MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(a(), X, X) -> f(X, b(), b()) , b() -> a() } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { f^#(a(), X, X) -> c_1(f^#(X, b(), b())) , b^#() -> c_2() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(a(), X, X) -> c_1(f^#(X, b(), b())) , b^#() -> c_2() } Strict Trs: { f(a(), X, X) -> f(X, b(), b()) , b() -> a() } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Strict Usable Rules: { b() -> a() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(a(), X, X) -> c_1(f^#(X, b(), b())) , b^#() -> c_2() } Strict Trs: { b() -> a() } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(f^#) = {2, 3}, Uargs(c_1) = {1} TcT has computed following constructor-restricted matrix interpretation. [a] = [0] [b] = [1] [f^#](x1, x2, x3) = [1] x1 + [1] x2 + [2] x3 + [1] [c_1](x1) = [1] x1 + [2] [b^#] = [1] [c_2] = [1] This order satisfies following ordering constraints: [b()] = [1] > [0] = [a()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(a(), X, X) -> c_1(f^#(X, b(), b())) , b^#() -> c_2() } Weak Trs: { b() -> a() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {2} by applications of Pre({2}) = {}. Here rules are labeled as follows: DPs: { 1: f^#(a(), X, X) -> c_1(f^#(X, b(), b())) , 2: b^#() -> c_2() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(a(), X, X) -> c_1(f^#(X, b(), b())) } Weak DPs: { b^#() -> c_2() } Weak Trs: { b() -> a() } 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. { b^#() -> c_2() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(a(), X, X) -> c_1(f^#(X, b(), b())) } Weak Trs: { b() -> a() } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..