YES(O(1),O(1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Strict Trs: { f(a(), b(), X) -> f(X, X, X) , c() -> a() , c() -> b() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) We add following weak dependency pairs: Strict DPs: { f^#(a(), b(), X) -> c_1(f^#(X, X, X)) , c^#() -> c_2() , c^#() -> c_3() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Strict DPs: { f^#(a(), b(), X) -> c_1(f^#(X, X, X)) , c^#() -> c_2() , c^#() -> c_3() } Strict Trs: { f(a(), b(), X) -> f(X, X, X) , c() -> a() , c() -> b() } Obligation: innermost runtime complexity Answer: YES(O(1),O(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(1)). Strict DPs: { f^#(a(), b(), X) -> c_1(f^#(X, X, X)) , c^#() -> c_2() , c^#() -> c_3() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: none TcT has computed following constructor-restricted matrix interpretation. [a] = [2] [b] = [1] [f^#](x1, x2, x3) = [2] x3 + [2] [c_1](x1) = [1] x1 + [2] [c^#] = [2] [c_2] = [1] [c_3] = [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(1),O(1)). Strict DPs: { f^#(a(), b(), X) -> c_1(f^#(X, X, X)) } Weak DPs: { c^#() -> c_2() , c^#() -> c_3() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) We estimate the number of application of {1} by applications of Pre({1}) = {}. Here rules are labeled as follows: DPs: { 1: f^#(a(), b(), X) -> c_1(f^#(X, X, X)) , 2: c^#() -> c_2() , 3: c^#() -> c_3() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { f^#(a(), b(), X) -> c_1(f^#(X, X, X)) , c^#() -> c_2() , c^#() -> c_3() } 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^#(a(), b(), X) -> c_1(f^#(X, X, X)) , c^#() -> c_2() , c^#() -> c_3() } 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(1))