MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(0()) -> cons(0()) , f(s(0())) -> f(p(s(0()))) , p(s(X)) -> X } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2(f^#(p(s(0())))) , p^#(s(X)) -> 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())) -> c_2(f^#(p(s(0())))) , p^#(s(X)) -> c_3() } Strict Trs: { f(0()) -> cons(0()) , f(s(0())) -> f(p(s(0()))) , p(s(X)) -> X } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Strict Usable Rules: { p(s(X)) -> X } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2(f^#(p(s(0())))) , p^#(s(X)) -> c_3() } Strict Trs: { p(s(X)) -> X } 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_2) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [0] [s](x1) = [1] x1 + [2] [p](x1) = [2] x1 + [0] [f^#](x1) = [1] x1 + [2] [c_1] = [1] [c_2](x1) = [1] x1 + [1] [p^#](x1) = [1] x1 + [1] [c_3] = [2] This order satisfies following ordering constraints: [p(s(X))] = [2] X + [4] > [1] X + [0] = [X] 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^#(s(0())) -> c_2(f^#(p(s(0())))) } Weak DPs: { f^#(0()) -> c_1() , p^#(s(X)) -> c_3() } Weak Trs: { p(s(X)) -> X } 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. { f^#(0()) -> c_1() , p^#(s(X)) -> c_3() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(0())) -> c_2(f^#(p(s(0())))) } Weak Trs: { p(s(X)) -> X } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..