MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__h(X) -> a__g(mark(X), X) , a__h(X) -> h(X) , a__g(X1, X2) -> g(X1, X2) , a__g(a(), X) -> a__f(b(), X) , mark(a()) -> a__a() , mark(b()) -> b() , mark(h(X)) -> a__h(mark(X)) , mark(g(X1, X2)) -> a__g(mark(X1), X2) , mark(f(X1, X2)) -> a__f(mark(X1), X2) , a__f(X, X) -> a__h(a__a()) , a__f(X1, X2) -> f(X1, X2) , a__a() -> a() , a__a() -> b() } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { a__h^#(X) -> c_1(a__g^#(mark(X), X), mark^#(X)) , a__h^#(X) -> c_2() , a__g^#(X1, X2) -> c_3() , a__g^#(a(), X) -> c_4(a__f^#(b(), X)) , mark^#(a()) -> c_5(a__a^#()) , mark^#(b()) -> c_6() , mark^#(h(X)) -> c_7(a__h^#(mark(X)), mark^#(X)) , mark^#(g(X1, X2)) -> c_8(a__g^#(mark(X1), X2), mark^#(X1)) , mark^#(f(X1, X2)) -> c_9(a__f^#(mark(X1), X2), mark^#(X1)) , a__f^#(X, X) -> c_10(a__h^#(a__a()), a__a^#()) , a__f^#(X1, X2) -> c_11() , a__a^#() -> c_12() , a__a^#() -> c_13() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__h^#(X) -> c_1(a__g^#(mark(X), X), mark^#(X)) , a__h^#(X) -> c_2() , a__g^#(X1, X2) -> c_3() , a__g^#(a(), X) -> c_4(a__f^#(b(), X)) , mark^#(a()) -> c_5(a__a^#()) , mark^#(b()) -> c_6() , mark^#(h(X)) -> c_7(a__h^#(mark(X)), mark^#(X)) , mark^#(g(X1, X2)) -> c_8(a__g^#(mark(X1), X2), mark^#(X1)) , mark^#(f(X1, X2)) -> c_9(a__f^#(mark(X1), X2), mark^#(X1)) , a__f^#(X, X) -> c_10(a__h^#(a__a()), a__a^#()) , a__f^#(X1, X2) -> c_11() , a__a^#() -> c_12() , a__a^#() -> c_13() } Weak Trs: { a__h(X) -> a__g(mark(X), X) , a__h(X) -> h(X) , a__g(X1, X2) -> g(X1, X2) , a__g(a(), X) -> a__f(b(), X) , mark(a()) -> a__a() , mark(b()) -> b() , mark(h(X)) -> a__h(mark(X)) , mark(g(X1, X2)) -> a__g(mark(X1), X2) , mark(f(X1, X2)) -> a__f(mark(X1), X2) , a__f(X, X) -> a__h(a__a()) , a__f(X1, X2) -> f(X1, X2) , a__a() -> a() , a__a() -> b() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {2,3,6,11,12,13} by applications of Pre({2,3,6,11,12,13}) = {1,4,5,7,8,9,10}. Here rules are labeled as follows: DPs: { 1: a__h^#(X) -> c_1(a__g^#(mark(X), X), mark^#(X)) , 2: a__h^#(X) -> c_2() , 3: a__g^#(X1, X2) -> c_3() , 4: a__g^#(a(), X) -> c_4(a__f^#(b(), X)) , 5: mark^#(a()) -> c_5(a__a^#()) , 6: mark^#(b()) -> c_6() , 7: mark^#(h(X)) -> c_7(a__h^#(mark(X)), mark^#(X)) , 8: mark^#(g(X1, X2)) -> c_8(a__g^#(mark(X1), X2), mark^#(X1)) , 9: mark^#(f(X1, X2)) -> c_9(a__f^#(mark(X1), X2), mark^#(X1)) , 10: a__f^#(X, X) -> c_10(a__h^#(a__a()), a__a^#()) , 11: a__f^#(X1, X2) -> c_11() , 12: a__a^#() -> c_12() , 13: a__a^#() -> c_13() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__h^#(X) -> c_1(a__g^#(mark(X), X), mark^#(X)) , a__g^#(a(), X) -> c_4(a__f^#(b(), X)) , mark^#(a()) -> c_5(a__a^#()) , mark^#(h(X)) -> c_7(a__h^#(mark(X)), mark^#(X)) , mark^#(g(X1, X2)) -> c_8(a__g^#(mark(X1), X2), mark^#(X1)) , mark^#(f(X1, X2)) -> c_9(a__f^#(mark(X1), X2), mark^#(X1)) , a__f^#(X, X) -> c_10(a__h^#(a__a()), a__a^#()) } Weak DPs: { a__h^#(X) -> c_2() , a__g^#(X1, X2) -> c_3() , mark^#(b()) -> c_6() , a__f^#(X1, X2) -> c_11() , a__a^#() -> c_12() , a__a^#() -> c_13() } Weak Trs: { a__h(X) -> a__g(mark(X), X) , a__h(X) -> h(X) , a__g(X1, X2) -> g(X1, X2) , a__g(a(), X) -> a__f(b(), X) , mark(a()) -> a__a() , mark(b()) -> b() , mark(h(X)) -> a__h(mark(X)) , mark(g(X1, X2)) -> a__g(mark(X1), X2) , mark(f(X1, X2)) -> a__f(mark(X1), X2) , a__f(X, X) -> a__h(a__a()) , a__f(X1, X2) -> f(X1, X2) , a__a() -> a() , a__a() -> b() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {3} by