MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(x, 0(), 0()) -> s(x) , f(0(), y, 0()) -> s(y) , f(0(), 0(), z) -> s(z) , f(0(), s(0()), s(0())) -> s(s(0())) , f(0(), s(0()), s(s(z))) -> f(0(), s(0()), z) , f(0(), s(s(y)), s(0())) -> f(0(), y, s(0())) , f(0(), s(s(y)), s(s(z))) -> f(0(), y, f(0(), s(s(y)), s(z))) , f(s(x), 0(), s(z)) -> f(x, s(0()), z) , f(s(x), s(y), 0()) -> f(x, y, s(0())) , f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) , f(s(0()), y, z) -> f(0(), s(y), s(z)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { f^#(x, 0(), 0()) -> c_1(x) , f^#(0(), y, 0()) -> c_2(y) , f^#(0(), 0(), z) -> c_3(z) , f^#(0(), s(0()), s(0())) -> c_4() , f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z)) , f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0()))) , f^#(0(), s(s(y)), s(s(z))) -> c_7(f^#(0(), y, f(0(), s(s(y)), s(z)))) , f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z)) , f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0()))) , f^#(s(x), s(y), s(z)) -> c_10(f^#(x, y, f(s(x), s(y), z))) , f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(x, 0(), 0()) -> c_1(x) , f^#(0(), y, 0()) -> c_2(y) , f^#(0(), 0(), z) -> c_3(z) , f^#(0(), s(0()), s(0())) -> c_4() , f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z)) , f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0()))) , f^#(0(), s(s(y)), s(s(z))) -> c_7(f^#(0(), y, f(0(), s(s(y)), s(z)))) , f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z)) , f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0()))) , f^#(s(x), s(y), s(z)) -> c_10(f^#(x, y, f(s(x), s(y), z))) , f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) } Strict Trs: { f(x, 0(), 0()) -> s(x) , f(0(), y, 0()) -> s(y) , f(0(), 0(), z) -> s(z) , f(0(), s(0()), s(0())) -> s(s(0())) , f(0(), s(0()), s(s(z))) -> f(0(), s(0()), z) , f(0(), s(s(y)), s(0())) -> f(0(), y, s(0())) , f(0(), s(s(y)), s(s(z))) -> f(0(), y, f(0(), s(s(y)), s(z))) , f(s(x), 0(), s(z)) -> f(x, s(0()), z) , f(s(x), s(y), 0()) -> f(x, y, s(0())) , f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) , f(s(0()), y, z) -> f(0(), s(y), s(z)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4} by applications of Pre({4}) = {1,2,3,5,6,7,8,9,10,11}. Here rules are labeled as follows: DPs: { 1: f^#(x, 0(), 0()) -> c_1(x) , 2: f^#(0(), y, 0()) -> c_2(y) , 3: f^#(0(), 0(), z) -> c_3(z) , 4: f^#(0(), s(0()), s(0())) -> c_4() , 5: f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z)) , 6: f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0()))) , 7: f^#(0(), s(s(y)), s(s(z))) -> c_7(f^#(0(), y, f(0(), s(s(y)), s(z)))) , 8: f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z)) , 9: f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0()))) , 10: f^#(s(x), s(y), s(z)) -> c_10(f^#(x, y, f(s(x), s(y), z))) , 11: f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(x, 0(), 0()) -> c_1(x) , f^#(0(), y, 0()) -> c_2(y) , f^#(0(), 0(), z) -> c_3(z) , f^#(0(), s(0()), s(s(z))) -> c_5(f^#(0(), s(0()), z)) , f^#(0(), s(s(y)), s(0())) -> c_6(f^#(0(), y, s(0()))) , f^#(0(), s(s(y)), s(s(z))) -> c_7(f^#(0(), y, f(0(), s(s(y)), s(z)))) , f^#(s(x), 0(), s(z)) -> c_8(f^#(x, s(0()), z)) , f^#(s(x), s(y), 0()) -> c_9(f^#(x, y, s(0()))) , f^#(s(x), s(y), s(z)) -> c_10(f^#(x, y, f(s(x), s(y), z))) , f^#(s(0()), y, z) -> c_11(f^#(0(), s(y), s(z))) } Strict Trs: { f(x, 0(), 0()) -> s(x) , f(0(), y, 0()) -> s(y) , f(0(), 0(), z) -> s(z) , f(0(), s(0()), s(0())) -> s(s(0())) , f(0(), s(0()), s(s(z))) -> f(0(), s(0()), z) , f(0(), s(s(y)), s(0())) -> f(0(), y, s(0())) , f(0(), s(s(y)), s(s(z))) -> f(0(), y, f(0(), s(s(y)), s(z))) , f(s(x), 0(), s(z)) -> f(x, s(0()), z) , f(s(x), s(y), 0()) -> f(x, y, s(0())) , f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z)) , f(s(0()), y, z) -> f(0(), s(y), s(z)) } Weak DPs: { f^#(0(), s(0()), s(0())) -> c_4() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..