MAYBE

Problem:
 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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
 app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
 app(app(plus(),0()),x) -> 0()
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))

Proof:
 DP Processor:
  DPs:
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
   app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs)
   app#(sum(),app(app(cons(),x),xs)) -> app#(plus(),x)
   app#(sum(),app(app(cons(),x),xs)) -> app#(app(plus(),x),app(sum(),xs))
   app#(size(),app(app(node(),x),xs)) -> app#(map(),size())
   app#(size(),app(app(node(),x),xs)) -> app#(app(map(),size()),xs)
   app#(size(),app(app(node(),x),xs)) -> app#(sum(),app(app(map(),size()),xs))
   app#(size(),app(app(node(),x),xs)) -> app#(s(),app(sum(),app(app(map(),size()),xs)))
   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))
  TRS:
   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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
   app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
   app(app(plus(),0()),x) -> 0()
   app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
  EDG Processor:
   DPs:
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs)
    app#(sum(),app(app(cons(),x),xs)) -> app#(plus(),x)
    app#(sum(),app(app(cons(),x),xs)) -> app#(app(plus(),x),app(sum(),xs))
    app#(size(),app(app(node(),x),xs)) -> app#(map(),size())
    app#(size(),app(app(node(),x),xs)) -> app#(app(map(),size()),xs)
    app#(size(),app(app(node(),x),xs)) -> app#(sum(),app(app(map(),size()),xs))
    app#(size(),app(app(node(),x),xs)) -> app#(s(),app(sum(),app(app(map(),size()),xs)))
    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))
   TRS:
    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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
    app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
    app(app(plus(),0()),x) -> 0()
    app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
   graph:
    app#(size(),app(app(node(),x),xs)) -> app#(sum(),app(app(map(),size()),xs)) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs)
    app#(size(),app(app(node(),x),xs)) -> app#(sum(),app(app(map(),size()),xs)) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(plus(),x)
    app#(size(),app(app(node(),x),xs)) -> app#(sum(),app(app(map(),size()),xs)) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(app(plus(),x),app(sum(),xs))
    app#(size(),app(app(node(),x),xs)) -> app#(app(map(),size()),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(size(),app(app(node(),x),xs)) -> app#(app(map(),size()),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(size(),app(app(node(),x),xs)) -> app#(app(map(),size()),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(size(),app(app(node(),x),xs)) -> app#(app(map(),size()),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs)
    app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(plus(),x)
    app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(app(plus(),x),app(sum(),xs))
    app#(sum(),app(app(cons(),x),xs)) -> app#(app(plus(),x),app(sum(),xs)) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(sum(),app(app(cons(),x),xs)) -> app#(app(plus(),x),app(sum(),xs)) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(sum(),app(app(cons(),x),xs)) -> app#(app(plus(),x),app(sum(),xs)) ->
    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),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))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(plus(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(sum(),app(app(cons(),x),xs)) -> app#(app(plus(),x),app(sum(),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(size(),app(app(node(),x),xs)) -> app#(map(),size())
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(size(),app(app(node(),x),xs)) -> app#(app(map(),size()),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(size(),app(app(node(),x),xs)) -> app#(sum(),app(app(map(),size()),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(size(),app(app(node(),x),xs)) -> app#(s(),app(sum(),app(app(map(),size()),xs)))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(plus(),app(s(),x)),y) -> app#(s(),app(app(plus(),x),y))
   SCC Processor:
    #sccs: 3
    #rules: 5
    #arcs: 34/196
    DPs:
     app#(size(),app(app(node(),x),xs)) -> app#(app(map(),size()),xs)
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    TRS:
     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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
     app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
     app(app(plus(),0()),x) -> 0()
     app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 1
     problem:
      DPs:
       
      TRS:
       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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
       app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
       app(app(plus(),0()),x) -> 0()
       app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
     Qed
    
    DPs:
     app#(sum(),app(app(cons(),x),xs)) -> app#(sum(),xs)
    TRS:
     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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
     app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
     app(app(plus(),0()),x) -> 0()
     app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 1
     problem:
      DPs:
       
      TRS:
       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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
       app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
       app(app(plus(),0()),x) -> 0()
       app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
     Qed
    
    DPs:
     app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    TRS:
     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(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
     app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
     app(app(plus(),0()),x) -> 0()
     app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
    Open