YES(O(1),O(n^2)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { f(X) -> n__f(X) , f(0()) -> cons(0(), n__f(s(0()))) , f(s(0())) -> f(p(s(0()))) , p(s(X)) -> X , activate(X) -> X , activate(n__f(X)) -> f(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add following weak dependency pairs: Strict DPs: { f^#(X) -> c_1() , f^#(0()) -> c_2() , f^#(s(0())) -> c_3(f^#(p(s(0())))) , p^#(s(X)) -> c_4() , activate^#(X) -> c_5() , activate^#(n__f(X)) -> c_6(f^#(X)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { f^#(X) -> c_1() , f^#(0()) -> c_2() , f^#(s(0())) -> c_3(f^#(p(s(0())))) , p^#(s(X)) -> c_4() , activate^#(X) -> c_5() , activate^#(n__f(X)) -> c_6(f^#(X)) } Strict Trs: { f(X) -> n__f(X) , f(0()) -> cons(0(), n__f(s(0()))) , f(s(0())) -> f(p(s(0()))) , p(s(X)) -> X , activate(X) -> X , activate(n__f(X)) -> f(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) 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 YES(O(1),O(n^2)). Strict DPs: { f^#(X) -> c_1() , f^#(0()) -> c_2() , f^#(s(0())) -> c_3(f^#(p(s(0())))) , p^#(s(X)) -> c_4() , activate^#(X) -> c_5() , activate^#(n__f(X)) -> c_6(f^#(X)) } Strict Trs: { p(s(X)) -> X } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(f^#) = {1}, Uargs(c_3) = {1}, Uargs(c_6) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [0] [n__f](x1) = [1] x1 + [1] [s](x1) = [1] x1 + [2] [p](x1) = [2] x1 + [0] [f^#](x1) = [1] x1 + [2] [c_1] = [1] [c_2] = [1] [c_3](x1) = [1] x1 + [1] [p^#](x1) = [1] x1 + [1] [c_4] = [2] [activate^#](x1) = [2] x1 + [2] [c_5] = [2] [c_6](x1) = [1] x1 + [1] 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 YES(?,O(n^2)). Strict DPs: { f^#(s(0())) -> c_3(f^#(p(s(0())))) , activate^#(X) -> c_5() } Weak DPs: { f^#(X) -> c_1() , f^#(0()) -> c_2() , p^#(s(X)) -> c_4() , activate^#(n__f(X)) -> c_6(f^#(X)) } Weak Trs: { p(s(X)) -> X } Obligation: innermost runtime complexity Answer: YES(?,O(n^2)) We estimate the number of application of {2} by applications of Pre({2}) = {}. Here rules are labeled as follows: DPs: { 1: f^#(s(0())) -> c_3(f^#(p(s(0())))) , 2: activate^#(X) -> c_5() , 3: f^#(X) -> c_1() , 4: f^#(0()) -> c_2() , 5: p^#(s(X)) -> c_4() , 6: activate^#(n__f(X)) -> c_6(f^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^2)). Strict DPs: { f^#(s(0())) -> c_3(f^#(p(s(0())))) } Weak DPs: { f^#(X) -> c_1() , f^#(0()) -> c_2() , p^#(s(X)) -> c_4() , activate^#(X) -> c_5() , activate^#(n__f(X)) -> c_6(f^#(X)) } Weak Trs: { p(s(X)) -> X } Obligation: innermost runtime complexity Answer: YES(?,O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { f^#(X) -> c_1() , f^#(0()) -> c_2() , p^#(s(X)) -> c_4() , activate^#(X) -> c_5() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^2)). Strict DPs: { f^#(s(0())) -> c_3(f^#(p(s(0())))) } Weak DPs: { activate^#(n__f(X)) -> c_6(f^#(X)) } Weak Trs: { p(s(X)) -> X } Obligation: innermost runtime complexity Answer: YES(?,O(n^2)) Consider the dependency graph 1: f^#(s(0())) -> c_3(f^#(p(s(0())))) -->_1 f^#(s(0())) -> c_3(f^#(p(s(0())))) :1 2: activate^#(n__f(X)) -> c_6(f^#(X)) -->_1 f^#(s(0())) -> c_3(f^#(p(s(0())))) :1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). { activate^#(n__f(X)) -> c_6(f^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^2)). Strict DPs: { f^#(s(0())) -> c_3(f^#(p(s(0())))) } Weak Trs: { p(s(X)) -> X } Obligation: innermost runtime complexity Answer: YES(?,O(n^2)) The following argument positions are usable: Uargs(c_3) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [0] = [0] [0] [s](x1) = [0 1] x1 + [1] [1 0] [0] [p](x1) = [0 1] x1 + [0] [2 0] [0] [f^#](x1) = [1 0] x1 + [0] [0 0] [0] [c_3](x1) = [1 0] x1 + [0] [0 0] [0] This order satisfies following ordering constraints: [p(s(X))] = [1 0] X + [0] [0 2] [2] >= [1 0] X + [0] [0 1] [0] = [X] [f^#(s(0()))] = [1] [0] > [0] [0] = [c_3(f^#(p(s(0()))))] Hurray, we answered YES(O(1),O(n^2))