TRS:
 {a__f(f(a())) -> a__f(g(f(a()))),
    mark(f(X)) -> a__f(mark(X)),
     mark(a()) -> a(),
    mark(g(X)) -> g(X),
       a__f(X) -> f(X)}
 POP* + Boolean Semantic Labelling:
  Normal positions:
  pi(mark_sl=1) = [1], pi(mark_sl=0) = [1]
  
Safe positions:
   pi(f_sl=1) = [1], pi(f_sl=0) = [1], pi(g_sl=1) = [1], pi(g_sl=0) = [1], pi(a__f_sl=1) = [1], pi(a__f_sl=0) = [1]
  
Precedence:
   mark_sl=0 > a__f_sl=0, 
   mark_sl=0 > a__f_sl=1, 
   mark_sl=0 > mark_sl=1, 
   a__f_sl=1 > a__f_sl=0, 
   a__f_sl=1 > g_sl=0, 
   a__f_sl=1 > f_sl=1, 
   a__f_sl=0 > f_sl=1
   empty
  
Interpretation:
   a__f^(1):
   0 | 0
   1 | 0
   g^(1):
   0 | 1
   1 | 0
   f^(1):
   0 | 0
   1 | 0
   a^(0):
    | 0
   mark^(1):
   0 | 0
   1 | 1
   
  
Labelling:
   a__f^(1):
   0 | 1
   1 | 0
   g^(1):
   0 | 0
   1 | 0
   f^(1):
   0 | 1
   1 | 1
   a^(0):
    | 0
   mark^(1):
   0 | 0
   1 | 1
   
  
Labelled predicative System:
   {a__f_sl=1(;f_sl=1(;a_sl=0())) -> a__f_sl=0(;g_sl=0(;f_sl=1(;a_sl=0()))),
           mark_sl=0(f_sl=1(;X);) -> a__f_sl=1(;mark_sl=0(X;)),
           mark_sl=0(f_sl=1(;X);) -> a__f_sl=0(;mark_sl=1(X;)),
             mark_sl=0(a_sl=0();) -> a_sl=0(),
           mark_sl=1(g_sl=0(;X);) -> g_sl=0(;X),
           mark_sl=0(g_sl=0(;X);) -> g_sl=0(;X),
                    a__f_sl=1(;X) -> f_sl=1(;X),
                    a__f_sl=0(;X) -> f_sl=1(;X)}
  

  Qed