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