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: { ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) , u21(ackout(X), Y) -> u22(ackin(Y, X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { ackin^#(s(X), s(Y)) -> c_1(u21^#(ackin(s(X), Y), X)) , u21^#(ackout(X), Y) -> c_2(ackin^#(Y, X)) } 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: { ackin^#(s(X), s(Y)) -> c_1(u21^#(ackin(s(X), Y), X)) , u21^#(ackout(X), Y) -> c_2(ackin^#(Y, X)) } Strict Trs: { ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) , u21(ackout(X), Y) -> u22(ackin(Y, X)) } 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(u21) = {1}, Uargs(u22) = {1}, Uargs(c_1) = {1}, Uargs(u21^#) = {1}, Uargs(c_2) = {1} TcT has computed following constructor-restricted matrix interpretation. [ackin](x1, x2) = [1] x2 + [0] [s](x1) = [1] x1 + [1] [u21](x1, x2) = [1] x1 + [0] [ackout](x1) = [1] x1 + [1] [u22](x1) = [1] x1 + [0] [ackin^#](x1, x2) = [2] x1 + [2] x2 + [2] [c_1](x1) = [1] x1 + [2] [u21^#](x1, x2) = [2] x1 + [2] x2 + [2] [c_2](x1) = [1] x1 + [1] This order satisfies following ordering constraints: [ackin(s(X), s(Y))] = [1] Y + [1] > [1] Y + [0] = [u21(ackin(s(X), Y), X)] [u21(ackout(X), Y)] = [1] X + [1] > [1] X + [0] = [u22(ackin(Y, X))] 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: { ackin^#(s(X), s(Y)) -> c_1(u21^#(ackin(s(X), Y), X)) , u21^#(ackout(X), Y) -> c_2(ackin^#(Y, X)) } Weak Trs: { ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) , u21(ackout(X), Y) -> u22(ackin(Y, X)) } 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. { ackin^#(s(X), s(Y)) -> c_1(u21^#(ackin(s(X), Y), X)) , u21^#(ackout(X), Y) -> c_2(ackin^#(Y, X)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) , u21(ackout(X), Y) -> u22(ackin(Y, X)) } 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)). 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))