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: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , after(s(N), cons(X, XS)) -> after(N, activate(XS)) , after(0(), XS) -> XS , activate(X) -> X , activate(n__from(X)) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { from^#(X) -> c_1() , from^#(X) -> c_2() , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4() , activate^#(X) -> c_5() , activate^#(n__from(X)) -> c_6(from^#(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: { from^#(X) -> c_1() , from^#(X) -> c_2() , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4() , activate^#(X) -> c_5() , activate^#(n__from(X)) -> c_6(from^#(X)) } Strict Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , after(s(N), cons(X, XS)) -> after(N, activate(XS)) , after(0(), XS) -> XS , activate(X) -> X , activate(n__from(X)) -> from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Strict Usable Rules: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { from^#(X) -> c_1() , from^#(X) -> c_2() , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4() , activate^#(X) -> c_5() , activate^#(n__from(X)) -> c_6(from^#(X)) } Strict Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(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(after^#) = {2}, Uargs(c_3) = {1}, Uargs(c_6) = {1} TcT has computed following constructor-restricted matrix interpretation. [from](x1) = [2] [cons](x1, x2) = [1] x2 + [0] [n__from](x1) = [1] [s](x1) = [1] x1 + [2] [0] = [2] [activate](x1) = [1] x1 + [2] [from^#](x1) = [2] [c_1] = [1] [c_2] = [1] [after^#](x1, x2) = [1] x1 + [1] x2 + [2] [c_3](x1) = [1] x1 + [2] [c_4] = [1] [activate^#](x1) = [2] x1 + [1] [c_5] = [1] [c_6](x1) = [1] x1 + [1] This order satisfies following ordering constraints: [from(X)] = [2] > [1] = [cons(X, n__from(s(X)))] [from(X)] = [2] > [1] = [n__from(X)] [activate(X)] = [1] X + [2] > [1] X + [0] = [X] [activate(n__from(X))] = [3] > [2] = [from(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(n^1)). Strict DPs: { after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , activate^#(X) -> c_5() , activate^#(n__from(X)) -> c_6(from^#(X)) } Weak DPs: { from^#(X) -> c_1() , from^#(X) -> c_2() , after^#(0(), XS) -> c_4() } Weak Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) We estimate the number of application of {2,3} by applications of Pre({2,3}) = {}. Here rules are labeled as follows: DPs: { 1: after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , 2: activate^#(X) -> c_5() , 3: activate^#(n__from(X)) -> c_6(from^#(X)) , 4: from^#(X) -> c_1() , 5: from^#(X) -> c_2() , 6: after^#(0(), XS) -> c_4() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) } Weak DPs: { from^#(X) -> c_1() , from^#(X) -> c_2() , after^#(0(), XS) -> c_4() , activate^#(X) -> c_5() , activate^#(n__from(X)) -> c_6(from^#(X)) } Weak Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } 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. { from^#(X) -> c_1() , from^#(X) -> c_2() , after^#(0(), XS) -> c_4() , activate^#(X) -> c_5() , activate^#(n__from(X)) -> c_6(from^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) } Weak Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } 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). [from](x1) = [3] x1 + [0] [cons](x1, x2) = [1] x1 + [0] [n__from](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [1] [activate](x1) = [0] [after^#](x1, x2) = [1] x1 + [0] [c_3](x1) = [1] x1 + [0] This order satisfies following ordering constraints: [after^#(s(N), cons(X, XS))] = [1] N + [1] > [1] N + [0] = [c_3(after^#(N, activate(XS)))] Hurray, we answered YES(O(1),O(n^1))