MAYBE

Problem:
 app(app(plus(),0()),y) -> y
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
 app(app(times(),0()),y) -> 0()
 app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
 app(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
 app(double(),xs) -> app(app(map(),app(times(),app(s(),app(s(),0())))),xs)
 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))

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(times(),app(s(),x)),y) -> app#(times(),x)
   app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
   app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
   app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
   app#(inc(),xs) -> app#(s(),0())
   app#(inc(),xs) -> app#(plus(),app(s(),0()))
   app#(inc(),xs) -> app#(map(),app(plus(),app(s(),0())))
   app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),xs)
   app#(double(),xs) -> app#(s(),0())
   app#(double(),xs) -> app#(s(),app(s(),0()))
   app#(double(),xs) -> app#(times(),app(s(),app(s(),0())))
   app#(double(),xs) -> app#(map(),app(times(),app(s(),app(s(),0()))))
   app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),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#(cons(),app(f,x))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),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(times(),0()),y) -> 0()
   app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
   app(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
   app(double(),xs) -> app(app(map(),app(times(),app(s(),app(s(),0())))),xs)
   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))
  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(times(),app(s(),x)),y) -> app#(times(),x)
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
    app#(inc(),xs) -> app#(s(),0())
    app#(inc(),xs) -> app#(plus(),app(s(),0()))
    app#(inc(),xs) -> app#(map(),app(plus(),app(s(),0())))
    app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),xs)
    app#(double(),xs) -> app#(s(),0())
    app#(double(),xs) -> app#(s(),app(s(),0()))
    app#(double(),xs) -> app#(times(),app(s(),app(s(),0())))
    app#(double(),xs) -> app#(map(),app(times(),app(s(),app(s(),0()))))
    app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),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#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),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(times(),0()),y) -> 0()
    app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
    app(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
    app(double(),xs) -> app(app(map(),app(times(),app(s(),app(s(),0())))),xs)
    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))
   graph:
    app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),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#(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(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))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(times(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(inc(),xs) -> app#(s(),0())
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(inc(),xs) -> app#(plus(),app(s(),0()))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(inc(),xs) -> app#(map(),app(plus(),app(s(),0())))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(double(),xs) -> app#(s(),0())
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(double(),xs) -> app#(s(),app(s(),0()))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(double(),xs) -> app#(times(),app(s(),app(s(),0())))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(double(),xs) -> app#(map(),app(times(),app(s(),app(s(),0()))))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),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(times(),app(s(),x)),y) -> app#(app(times(),x),y) ->
    app#(app(times(),app(s(),x)),y) -> app#(times(),x)
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y)
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y) ->
    app#(app(times(),app(s(),x)),y) -> app#(plus(),app(app(times(),x),y))
    app#(app(times(),app(s(),x)),y) -> app#(app(times(),x),y) ->
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y)
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(plus(),x)
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y) ->
    app#(app(plus(),app(s(),x)),y) -> app#(app(plus(),x),y)
    app#(app(times(),app(s(),x)),y) -> app#(app(plus(),app(app(times(),x),y)),y) ->
    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))
   SCC Processor:
    #sccs: 3
    #rules: 6
    #arcs: 42/400
    DPs:
     app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),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#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),xs)
    TRS:
     app(app(plus(),0()),y) -> y
     app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
     app(app(times(),0()),y) -> 0()
     app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
     app(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
     app(double(),xs) -> app(app(map(),app(times(),app(s(),app(s(),0())))),xs)
     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))
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 1
     problem:
      DPs:
       app#(double(),xs) -> app#(app(map(),app(times(),app(s(),app(s(),0())))),xs)
       app#(inc(),xs) -> app#(app(map(),app(plus(),app(s(),0()))),xs)
      TRS:
       app(app(plus(),0()),y) -> y
       app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
       app(app(times(),0()),y) -> 0()
       app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
       app(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
       app(double(),xs) -> app(app(map(),app(times(),app(s(),app(s(),0())))),xs)
       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))
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 10/4
      
    
    DPs:
     app#(app(times(),app(s(),x)),y) -> app#(app(times(),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(times(),0()),y) -> 0()
     app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
     app(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
     app(double(),xs) -> app(app(map(),app(times(),app(s(),app(s(),0())))),xs)
     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))
    Open
    
    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(times(),0()),y) -> 0()
     app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
     app(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
     app(double(),xs) -> app(app(map(),app(times(),app(s(),app(s(),0())))),xs)
     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))
    Open