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: { 2nd(cons1(X, cons(Y, Z))) -> Y , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) , activate(X) -> X , activate(n__from(X)) -> from(X) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { 2nd^#(cons1(X, cons(Y, Z))) -> c_1() , 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) , activate^#(X) -> c_3() , activate^#(n__from(X)) -> c_4(from^#(X)) , from^#(X) -> c_5() , from^#(X) -> c_6() } 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: { 2nd^#(cons1(X, cons(Y, Z))) -> c_1() , 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) , activate^#(X) -> c_3() , activate^#(n__from(X)) -> c_4(from^#(X)) , from^#(X) -> c_5() , from^#(X) -> c_6() } Strict Trs: { 2nd(cons1(X, cons(Y, Z))) -> Y , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1))) , activate(X) -> X , activate(n__from(X)) -> from(X) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Strict Usable Rules: { activate(X) -> X , activate(n__from(X)) -> from(X) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { 2nd^#(cons1(X, cons(Y, Z))) -> c_1() , 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) , activate^#(X) -> c_3() , activate^#(n__from(X)) -> c_4(from^#(X)) , from^#(X) -> c_5() , from^#(X) -> c_6() } Strict Trs: { activate(X) -> X , activate(n__from(X)) -> from(X) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__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(cons1) = {2}, Uargs(2nd^#) = {1}, Uargs(c_2) = {1}, Uargs(c_4) = {1} TcT has computed following constructor-restricted matrix interpretation. [cons1](x1, x2) = [1] x1 + [1] x2 + [2] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [activate](x1) = [1] x1 + [2] [from](x1) = [1] x1 + [2] [n__from](x1) = [1] x1 + [1] [s](x1) = [0] [2nd^#](x1) = [1] x1 + [2] [c_1] = [2] [c_2](x1) = [1] x1 + [1] [activate^#](x1) = [2] x1 + [1] [c_3] = [0] [c_4](x1) = [1] x1 + [2] [from^#](x1) = [2] x1 + [1] [c_5] = [0] [c_6] = [0] This order satisfies following ordering constraints: [activate(X)] = [1] X + [2] > [1] X + [0] = [X] [activate(n__from(X))] = [1] X + [3] > [1] X + [2] = [from(X)] [from(X)] = [1] X + [2] > [1] X + [1] = [cons(X, n__from(s(X)))] [from(X)] = [1] X + [2] > [1] X + [1] = [n__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(1)). Strict DPs: { 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) , activate^#(n__from(X)) -> c_4(from^#(X)) } Weak DPs: { 2nd^#(cons1(X, cons(Y, Z))) -> c_1() , activate^#(X) -> c_3() , from^#(X) -> c_5() , from^#(X) -> c_6() } Weak Trs: { activate(X) -> X , activate(n__from(X)) -> from(X) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: YES(?,O(1)) We estimate the number of application of {1,2} by applications of Pre({1,2}) = {}. Here rules are labeled as follows: DPs: { 1: 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) , 2: activate^#(n__from(X)) -> c_4(from^#(X)) , 3: 2nd^#(cons1(X, cons(Y, Z))) -> c_1() , 4: activate^#(X) -> c_3() , 5: from^#(X) -> c_5() , 6: from^#(X) -> c_6() } We are left with following problem, upon which TcT provides the certificate YES(?,O(1)). Weak DPs: { 2nd^#(cons1(X, cons(Y, Z))) -> c_1() , 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) , activate^#(X) -> c_3() , activate^#(n__from(X)) -> c_4(from^#(X)) , from^#(X) -> c_5() , from^#(X) -> c_6() } Weak Trs: { activate(X) -> X , activate(n__from(X)) -> from(X) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: YES(?,O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { 2nd^#(cons1(X, cons(Y, Z))) -> c_1() , 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) , activate^#(X) -> c_3() , activate^#(n__from(X)) -> c_4(from^#(X)) , from^#(X) -> c_5() , from^#(X) -> c_6() } We are left with following problem, upon which TcT provides the certificate YES(?,O(1)). Weak Trs: { activate(X) -> X , activate(n__from(X)) -> from(X) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: YES(?,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)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(?,O(1)) The input was oriented with the instance of 'Small Polynomial Path Order (PS)' as induced by the safe mapping and precedence empty . Following symbols are considered recursive: {} The recursion depth is 0. Further, following argument filtering is employed: empty Usable defined function symbols are a subset of: {} For your convenience, here are the satisfied ordering constraints: Hurray, we answered YES(O(1),O(n^1))