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, x) -> a() , f(g(x), y) -> f(x, y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { f^#(x, x) -> c_1() , f^#(g(x), y) -> c_2(f^#(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, x) -> c_1() , f^#(g(x), y) -> c_2(f^#(x, y)) } Strict Trs: { f(x, x) -> a() , f(g(x), y) -> f(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, x) -> c_1() , f^#(g(x), y) -> c_2(f^#(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) = [1] x1 + [1] [f^#](x1, x2) = [1] x1 + [2] x2 + [2] [c_1] = [1] [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(1),O(1)). Weak DPs: { f^#(x, x) -> c_1() , f^#(g(x), y) -> c_2(f^#(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { f^#(x, x) -> c_1() , f^#(g(x), y) -> c_2(f^#(x, y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))