MAYBE

Problem:
 f(X) -> cons(X,n__f(g(X)))
 g(0()) -> s(0())
 g(s(X)) -> s(s(g(X)))
 sel(0(),cons(X,Y)) -> X
 sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
 f(X) -> n__f(X)
 activate(n__f(X)) -> f(X)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   f#(X) -> g#(X)
   g#(s(X)) -> g#(X)
   sel#(s(X),cons(Y,Z)) -> activate#(Z)
   sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
   activate#(n__f(X)) -> f#(X)
  TRS:
   f(X) -> cons(X,n__f(g(X)))
   g(0()) -> s(0())
   g(s(X)) -> s(s(g(X)))
   sel(0(),cons(X,Y)) -> X
   sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X
  Usable Rule Processor:
   DPs:
    f#(X) -> g#(X)
    g#(s(X)) -> g#(X)
    sel#(s(X),cons(Y,Z)) -> activate#(Z)
    sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
    activate#(n__f(X)) -> f#(X)
   TRS:
    f12(x,y) -> x
    f12(x,y) -> y
    activate(n__f(X)) -> f(X)
    activate(X) -> X
    f(X) -> cons(X,n__f(g(X)))
    f(X) -> n__f(X)
    g(0()) -> s(0())
    g(s(X)) -> s(s(g(X)))
   EDG Processor:
    DPs:
     f#(X) -> g#(X)
     g#(s(X)) -> g#(X)
     sel#(s(X),cons(Y,Z)) -> activate#(Z)
     sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
     activate#(n__f(X)) -> f#(X)
    TRS:
     f12(x,y) -> x
     f12(x,y) -> y
     activate(n__f(X)) -> f(X)
     activate(X) -> X
     f(X) -> cons(X,n__f(g(X)))
     f(X) -> n__f(X)
     g(0()) -> s(0())
     g(s(X)) -> s(s(g(X)))
    graph:
     activate#(n__f(X)) -> f#(X) -> f#(X) -> g#(X)
     sel#(s(X),cons(Y,Z)) -> activate#(Z) ->
     activate#(n__f(X)) -> f#(X)
     sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) ->
     sel#(s(X),cons(Y,Z)) -> activate#(Z)
     sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) ->
     sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
     g#(s(X)) -> g#(X) -> g#(s(X)) -> g#(X)
     f#(X) -> g#(X) -> g#(s(X)) -> g#(X)
    Restore Modifier:
     DPs:
      f#(X) -> g#(X)
      g#(s(X)) -> g#(X)
      sel#(s(X),cons(Y,Z)) -> activate#(Z)
      sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
      activate#(n__f(X)) -> f#(X)
     TRS:
      f(X) -> cons(X,n__f(g(X)))
      g(0()) -> s(0())
      g(s(X)) -> s(s(g(X)))
      sel(0(),cons(X,Y)) -> X
      sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
      f(X) -> n__f(X)
      activate(n__f(X)) -> f(X)
      activate(X) -> X
     SCC Processor:
      #sccs: 2
      #rules: 2
      #arcs: 6/25
      DPs:
       sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
      TRS:
       f(X) -> cons(X,n__f(g(X)))
       g(0()) -> s(0())
       g(s(X)) -> s(s(g(X)))
       sel(0(),cons(X,Y)) -> X
       sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
       f(X) -> n__f(X)
       activate(n__f(X)) -> f(X)
       activate(X) -> X
      Open
      
      DPs:
       g#(s(X)) -> g#(X)
      TRS:
       f(X) -> cons(X,n__f(g(X)))
       g(0()) -> s(0())
       g(s(X)) -> s(s(g(X)))
       sel(0(),cons(X,Y)) -> X
       sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
       f(X) -> n__f(X)
       activate(n__f(X)) -> f(X)
       activate(X) -> X
      Open