applications of Pre({3}) = {1,4,5,6}. Here rules are labeled as follows: DPs: { 1: a__h^#(X) -> c_1(a__g^#(mark(X), X), mark^#(X)) , 2: a__g^#(a(), X) -> c_4(a__f^#(b(), X)) , 3: mark^#(a()) -> c_5(a__a^#()) , 4: mark^#(h(X)) -> c_7(a__h^#(mark(X)), mark^#(X)) , 5: mark^#(g(X1, X2)) -> c_8(a__g^#(mark(X1), X2), mark^#(X1)) , 6: mark^#(f(X1, X2)) -> c_9(a__f^#(mark(X1), X2), mark^#(X1)) , 7: a__f^#(X, X) -> c_10(a__h^#(a__a()), a__a^#()) , 8: a__h^#(X) -> c_2() , 9: a__g^#(X1, X2) -> c_3() , 10: mark^#(b()) -> c_6() , 11: a__f^#(X1, X2) -> c_11() , 12: a__a^#() -> c_12() , 13: a__a^#() -> c_13() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__h^#(X) -> c_1(a__g^#(mark(X), X), mark^#(X)) , a__g^#(a(), X) -> c_4(a__f^#(b(), X)) , mark^#(h(X)) -> c_7(a__h^#(mark(X)), mark^#(X)) , mark^#(g(X1, X2)) -> c_8(a__g^#(mark(X1), X2), mark^#(X1)) , mark^#(f(X1, X2)) -> c_9(a__f^#(mark(X1), X2), mark^#(X1)) , a__f^#(X, X) -> c_10(a__h^#(a__a()), a__a^#()) } Weak DPs: { a__h^#(X) -> c_2() , a__g^#(X1, X2) -> c_3() , mark^#(a()) -> c_5(a__a^#()) , mark^#(b()) -> c_6() , a__f^#(X1, X2) -> c_11() , a__a^#() -> c_12() , a__a^#() -> c_13() } Weak Trs: { a__h(X) -> a__g(mark(X), X) , a__h(X) -> h(X) , a__g(X1, X2) -> g(X1, X2) , a__g(a(), X) -> a__f(b(), X) , mark(a()) -> a__a() , mark(b()) -> b() , mark(h(X)) -> a__h(mark(X)) , mark(g(X1, X2)) -> a__g(mark(X1), X2) , mark(f(X1, X2)) -> a__f(mark(X1), X2) , a__f(X, X) -> a__h(a__a()) , a__f(X1, X2) -> f(X1, X2) , a__a() -> a() , a__a() -> b() } 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. { a__h^#(X) -> c_2() , a__g^#(X1, X2) -> c_3() , mark^#(a()) -> c_5(a__a^#()) , mark^#(b()) -> c_6() , a__f^#(X1, X2) -> c_11() , a__a^#() -> c_12() , a__a^#() -> c_13() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__h^#(X) -> c_1(a__g^#(mark(X), X), mark^#(X)) , a__g^#(a(), X) -> c_4(a__f^#(b(), X)) , mark^#(h(X)) -> c_7(a__h^#(mark(X)), mark^#(X)) , mark^#(g(X1, X2)) -> c_8(a__g^#(mark(X1), X2), mark^#(X1)) , mark^#(f(X1, X2)) -> c_9(a__f^#(mark(X1), X2), mark^#(X1)) , a__f^#(X, X) -> c_10(a__h^#(a__a()), a__a^#()) } Weak Trs: { a__h(X) -> a__g(mark(X), X) , a__h(X) -> h(X) , a__g(X1, X2) -> g(X1, X2) , a__g(a(), X) -> a__f(b(), X) , mark(a()) -> a__a() , mark(b()) -> b() , mark(h(X)) -> a__h(mark(X)) , mark(g(X1, X2)) -> a__g(mark(X1), X2) , mark(f(X1, X2)) -> a__f(mark(X1), X2) , a__f(X, X) -> a__h(a__a()) , a__f(X1, X2) -> f(X1, X2) , a__a() -> a() , a__a() -> b() } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { a__f^#(X, X) -> c_10(a__h^#(a__a()), a__a^#()) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__h^#(X) -> c_1(a__g^#(mark(X), X), mark^#(X)) , a__g^#(a(), X) -> c_2(a__f^#(b(), X)) , mark^#(h(X)) -> c_3(a__h^#(mark(X)), mark^#(X)) , mark^#(g(X1, X2)) -> c_4(a__g^#(mark(X1), X2), mark^#(X1)) , mark^#(f(X1, X2)) -> c_5(a__f^#(mark(X1), X2), mark^#(X1)) , a__f^#(X, X) -> c_6(a__h^#(a__a())) } Weak Trs: { a__h(X) -> a__g(mark(X), X) , a__h(X) -> h(X) , a__g(X1, X2) -> g(X1, X2) , a__g(a(), X) -> a__f(b(), X) , mark(a()) -> a__a() , mark(b()) -> b() , mark(h(X)) -> a__h(mark(X)) , mark(g(X1, X2)) -> a__g(mark(X1), X2) , mark(f(X1, X2)) -> a__f(mark(X1), X2) , a__f(X, X) -> a__h(a__a()) , a__f(X1, X2) -> f(X1, X2) , a__a() -> a() , a__a() -> b() } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..