TRS:
 {  sum(0()) -> 0(),
   sum(s(x)) -> +(sum(x), s(x)),
   +(x, 0()) -> x,
  +(x, s(y)) -> s(+(x, y))}
 POP* + Boolean Semantic Labelling:
  Normal positions:
  pi(+_sl=1) = [1,2], pi(+_sl=0) = [2], pi(sum_sl=1) = [1], pi(sum_sl=0) = [1]
  
Safe positions:
   pi(s_sl=1) = [1], pi(s_sl=0) = [1], pi(+_sl=0) = [1]
  
Precedence:
   +_sl=0 > s_sl=0, 
   sum_sl=0 > +_sl=0
   empty
  
Interpretation:
   0^(0):
    | 1
   sum^(1):
   0 | 0
   1 | 1
   +^(2):
   00 | 0
   01 | 0
   10 | 0
   11 | 1
   s^(1):
   0 | 0
   1 | 0
   
  
Labelling:
   0^(0):
    | 0
   sum^(1):
   0 | 0
   1 | 0
   +^(2):
   00 | 0
   01 | 0
   10 | 0
   11 | 0
   s^(1):
   0 | 0
   1 | 0
   
  
Labelled predicative System:
   {  sum_sl=0(0_sl=0();) -> 0_sl=0(),
    sum_sl=0(s_sl=0(;x);) -> +_sl=0(s_sl=0(;x);sum_sl=0(x;)),
    sum_sl=0(s_sl=0(;x);) -> +_sl=0(s_sl=0(;x);sum_sl=0(x;)),
       +_sl=0(0_sl=0();x) -> x,
       +_sl=0(0_sl=0();x) -> x,
     +_sl=0(s_sl=0(;y);x) -> s_sl=0(;+_sl=0(y;x)),
     +_sl=0(s_sl=0(;y);x) -> s_sl=0(;+_sl=0(y;x)),
     +_sl=0(s_sl=0(;y);x) -> s_sl=0(;+_sl=0(y;x)),
     +_sl=0(s_sl=0(;y);x) -> s_sl=0(;+_sl=0(y;x))}
  

  Qed