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(x, g(x)) -> x , f(x, h(y)) -> f(h(x), y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { f^#(x, g(x)) -> c_1() , f^#(x, h(y)) -> c_2(f^#(h(x), y)) } 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^#(x, g(x)) -> c_1() , f^#(x, h(y)) -> c_2(f^#(h(x), y)) } Strict Trs: { f(x, g(x)) -> x , f(x, h(y)) -> f(h(x), y) } 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^#(x, g(x)) -> c_1() , f^#(x, h(y)) -> c_2(f^#(h(x), y)) } 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_2) = {1} TcT has computed following constructor-restricted matrix interpretation. [g](x1) = [2] [h](x1) = [1] x1 + [0] [f^#](x1, x2) = [2] x1 + [1] [c_1] = [0] [c_2](x1) = [1] x1 + [0] 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^#(x, h(y)) -> c_2(f^#(h(x), y)) } Weak DPs: { f^#(x, g(x)) -> c_1() } 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^#(x, g(x)) -> c_1() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { f^#(x, h(y)) -> c_2(f^#(h(x), y)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following argument positions are usable: Uargs(c_2) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [h](x1) = [1] x1 + [2] [f^#](x1, x2) = [2] x2 + [0] [c_2](x1) = [1] x1 + [3] This order satisfies following ordering constraints: [f^#(x, h(y))] = [2] y + [4] > [2] y + [3] = [c_2(f^#(h(x), y))] Hurray, we answered YES(O(1),O(n^1))