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