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) -> 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() , plus^#(N, 0()) -> c_2() , plus^#(N, s(M)) -> c_3(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() , plus^#(N, 0()) -> c_2() , plus^#(N, s(M)) -> c_3(plus^#(N, M)) } Strict Trs: { and(tt(), 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() , plus^#(N, 0()) -> c_2() , plus^#(N, s(M)) -> c_3(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_3) = {1} TcT has computed following constructor-restricted matrix interpretation. [tt] = [1] [0] = [2] [s](x1) = [1] x1 + [1] [and^#](x1, x2) = [1] x1 + [0] [c_1] = [0] [plus^#](x1, x2) = [2] x1 + [0] [c_2] = [1] [c_3](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(n^1)). Strict DPs: { plus^#(N, 0()) -> c_2() , plus^#(N, s(M)) -> c_3(plus^#(N, M)) } Weak DPs: { and^#(tt(), X) -> c_1() } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) We estimate the number of application of {1} by applications of Pre({1}) = {2}. Here rules are labeled as follows: DPs: { 1: plus^#(N, 0()) -> c_2() , 2: plus^#(N, s(M)) -> c_3(plus^#(N, M)) , 3: and^#(tt(), X) -> c_1() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { plus^#(N, s(M)) -> c_3(plus^#(N, M)) } Weak DPs: { and^#(tt(), X) -> c_1() , plus^#(N, 0()) -> 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. { and^#(tt(), X) -> c_1() , plus^#(N, 0()) -> c_2() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { plus^#(N, s(M)) -> c_3(plus^#(N, M)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following argument positions are usable: Uargs(c_3) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [s](x1) = [1] x1 + [3] [plus^#](x1, x2) = [3] x1 + [1] x2 + [3] [c_3](x1) = [1] x1 + [2] This order satisfies following ordering constraints: [plus^#(N, s(M))] = [3] N + [1] M + [6] > [3] N + [1] M + [5] = [c_3(plus^#(N, M))] Hurray, we answered YES(O(1),O(n^1))