MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(a(), a()) -> f(a(), b()) , f(a(), b()) -> f(s(a()), c()) , f(s(X), c()) -> f(X, c()) , f(c(), c()) -> f(a(), a()) } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { f^#(a(), a()) -> c_1(f^#(a(), b())) , f^#(a(), b()) -> c_2(f^#(s(a()), c())) , f^#(s(X), c()) -> c_3(f^#(X, c())) , f^#(c(), c()) -> c_4(f^#(a(), a())) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(a(), a()) -> c_1(f^#(a(), b())) , f^#(a(), b()) -> c_2(f^#(s(a()), c())) , f^#(s(X), c()) -> c_3(f^#(X, c())) , f^#(c(), c()) -> c_4(f^#(a(), a())) } Strict Trs: { f(a(), a()) -> f(a(), b()) , f(a(), b()) -> f(s(a()), c()) , f(s(X), c()) -> f(X, c()) , f(c(), c()) -> f(a(), a()) } Obligation: innermost runtime complexity Answer: MAYBE No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(a(), a()) -> c_1(f^#(a(), b())) , f^#(a(), b()) -> c_2(f^#(s(a()), c())) , f^#(s(X), c()) -> c_3(f^#(X, c())) , f^#(c(), c()) -> c_4(f^#(a(), 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(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed following constructor-restricted matrix interpretation. [a] = [2] [b] = [0] [s](x1) = [1] x1 + [0] [c] = [0] [f^#](x1, x2) = [2] x2 + [0] [c_1](x1) = [1] x1 + [1] [c_2](x1) = [1] x1 + [1] [c_3](x1) = [1] x1 + [1] [c_4](x1) = [1] x1 + [1] This order satisfies following ordering constraints: 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(), b()) -> c_2(f^#(s(a()), c())) , f^#(s(X), c()) -> c_3(f^#(X, c())) , f^#(c(), c()) -> c_4(f^#(a(), a())) } Weak DPs: { f^#(a(), a()) -> c_1(f^#(a(), b())) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {3} by applications of Pre({3}) = {2}. Here rules are labeled as follows: DPs: { 1: f^#(a(), b()) -> c_2(f^#(s(a()), c())) , 2: f^#(s(X), c()) -> c_3(f^#(X, c())) , 3: f^#(c(), c()) -> c_4(f^#(a(), a())) , 4: f^#(a(), a()) -> c_1(f^#(a(), b())) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(a(), b()) -> c_2(f^#(s(a()), c())) , f^#(s(X), c()) -> c_3(f^#(X, c())) } Weak DPs: { f^#(a(), a()) -> c_1(f^#(a(), b())) , f^#(c(), c()) -> c_4(f^#(a(), a())) } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..