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(g(x)) -> g(g(f(x))) , f(g(x)) -> g(g(g(x))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { f^#(g(x)) -> c_1(f^#(x)) , f^#(g(x)) -> c_2() } 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^#(g(x)) -> c_1(f^#(x)) , f^#(g(x)) -> c_2() } Strict Trs: { f(g(x)) -> g(g(f(x))) , f(g(x)) -> g(g(g(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^#(g(x)) -> c_1(f^#(x)) , f^#(g(x)) -> c_2() } 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_1) = {1} TcT has computed following constructor-restricted matrix interpretation. [g](x1) = [1] x1 + [1] [f^#](x1) = [1] x1 + [2] [c_1](x1) = [1] x1 + [1] [c_2] = [2] 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^#(g(x)) -> c_1(f^#(x)) } Weak DPs: { f^#(g(x)) -> c_2() } 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^#(g(x)) -> c_2() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { f^#(g(x)) -> c_1(f^#(x)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following argument positions are usable: Uargs(c_1) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [g](x1) = [1] x1 + [3] [f^#](x1) = [1] x1 + [3] [c_1](x1) = [1] x1 + [2] This order satisfies following ordering constraints: [f^#(g(x))] = [1] x + [6] > [1] x + [5] = [c_1(f^#(x))] Hurray, we answered YES(O(1),O(n^1))