YES

Problem:
 f(0()) -> cons(0(),n__f(n__s(n__0())))
 f(s(0())) -> f(p(s(0())))
 p(s(0())) -> 0()
 f(X) -> n__f(X)
 s(X) -> n__s(X)
 0() -> n__0()
 activate(n__f(X)) -> f(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__0()) -> 0()
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   f#(s(0())) -> p#(s(0()))
   f#(s(0())) -> f#(p(s(0())))
   activate#(n__f(X)) -> activate#(X)
   activate#(n__f(X)) -> f#(activate(X))
   activate#(n__s(X)) -> activate#(X)
   activate#(n__s(X)) -> s#(activate(X))
   activate#(n__0()) -> 0#()
  TRS:
   f(0()) -> cons(0(),n__f(n__s(n__0())))
   f(s(0())) -> f(p(s(0())))
   p(s(0())) -> 0()
   f(X) -> n__f(X)
   s(X) -> n__s(X)
   0() -> n__0()
   activate(n__f(X)) -> f(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__0()) -> 0()
   activate(X) -> X
  Matrix Interpretation Processor: dim=1
   
   usable rules:
    f(0()) -> cons(0(),n__f(n__s(n__0())))
    f(s(0())) -> f(p(s(0())))
    p(s(0())) -> 0()
    f(X) -> n__f(X)
    s(X) -> n__s(X)
    0() -> n__0()
    activate(n__f(X)) -> f(activate(X))
    activate(n__s(X)) -> s(activate(X))
    activate(n__0()) -> 0()
    activate(X) -> X
   interpretation:
    [activate#](x0) = x0 + 2,
    
    [0#] = 0,
    
    [s#](x0) = x0,
    
    [p#](x0) = 0,
    
    [f#](x0) = x0 + 2,
    
    [activate](x0) = x0,
    
    [p](x0) = 4,
    
    [s](x0) = x0 + 2,
    
    [cons](x0, x1) = 4,
    
    [n__f](x0) = x0 + 1,
    
    [n__s](x0) = x0 + 2,
    
    [n__0] = 3,
    
    [f](x0) = x0 + 1,
    
    [0] = 3
   orientation:
    f#(s(0())) = 7 >= 0 = p#(s(0()))
    
    f#(s(0())) = 7 >= 6 = f#(p(s(0())))
    
    activate#(n__f(X)) = X + 3 >= X + 2 = activate#(X)
    
    activate#(n__f(X)) = X + 3 >= X + 2 = f#(activate(X))
    
    activate#(n__s(X)) = X + 4 >= X + 2 = activate#(X)
    
    activate#(n__s(X)) = X + 4 >= X = s#(activate(X))
    
    activate#(n__0()) = 5 >= 0 = 0#()
    
    f(0()) = 4 >= 4 = cons(0(),n__f(n__s(n__0())))
    
    f(s(0())) = 6 >= 5 = f(p(s(0())))
    
    p(s(0())) = 4 >= 3 = 0()
    
    f(X) = X + 1 >= X + 1 = n__f(X)
    
    s(X) = X + 2 >= X + 2 = n__s(X)
    
    0() = 3 >= 3 = n__0()
    
    activate(n__f(X)) = X + 1 >= X + 1 = f(activate(X))
    
    activate(n__s(X)) = X + 2 >= X + 2 = s(activate(X))
    
    activate(n__0()) = 3 >= 3 = 0()
    
    activate(X) = X >= X = X
   problem:
    DPs:
     
    TRS:
     f(0()) -> cons(0(),n__f(n__s(n__0())))
     f(s(0())) -> f(p(s(0())))
     p(s(0())) -> 0()
     f(X) -> n__f(X)
     s(X) -> n__s(X)
     0() -> n__0()
     activate(n__f(X)) -> f(activate(X))
     activate(n__s(X)) -> s(activate(X))
     activate(n__0()) -> 0()
     activate(X) -> X
   Qed