TRS:
 {   f(x, x) -> a(),
  f(g(x), y) -> f(x, y)}
 POP* + Boolean Semantic Labelling:
  Normal positions:
  pi(f_sl=1) = [1], pi(f_sl=0) = [1,2]
  
Safe positions:
   pi(g_sl=1) = [1], pi(g_sl=0) = [1], pi(f_sl=1) = [2]
  
Precedence:
   f_sl=0 > a_sl=0, 
   f_sl=0 > f_sl=1, 
   f_sl=1 > a_sl=0
   empty
  
Interpretation:
   a^(0):
    | 0
   f^(2):
   00 | 0
   01 | 0
   10 | 0
   11 | 0
   g^(1):
   0 | 1
   1 | 1
   
  
Labelling:
   a^(0):
    | 0
   f^(2):
   00 | 1
   01 | 1
   10 | 0
   11 | 0
   g^(1):
   0 | 0
   1 | 0
   
  
Labelled predicative System:
   {          f_sl=1(x;x) -> a_sl=0(),
             f_sl=0(x,x;) -> a_sl=0(),
    f_sl=0(g_sl=0(;x),y;) -> f_sl=1(x;y),
    f_sl=0(g_sl=0(;x),y;) -> f_sl=1(x;y),
    f_sl=0(g_sl=0(;x),y;) -> f_sl=0(x,y;),
    f_sl=0(g_sl=0(;x),y;) -> f_sl=0(x,y;)}
  

  Qed