TRS:
 {    f(x, y, z) -> g(x, y, z),
  g(0(), 1(), x) -> f(x, x, x),
             a() -> b(),
             a() -> c()}
 POP* + Boolean Semantic Labelling:
  Normal positions:
  pi(f_sl=1) = [1,2,3], pi(f_sl=0) = [1,2,3], pi(g_sl=1) = [1,3], pi(g_sl=0) = [1]
  
Safe positions:
   pi(g_sl=1) = [2], pi(g_sl=0) = [2,3]
  
Precedence:
   a_sl=0 > b_sl=1, 
   a_sl=0 > c_sl=1, 
   f_sl=1 > g_sl=1, 
   g_sl=1 > f_sl=0, 
   f_sl=0 > g_sl=0
   empty
  
Interpretation:
   g^(3):
   000 | 1
   001 | 0
   010 | 1
   011 | 0
   100 | 0
   101 | 0
   110 | 0
   111 | 0
   f^(3):
   000 | 1
   001 | 0
   010 | 1
   011 | 0
   100 | 0
   101 | 0
   110 | 0
   111 | 0
   0^(0):
    | 0
   1^(0):
    | 1
   b^(0):
    | 0
   a^(0):
    | 0
   c^(0):
    | 0
   
  
Labelling:
   g^(3):
   000 | 0
   001 | 0
   010 | 1
   011 | 1
   100 | 0
   101 | 0
   110 | 0
   111 | 0
   f^(3):
   000 | 0
   001 | 0
   010 | 1
   011 | 1
   100 | 0
   101 | 0
   110 | 0
   111 | 0
   0^(0):
    | 0
   1^(0):
    | 0
   b^(0):
    | 1
   a^(0):
    | 0
   c^(0):
    | 1
   
  
Labelled predicative System:
   {             f_sl=0(x,y,z;) -> g_sl=0(x;y,z),
                 f_sl=0(x,y,z;) -> g_sl=0(x;y,z),
                 f_sl=1(x,y,z;) -> g_sl=1(x,z;y),
                 f_sl=1(x,y,z;) -> g_sl=1(x,z;y),
                 f_sl=0(x,y,z;) -> g_sl=0(x;y,z),
                 f_sl=0(x,y,z;) -> g_sl=0(x;y,z),
                 f_sl=0(x,y,z;) -> g_sl=0(x;y,z),
                 f_sl=0(x,y,z;) -> g_sl=0(x;y,z),
    g_sl=1(0_sl=0(),x;1_sl=0()) -> f_sl=0(x,x,x;),
    g_sl=1(0_sl=0(),x;1_sl=0()) -> f_sl=0(x,x,x;),
                       a_sl=0() -> b_sl=1(),
                       a_sl=0() -> c_sl=1()}
  

  Qed