YES

Problem:
 f(s(x)) -> s(f(f(p(s(x)))))
 f(0()) -> 0()
 p(s(x)) -> x

Proof:
 DP Processor:
  DPs:
   f#(s(x)) -> p#(s(x))
   f#(s(x)) -> f#(p(s(x)))
   f#(s(x)) -> f#(f(p(s(x))))
  TRS:
   f(s(x)) -> s(f(f(p(s(x)))))
   f(0()) -> 0()
   p(s(x)) -> x
  TDG Processor:
   DPs:
    f#(s(x)) -> p#(s(x))
    f#(s(x)) -> f#(p(s(x)))
    f#(s(x)) -> f#(f(p(s(x))))
   TRS:
    f(s(x)) -> s(f(f(p(s(x)))))
    f(0()) -> 0()
    p(s(x)) -> x
   graph:
    f#(s(x)) -> f#(p(s(x))) -> f#(s(x)) -> f#(f(p(s(x))))
    f#(s(x)) -> f#(p(s(x))) -> f#(s(x)) -> f#(p(s(x)))
    f#(s(x)) -> f#(p(s(x))) -> f#(s(x)) -> p#(s(x))
    f#(s(x)) -> f#(f(p(s(x)))) -> f#(s(x)) -> f#(f(p(s(x))))
    f#(s(x)) -> f#(f(p(s(x)))) -> f#(s(x)) -> f#(p(s(x)))
    f#(s(x)) -> f#(f(p(s(x)))) -> f#(s(x)) -> p#(s(x))
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 6/9
    DPs:
     f#(s(x)) -> f#(p(s(x)))
     f#(s(x)) -> f#(f(p(s(x))))
    TRS:
     f(s(x)) -> s(f(f(p(s(x)))))
     f(0()) -> 0()
     p(s(x)) -> x
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [f#](x0) = x0 + 0,
      
      [0] = 5,
      
      [p](x0) = -4x0 + 0,
      
      [f](x0) = x0 + 0,
      
      [s](x0) = 4x0 + 4
     orientation:
      f#(s(x)) = 4x + 4 >= x + 0 = f#(p(s(x)))
      
      f#(s(x)) = 4x + 4 >= x + 0 = f#(f(p(s(x))))
      
      f(s(x)) = 4x + 4 >= 4x + 4 = s(f(f(p(s(x)))))
      
      f(0()) = 5 >= 5 = 0()
      
      p(s(x)) = x + 0 >= x = x
     problem:
      DPs:
       
      TRS:
       f(s(x)) -> s(f(f(p(s(x)))))
       f(0()) -> 0()
       p(s(x)) -> x
     Qed