YES Time: 0.249605 TRS: { f(f(f(a(), x), b()), b()) -> f(f(a(), f(f(x, b()), b())), b()), f(a(), f(a(), f(a(), f(x, b())))) -> f(f(a(), f(a(), f(a(), x))), b())} DP: DP: { f#(f(f(a(), x), b()), b()) -> f#(x, b()), f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b()), f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b()), f#(f(f(a(), x), b()), b()) -> f#(a(), f(f(x, b()), b())), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(f(a(), f(a(), f(a(), x))), b()), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x)), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x)))} TRS: { f(f(f(a(), x), b()), b()) -> f(f(a(), f(f(x, b()), b())), b()), f(a(), f(a(), f(a(), f(x, b())))) -> f(f(a(), f(a(), f(a(), x))), b())} EDG: {(f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x))), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x)))) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x))), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x))) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x))), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x)) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x))), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(f(a(), f(a(), f(a(), x))), b())) (f#(f(f(a(), x), b()), b()) -> f#(x, b()), f#(f(f(a(), x), b()), b()) -> f#(a(), f(f(x, b()), b()))) (f#(f(f(a(), x), b()), b()) -> f#(x, b()), f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b())) (f#(f(f(a(), x), b()), b()) -> f#(x, b()), f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b())) (f#(f(f(a(), x), b()), b()) -> f#(x, b()), f#(f(f(a(), x), b()), b()) -> f#(x, b())) (f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b()), f#(f(f(a(), x), b()), b()) -> f#(a(), f(f(x, b()), b()))) (f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b()), f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b())) (f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b()), f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b())) (f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b()), f#(f(f(a(), x), b()), b()) -> f#(x, b())) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(f(a(), f(a(), f(a(), x))), b()), f#(f(f(a(), x), b()), b()) -> f#(x, b())) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(f(a(), f(a(), f(a(), x))), b()), f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b())) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(f(a(), f(a(), f(a(), x))), b()), f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b())) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(f(a(), f(a(), f(a(), x))), b()), f#(f(f(a(), x), b()), b()) -> f#(a(), f(f(x, b()), b()))) (f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b()), f#(f(f(a(), x), b()), b()) -> f#(x, b())) (f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b()), f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b())) (f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b()), f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b())) (f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b()), f#(f(f(a(), x), b()), b()) -> f#(a(), f(f(x, b()), b()))) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x)), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(f(a(), f(a(), f(a(), x))), b())) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x)), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x)) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x)), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x))) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x)), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x)))) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(f(a(), f(a(), f(a(), x))), b())) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x)) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x))) (f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x))))} SCCS (2): Scc: {f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x)), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x)))} Scc: {f#(f(f(a(), x), b()), b()) -> f#(x, b()), f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b()), f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b())} SCC (3): Strict: {f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x)), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x)))} Weak: { f(f(f(a(), x), b()), b()) -> f(f(a(), f(f(x, b()), b())), b()), f(a(), f(a(), f(a(), f(x, b())))) -> f(f(a(), f(a(), f(a(), x))), b())} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [f](x0, x1) = x0 + x1, [a] = 1, [b] = 0, [f#](x0, x1) = x0 + x1 + 1 Strict: f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x))) 4 + 1x >= 4 + 1x f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), x)) 4 + 1x >= 3 + 1x f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), x) 4 + 1x >= 2 + 1x Weak: EDG: {(f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x))), f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x))))} SCCS (1): Scc: {f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x)))} SCC (1): Strict: {f#(a(), f(a(), f(a(), f(x, b())))) -> f#(a(), f(a(), f(a(), x)))} Weak: { f(f(f(a(), x), b()), b()) -> f(f(a(), f(f(x, b()), b())), b()), f(a(), f(a(), f(a(), f(x, b())))) -> f(f(a(), f(a(), f(a(), x))), b())} BOUND: Bound: match(-raise)-DP-bounded by 1 Automaton: { b_1() -> 12* b_0() -> 10* a_1() -> 13* a_0() -> 9* f#_1(13, 15) -> 5* f#_0(9, 12) -> 5* f_1(15, 12) -> 13* f_1(14, 12) -> 13* f_1(13, 15) -> 13* f_1(13, 14) -> 15* f_1(13, 12) -> 15 | 14 | 13 f_1(13, 11) -> 14* f_1(13, 10) -> 14* f_1(13, 9) -> 14* f_1(13, 8) -> 14* f_1(12, 12) -> 13* f_1(11, 12) -> 13* f_1(10, 12) -> 13* f_1(9, 12) -> 13* f_1(8, 12) -> 13* f_0(12, 12) -> 8* f_0(12, 11) -> 8* f_0(12, 10) -> 12 | 8 f_0(12, 9) -> 8* f_0(12, 8) -> 8* f_0(11, 12) -> 8* f_0(11, 11) -> 8* f_0(11, 10) -> 12 | 8 f_0(11, 9) -> 8* f_0(11, 8) -> 8* f_0(10, 12) -> 8* f_0(10, 11) -> 8* f_0(10, 10) -> 8* f_0(10, 9) -> 8* f_0(10, 8) -> 8* f_0(9, 12) -> 12* f_0(9, 11) -> 12* f_0(9, 10) -> 11* f_0(9, 9) -> 11* f_0(9, 8) -> 11* f_0(8, 12) -> 8* f_0(8, 11) -> 8* f_0(8, 10) -> 8* f_0(8, 9) -> 8* f_0(8, 8) -> 8*} Strict: {} Qed SCC (3): Strict: {f#(f(f(a(), x), b()), b()) -> f#(x, b()), f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b()), f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b())} Weak: { f(f(f(a(), x), b()), b()) -> f(f(a(), f(f(x, b()), b())), b()), f(a(), f(a(), f(a(), f(x, b())))) -> f(f(a(), f(a(), f(a(), x))), b())} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [f](x0, x1) = x0 + x1, [a] = 1, [b] = 0, [f#](x0, x1) = x0 Strict: f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b()) 1 + 1x >= 1 + 1x f#(f(f(a(), x), b()), b()) -> f#(f(x, b()), b()) 1 + 1x >= 0 + 1x f#(f(f(a(), x), b()), b()) -> f#(x, b()) 1 + 1x >= 0 + 1x Weak: EDG: {(f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b()), f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b()))} SCCS (1): Scc: {f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b())} SCC (1): Strict: {f#(f(f(a(), x), b()), b()) -> f#(f(a(), f(f(x, b()), b())), b())} Weak: { f(f(f(a(), x), b()), b()) -> f(f(a(), f(f(x, b()), b())), b()), f(a(), f(a(), f(a(), f(x, b())))) -> f(f(a(), f(a(), f(a(), x))), b())} BOUND: Bound: match(-raise)-DP-bounded by 0 Automaton: { b_0() -> 11* a_0() -> 10* f#_0(14, 11) -> 5* f_0(14, 14) -> 9* f_0(14, 13) -> 9* f_0(14, 12) -> 9* f_0(14, 11) -> 13 | 12 | 9 f_0(14, 10) -> 9* f_0(14, 9) -> 9* f_0(13, 14) -> 9* f_0(13, 13) -> 9* f_0(13, 12) -> 9* f_0(13, 11) -> 13* f_0(13, 10) -> 9* f_0(13, 9) -> 9* f_0(12, 14) -> 9* f_0(12, 13) -> 9* f_0(12, 12) -> 9* f_0(12, 11) -> 13* f_0(12, 10) -> 9* f_0(12, 9) -> 9* f_0(11, 14) -> 9* f_0(11, 13) -> 9* f_0(11, 12) -> 9* f_0(11, 11) -> 12* f_0(11, 10) -> 9* f_0(11, 9) -> 9* f_0(10, 14) -> 9* f_0(10, 13) -> 14* f_0(10, 12) -> 9* f_0(10, 11) -> 12* f_0(10, 10) -> 9* f_0(10, 9) -> 9* f_0(9, 14) -> 9* f_0(9, 13) -> 9* f_0(9, 12) -> 9* f_0(9, 11) -> 12 | 9 f_0(9, 10) -> 9* f_0(9, 9) -> 9*} Strict: {} Qed