YES(O(1),O(n^1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { f(0()) -> s(0()) , f(s(0())) -> s(s(0())) , f(s(0())) -> *(s(s(0())), f(0())) , f(+(x, y)) -> *(f(x), f(y)) , f(+(x, s(0()))) -> +(s(s(0())), f(x)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , f^#(s(0())) -> c_3(f^#(0())) , f^#(+(x, y)) -> c_4(f^#(x), f^#(y)) , f^#(+(x, s(0()))) -> c_5(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^1)). Strict DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , f^#(s(0())) -> c_3(f^#(0())) , f^#(+(x, y)) -> c_4(f^#(x), f^#(y)) , f^#(+(x, s(0()))) -> c_5(f^#(x)) } Strict Trs: { f(0()) -> s(0()) , f(s(0())) -> s(s(0())) , f(s(0())) -> *(s(s(0())), f(0())) , f(+(x, y)) -> *(f(x), f(y)) , f(+(x, s(0()))) -> +(s(s(0())), f(x)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , f^#(s(0())) -> c_3(f^#(0())) , f^#(+(x, y)) -> c_4(f^#(x), f^#(y)) , f^#(+(x, s(0()))) -> c_5(f^#(x)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [0] [s](x1) = [0] [+](x1, x2) = [1] x1 + [1] x2 + [2] [f^#](x1) = [1] [c_1] = [0] [c_2] = [0] [c_3](x1) = [1] x1 + [1] [c_4](x1, x2) = [1] x1 + [1] x2 + [2] [c_5](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 YES(?,O(n^1)). Strict DPs: { f^#(s(0())) -> c_3(f^#(0())) , f^#(+(x, y)) -> c_4(f^#(x), f^#(y)) , f^#(+(x, s(0()))) -> c_5(f^#(x)) } Weak DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) We estimate the number of application of {1} by applications of Pre({1}) = {2,3}. Here rules are labeled as follows: DPs: { 1: f^#(s(0())) -> c_3(f^#(0())) , 2: f^#(+(x, y)) -> c_4(f^#(x), f^#(y)) , 3: f^#(+(x, s(0()))) -> c_5(f^#(x)) , 4: f^#(0()) -> c_1() , 5: f^#(s(0())) -> c_2() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { f^#(+(x, y)) -> c_4(f^#(x), f^#(y)) , f^#(+(x, s(0()))) -> c_5(f^#(x)) } Weak DPs: { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , f^#(s(0())) -> c_3(f^#(0())) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { f^#(0()) -> c_1() , f^#(s(0())) -> c_2() , f^#(s(0())) -> c_3(f^#(0())) } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { f^#(+(x, y)) -> c_4(f^#(x), f^#(y)) , f^#(+(x, s(0()))) -> c_5(f^#(x)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following argument positions are usable: Uargs(c_4) = {1, 2}, Uargs(c_5) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [0] = [0] [s](x1) = [1] x1 + [0] [+](x1, x2) = [1] x1 + [1] x2 + [1] [f^#](x1) = [1] x1 + [0] [c_4](x1, x2) = [1] x1 + [1] x2 + [0] [c_5](x1) = [1] x1 + [0] This order satisfies following ordering constraints: [f^#(+(x, y))] = [1] x + [1] y + [1] > [1] x + [1] y + [0] = [c_4(f^#(x), f^#(y))] [f^#(+(x, s(0())))] = [1] x + [1] > [1] x + [0] = [c_5(f^#(x))] Hurray, we answered YES(O(1),O(n^1))