MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(0()) -> 0() , f(s(0())) -> s(0()) , f(s(s(x))) -> p(h(g(x))) , f(s(s(x))) -> +(p(g(x)), q(g(x))) , p(pair(x, y)) -> x , h(x) -> pair(+(p(x), q(x)), p(x)) , g(0()) -> pair(s(0()), s(0())) , g(s(x)) -> h(g(x)) , g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , q(pair(x, y)) -> y } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , f^#(s(s(x))) -> c_3(p^#(h(g(x))), h^#(g(x)), g^#(x)) , f^#(s(s(x))) -> c_4(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x)) , p^#(pair(x, y)) -> c_5() , h^#(x) -> c_6(+^#(p(x), q(x)), p^#(x), q^#(x), p^#(x)) , g^#(0()) -> c_7() , g^#(s(x)) -> c_8(h^#(g(x)), g^#(x)) , g^#(s(x)) -> c_9(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x), p^#(g(x)), g^#(x)) , +^#(x, 0()) -> c_10() , +^#(x, s(y)) -> c_11(+^#(x, y)) , q^#(pair(x, y)) -> c_12() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , f^#(s(s(x))) -> c_3(p^#(h(g(x))), h^#(g(x)), g^#(x)) , f^#(s(s(x))) -> c_4(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x)) , p^#(pair(x, y)) -> c_5() , h^#(x) -> c_6(+^#(p(x), q(x)), p^#(x), q^#(x), p^#(x)) , g^#(0()) -> c_7() , g^#(s(x)) -> c_8(h^#(g(x)), g^#(x)) , g^#(s(x)) -> c_9(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x), p^#(g(x)), g^#(x)) , +^#(x, 0()) -> c_10() , +^#(x, s(y)) -> c_11(+^#(x, y)) , q^#(pair(x, y)) -> c_12() } Weak Trs: { f(0()) -> 0() , f(s(0())) -> s(0()) , f(s(s(x))) -> p(h(g(x))) , f(s(s(x))) -> +(p(g(x)), q(g(x))) , p(pair(x, y)) -> x , h(x) -> pair(+(p(x), q(x)), p(x)) , g(0()) -> pair(s(0()), s(0())) , g(s(x)) -> h(g(x)) , g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , q(pair(x, y)) -> y } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,5,7,10,12} by applications of Pre({1,2,5,7,10,12}) = {3,4,6,8,9,11}. Here rules are labeled as follows: DPs: { 1: f^#(0()) -> c_1() , 2: f^#(s(0())) -> c_2() , 3: f^#(s(s(x))) -> c_3(p^#(h(g(x))), h^#(g(x)), g^#(x)) , 4: f^#(s(s(x))) -> c_4(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x)) , 5: p^#(pair(x, y)) -> c_5() , 6: h^#(x) -> c_6(+^#(p(x), q(x)), p^#(x), q^#(x), p^#(x)) , 7: g^#(0()) -> c_7() , 8: g^#(s(x)) -> c_8(h^#(g(x)), g^#(x)) , 9: g^#(s(x)) -> c_9(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x), p^#(g(x)), g^#(x)) , 10: +^#(x, 0()) -> c_10() , 11: +^#(x, s(y)) -> c_11(+^#(x, y)) , 12: q^#(pair(x, y)) -> c_12() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(s(x))) -> c_3(p^#(h(g(x))), h^#(g(x)), g^#(x)) , f^#(s(s(x))) -> c_4(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x)) , h^#(x) -> c_6(+^#(p(x), q(x)), p^#(x), q^#(x), p^#(x)) , g^#(s(x)) -> c_8(h^#(g(x)), g^#(x)) , g^#(s(x)) -> c_9(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x), p^#(g(x)), g^#(x)) , +^#(x, s(y)) -> c_11(+^#(x, y)) } Weak DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , p^#(pair(x, y)) -> c_5() , g^#(0()) -> c_7() , +^#(x, 0()) -> c_10() , q^#(pair(x, y)) -> c_12() } Weak