),xs)) ->
    app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
   graph:
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs) ->
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs) ->
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs) ->
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(plus(),app(f,x))
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs) ->
    app#(app(sumwith(),f),app(app(cons(),x),xs)) ->
    app#(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
    app#(app(sumwith(),f),app(app(cons(),x),xs)) ->
    app#(app(plus(),app(f,x)),app(app(sumwith(),f),xs)) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) ->
    app#(app(plus(),app(f,x)),app(app(sumwith(),f),xs)) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) ->
    app#(app(plus(),app(f,x)),app(app(sumwith(),f),xs)) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(plus(),app(f,x))
    app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(sumwith(),f),app(app(cons(),x),xs)) ->
    app#(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 17/49
    DPs:
     app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(app(sumwith(),f),xs)
     app#(app(sumwith(),f),app(app(cons(),x),xs)) -> app#(f,x)
    TRS:
     app(app(plus(),0()),y) -> y
     app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
     app(app(sumwith(),f),nil()) -> nil()
     app(app(sumwith(),f),app(app(cons(),x),xs)) ->
     app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 1
     problem:
      DPs:
       
      TRS:
       app(app(plus(),0()),y) -> y
       app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
       app(app(sumwith(),f),nil()) -> nil()
       app(app(sumwith(),f),app(app(cons(),x),xs)) ->
       app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
     Qed
    
    DPs:
     app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    TRS:
     app(app(plus(),0()),y) -> y
     app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
     app(app(sumwith(),f),nil()) -> nil()
     app(app(sumwith(),f),app(app(cons(),x),xs)) ->
     app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))
    Open