MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(0()) -> cons(0(), f(s(0()))) , f(s(0())) -> f(p(s(0()))) , p(s(0())) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { f^#(0()) -> c_1(f^#(s(0()))) , f^#(s(0())) -> c_2(f^#(p(s(0())))) , p^#(s(0())) -> c_3() } 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()))) , f^#(s(0())) -> c_2(f^#(p(s(0())))) , p^#(s(0())) -> c_3() } Strict Trs: { f(0()) -> cons(0(), f(s(0()))) , f(s(0())) -> f(p(s(0()))) , p(s(0())) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Strict Usable Rules: { p(s(0())) -> 0() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(0()) -> c_1(f^#(s(0()))) , f^#(s(0())) -> c_2(f^#(p(s(0())))) , p^#(s(0())) -> c_3() } Strict Trs: { p(s(0())) -> 0() } 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^#) = {1}, Uargs(c_1) = {1}, Uargs(c_2) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [0] [s](x1) = [2] [p](x1) = [2] x1 + [0] [f^#](x1) = [1] x1 + [0] [c_1](x1) = [1] x1 + [2] [c_2](x1) = [1] x1 + [2] [p^#](x1) = [2] x1 + [2] [c_3] = [1] This order satisfies following ordering constraints: [p(s(0()))] = [4] > [0] = [0()] 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^#(0()) -> c_1(f^#(s(0()))) , f^#(s(0())) -> c_2(f^#(p(s(0())))) } Weak DPs: { p^#(s(0())) -> c_3() } Weak Trs: { p(s(0())) -> 0() } 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. { p^#(s(0())) -> c_3() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(0()) -> c_1(f^#(s(0()))) , f^#(s(0())) -> c_2(f^#(p(s(0())))) } Weak Trs: { p(s(0())) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrices' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrix interpretation of dimension 4' failed due to the following reason: The input cannot be shown compatible 2) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 3) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 4) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 5) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 6) 'matrix interpretation of dimension 1' failed due to the following reason: The input cannot be shown compatible 2) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. Arrrr..