Trs: { f(0()) -> 0() , f(s(0())) -> s(0()) , f(s(s(x))) -> p(h(g(x))) , f(s(s(x))) -> +(p(g(x)), q(g(x))) , p(pair(x, y)) -> x , h(x) -> pair(+(p(x), q(x)), p(x)) , g(0()) -> pair(s(0()), s(0())) , g(s(x)) -> h(g(x)) , g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , q(pair(x, y)) -> y } Obligation: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , p^#(pair(x, y)) -> c_5() , g^#(0()) -> c_7() , +^#(x, 0()) -> c_10() , q^#(pair(x, y)) -> c_12() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(s(x))) -> c_3(p^#(h(g(x))), h^#(g(x)), g^#(x)) , f^#(s(s(x))) -> c_4(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x)) , h^#(x) -> c_6(+^#(p(x), q(x)), p^#(x), q^#(x), p^#(x)) , g^#(s(x)) -> c_8(h^#(g(x)), g^#(x)) , g^#(s(x)) -> c_9(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x), p^#(g(x)), g^#(x)) , +^#(x, s(y)) -> c_11(+^#(x, y)) } Weak Trs: { f(0()) -> 0() , f(s(0())) -> s(0()) , f(s(s(x))) -> p(h(g(x))) , f(s(s(x))) -> +(p(g(x)), q(g(x))) , p(pair(x, y)) -> x , h(x) -> pair(+(p(x), q(x)), p(x)) , g(0()) -> pair(s(0()), s(0())) , g(s(x)) -> h(g(x)) , g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , q(pair(x, y)) -> y } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { f^#(s(s(x))) -> c_3(p^#(h(g(x))), h^#(g(x)), g^#(x)) , f^#(s(s(x))) -> c_4(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x)) , h^#(x) -> c_6(+^#(p(x), q(x)), p^#(x), q^#(x), p^#(x)) , g^#(s(x)) -> c_9(+^#(p(g(x)), q(g(x))), p^#(g(x)), g^#(x), q^#(g(x)), g^#(x), p^#(g(x)), g^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(s(x))) -> c_1(h^#(g(x)), g^#(x)) , f^#(s(s(x))) -> c_2(+^#(p(g(x)), q(g(x))), g^#(x), g^#(x)) , h^#(x) -> c_3(+^#(p(x), q(x))) , g^#(s(x)) -> c_4(h^#(g(x)), g^#(x)) , g^#(s(x)) -> c_5(+^#(p(g(x)), q(g(x))), g^#(x), g^#(x), g^#(x)) , +^#(x, s(y)) -> c_6(+^#(x, y)) } Weak Trs: { f(0()) -> 0() , f(s(0())) -> s(0()) , f(s(s(x))) -> p(h(g(x))) , f(s(s(x))) -> +(p(g(x)), q(g(x))) , p(pair(x, y)) -> x , h(x) -> pair(+(p(x), q(x)), p(x)) , g(0()) -> pair(s(0()), s(0())) , g(s(x)) -> h(g(x)) , g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , q(pair(x, y)) -> y } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { p(pair(x, y)) -> x , h(x) -> pair(+(p(x), q(x)), p(x)) , g(0()) -> pair(s(0()), s(0())) , g(s(x)) -> h(g(x)) , g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , q(pair(x, y)) -> y } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(s(x))) -> c_1(h^#(g(x)), g^#(x)) , f^#(s(s(x))) -> c_2(+^#(p(g(x)), q(g(x))), g^#(x), g^#(x)) , h^#(x) -> c_3(+^#(p(x), q(x))) , g^#(s(x)) -> c_4(h^#(g(x)), g^#(x)) , g^#(s(x)) -> c_5(+^#(p(g(x)), q(g(x))), g^#(x), g^#(x), g^#(x)) , +^#(x, s(y)) -> c_6(+^#(x, y)) } Weak Trs: { p(pair(x, y)) -> x , h(x) -> pair(+(p(x), q(x)), p(x)) , g(0()) -> pair(s(0()), s(0())) , g(s(x)) -> h(g(x)) , g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , q(pair(x, y)) -> y } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..