TRS: {f(g(i(a(), b(), b'()), c()), d()) -> if(e(), f(.(b(), c()), d'()), f(.(b'(), c()), d'())), f(g(h(a(), b()), c()), d()) -> if(e(), f(.(b(), g(h(a(), b()), c())), d()), f(c(), d'()))} POP* + Boolean Semantic Labelling: Normal positions: pi(f_sl=1) = [1,2], pi(f_sl=0) = [1,2] Safe positions: pi(h_sl=1) = [1,2], pi(h_sl=0) = [1,2], pi(i_sl=1) = [1,2,3], pi(i_sl=0) = [1,2,3], pi(g_sl=1) = [1,2], pi(g_sl=0) = [1,2], pi(._sl=1) = [1,2], pi(._sl=0) = [1,2], pi(if_sl=1) = [1,2,3], pi(if_sl=0) = [1,2,3] Precedence: f_sl=1 > if_sl=0, f_sl=1 > f_sl=0, f_sl=1 > e_sl=1, f_sl=1 > ._sl=0, f_sl=1 > d'_sl=0 empty Interpretation: if^(3): 000 | 0 001 | 0 010 | 0 011 | 0 100 | 0 101 | 0 110 | 0 111 | 0 e^(0): | 0 f^(2): 00 | 0 01 | 0 10 | 0 11 | 0 .^(2): 00 | 0 01 | 0 10 | 0 11 | 0 b^(0): | 0 c^(0): | 0 d'^(0): | 0 b'^(0): | 0 g^(2): 00 | 1 01 | 0 10 | 0 11 | 0 i^(3): 000 | 0 001 | 0 010 | 0 011 | 0 100 | 0 101 | 0 110 | 0 111 | 0 a^(0): | 0 d^(0): | 0 h^(2): 00 | 0 01 | 0 10 | 0 11 | 0 Labelling: if^(3): 000 | 0 001 | 0 010 | 0 011 | 0 100 | 0 101 | 0 110 | 0 111 | 0 e^(0): | 1 f^(2): 00 | 0 01 | 0 10 | 1 11 | 0 .^(2): 00 | 0 01 | 0 10 | 0 11 | 0 b^(0): | 0 c^(0): | 0 d'^(0): | 0 b'^(0): | 0 g^(2): 00 | 0 01 | 0 10 | 0 11 | 0 i^(3): 000 | 0 001 | 0 010 | 0 011 | 0 100 | 0 101 | 0 110 | 0 111 | 0 a^(0): | 0 d^(0): | 0 h^(2): 00 | 0 01 | 0 10 | 0 11 | 0 Labelled predicative System: {f_sl=1(g_sl=0(;i_sl=0(;a_sl=0(),b_sl=0(),b'_sl=0()),c_sl=0()),d_sl=0();) -> if_sl=0(;e_sl=1(),f_sl=0(._sl=0(;b_sl=0(),c_sl=0()),d'_sl=0();),f_sl=0(._sl=0(;b'_sl=0(),c_sl=0()),d'_sl=0();)), f_sl=1(g_sl=0(;h_sl=0(;a_sl=0(),b_sl=0()),c_sl=0()),d_sl=0();) -> if_sl=0(;e_sl=1(),f_sl=0(._sl=0(;b_sl=0(),g_sl=0(;h_sl=0(;a_sl=0(),b_sl=0()),c_sl=0())),d_sl=0();),f_sl=0(c_sl=0(),d'_sl=0();))} Qed