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: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , tail(cons(X, XS)) -> activate(XS) , activate(X) -> X , activate(n__zeros()) -> zeros() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { zeros^#() -> c_1() , zeros^#() -> c_2() , tail^#(cons(X, XS)) -> c_3(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__zeros()) -> c_5(zeros^#()) } 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: { zeros^#() -> c_1() , zeros^#() -> c_2() , tail^#(cons(X, XS)) -> c_3(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__zeros()) -> c_5(zeros^#()) } Strict Trs: { zeros() -> cons(0(), n__zeros()) , zeros() -> n__zeros() , tail(cons(X, XS)) -> activate(XS) , activate(X) -> X , activate(n__zeros()) -> zeros() } 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: { zeros^#() -> c_1() , zeros^#() -> c_2() , tail^#(cons(X, XS)) -> c_3(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__zeros()) -> c_5(zeros^#()) } 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}, Uargs(c_5) = {1} TcT has computed following constructor-restricted matrix interpretation. [cons](x1, x2) = [1] x2 + [2] [n__zeros] = [1] [zeros^#] = [1] [c_1] = [0] [c_2] = [0] [tail^#](x1) = [2] x1 + [2] [c_3](x1) = [1] x1 + [1] [activate^#](x1) = [1] x1 + [2] [c_4] = [1] [c_5](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)). Weak DPs: { zeros^#() -> c_1() , zeros^#() -> c_2() , tail^#(cons(X, XS)) -> c_3(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__zeros()) -> c_5(zeros^#()) } 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. { zeros^#() -> c_1() , zeros^#() -> c_2() , tail^#(cons(X, XS)) -> c_3(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__zeros()) -> c_5(zeros^#()) } 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))