YES

Problem:
 f(0()) -> cons(0(),n__f(n__s(n__0())))
 f(s(0())) -> f(p(s(0())))
 p(s(X)) -> X
 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(X)) -> X
   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=3
   
   usable rules:
    f(0()) -> cons(0(),n__f(n__s(n__0())))
    f(s(0())) -> f(p(s(0())))
    p(s(X)) -> X
    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) = [1 1 1]x0,
    
    [0#] = 0,
    
    [s#](x0) = [0],
    
    [p#](x0) = [0],
    
    [f#](x0) = [0 1 0]x0,
    
                     [1 0 1]     [0]
    [activate](x0) = [0 1 1]x0 + [1]
                     [0 0 1]     [0],
    
              [0 1 0]  
    [p](x0) = [1 0 0]x0
              [1 0 0]  ,
    
              [0 1 1]     [0]
    [s](x0) = [1 0 0]x0 + [0]
              [0 0 1]     [1],
    
                     [1 0 0]  
    [cons](x0, x1) = [0 0 0]x0
                     [0 0 0]  ,
    
                 [1 0 1]     [1]
    [n__f](x0) = [0 1 0]x0 + [1]
                 [0 0 1]     [0],
    
                 [0 1 1]     [0]
    [n__s](x0) = [1 0 0]x0 + [0]
                 [0 0 1]     [1],
    
             [1]
    [n__0] = [0]
             [0],
    
              [1 0 1]     [1]
    [f](x0) = [0 1 0]x0 + [1]
              [0 0 1]     [0],
    
          [1]
    [0] = [0]
          [0]
   orientation:
    f#(s(0())) = 1 >= 0 = p#(s(0()))
    
    f#(s(0())) = 1 >= 0 = f#(p(s(0())))
    
    activate#(n__f(X)) = [1 1 2]X + [2] >= [1 1 1]X = activate#(X)
    
    activate#(n__f(X)) = [1 1 2]X + [2] >= [0 1 1]X + [1] = f#(activate(X))
    
    activate#(n__s(X)) = [1 1 2]X + [1] >= [1 1 1]X = activate#(X)
    
    activate#(n__s(X)) = [1 1 2]X + [1] >= [0] = s#(activate(X))
    
    activate#(n__0()) = 1 >= 0 = 0#()
    
             [2]    [1]                               
    f(0()) = [1] >= [0] = cons(0(),n__f(n__s(n__0())))
             [0]    [0]                               
    
                [2]    [2]               
    f(s(0())) = [2] >= [1] = f(p(s(0())))
                [1]    [0]               
    
              [1 0 0]          
    p(s(X)) = [0 1 1]X >= X = X
              [0 1 1]          
    
           [1 0 1]    [1]    [1 0 1]    [1]          
    f(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = n__f(X)
           [0 0 1]    [0]    [0 0 1]    [0]          
    
           [0 1 1]    [0]    [0 1 1]    [0]          
    s(X) = [1 0 0]X + [0] >= [1 0 0]X + [0] = n__s(X)
           [0 0 1]    [1]    [0 0 1]    [1]          
    
          [1]    [1]         
    0() = [0] >= [0] = n__0()
          [0]    [0]         
    
                        [1 0 2]    [1]    [1 0 2]    [1]                 
    activate(n__f(X)) = [0 1 1]X + [2] >= [0 1 1]X + [2] = f(activate(X))
                        [0 0 1]    [0]    [0 0 1]    [0]                 
    
                        [0 1 2]    [1]    [0 1 2]    [1]                 
    activate(n__s(X)) = [1 0 1]X + [2] >= [1 0 1]X + [0] = s(activate(X))
                        [0 0 1]    [1]    [0 0 1]    [1]                 
    
                       [1]    [1]      
    activate(n__0()) = [1] >= [0] = 0()
                       [0]    [0]      
    
                  [1 0 1]    [0]         
    activate(X) = [0 1 1]X + [1] >= X = X
                  [0 0 1]    [0]         
   problem:
    DPs:
     
    TRS:
     f(0()) -> cons(0(),n__f(n__s(n__0())))
     f(s(0())) -> f(p(s(0())))
     p(s(X)) -> X
     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