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: { and(tt(), X) -> activate(X) , activate(X) -> X , plus(N, 0()) -> N , plus(N, s(M)) -> s(plus(N, M)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { and^#(tt(), X) -> c_1(activate^#(X)) , activate^#(X) -> c_2() , plus^#(N, 0()) -> c_3() , plus^#(N, s(M)) -> c_4(plus^#(N, M)) } 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: { and^#(tt(), X) -> c_1(activate^#(X)) , activate^#(X) -> c_2() , plus^#(N, 0()) -> c_3() , plus^#(N, s(M)) -> c_4(plus^#(N, M)) } Strict Trs: { and(tt(), X) -> activate(X) , activate(X) -> X , plus(N, 0()) -> N , plus(N, s(M)) -> s(plus(N, M)) } 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: { and^#(tt(), X) -> c_1(activate^#(X)) , activate^#(X) -> c_2() , plus^#(N, 0()) -> c_3() , plus^#(N, s(M)) -> c_4(plus^#(N, M)) } 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}, Uargs(c_4) = {1} TcT has computed following constructor-restricted matrix interpretation. [tt] = [1] [0] = [2] [s](x1) = [1] x1 + [2] [and^#](x1, x2) = [1] x1 + [2] x2 + [2] [c_1](x1) = [1] x1 + [1] [activate^#](x1) = [1] x1 + [1] [c_2] = [0] [plus^#](x1, x2) = [2] x1 + [2] x2 + [2] [c_3] = [1] [c_4](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(1),O(1)). Weak DPs: { and^#(tt(), X) -> c_1(activate^#(X)) , activate^#(X) -> c_2() , plus^#(N, 0()) -> c_3() , plus^#(N, s(M)) -> c_4(plus^#(N, M)) } 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. { and^#(tt(), X) -> c_1(activate^#(X)) , activate^#(X) -> c_2() , plus^#(N, 0()) -> c_3() , plus^#(N, s(M)) -> c_4(plus^#(N, M)) } 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))