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: { g(X) -> u(h(X), h(X), X) , u(d(), c(Y), X) -> k(Y) , h(d()) -> c(a()) , h(d()) -> c(b()) , f(k(a()), k(b()), X) -> f(X, X, X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { g^#(X) -> c_1(u^#(h(X), h(X), X)) , u^#(d(), c(Y), X) -> c_2() , h^#(d()) -> c_3() , h^#(d()) -> c_4() , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, 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: { g^#(X) -> c_1(u^#(h(X), h(X), X)) , u^#(d(), c(Y), X) -> c_2() , h^#(d()) -> c_3() , h^#(d()) -> c_4() , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) } Strict Trs: { g(X) -> u(h(X), h(X), X) , u(d(), c(Y), X) -> k(Y) , h(d()) -> c(a()) , h(d()) -> c(b()) , f(k(a()), k(b()), X) -> f(X, X, X) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Strict Usable Rules: { h(d()) -> c(a()) , h(d()) -> c(b()) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { g^#(X) -> c_1(u^#(h(X), h(X), X)) , u^#(d(), c(Y), X) -> c_2() , h^#(d()) -> c_3() , h^#(d()) -> c_4() , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) } Strict Trs: { h(d()) -> c(a()) , h(d()) -> c(b()) } 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_1) = {1}, Uargs(u^#) = {1, 2} TcT has computed following constructor-restricted matrix interpretation. [h](x1) = [2] [d] = [1] [c](x1) = [0] [k](x1) = [1] x1 + [0] [a] = [0] [b] = [0] [g^#](x1) = [2] x1 + [1] [c_1](x1) = [1] x1 + [0] [u^#](x1, x2, x3) = [1] x1 + [2] x2 + [1] x3 + [1] [c_2] = [1] [h^#](x1) = [2] x1 + [1] [c_3] = [2] [c_4] = [2] [f^#](x1, x2, x3) = [2] [c_5](x1) = [2] This order satisfies following ordering constraints: [h(d())] = [2] > [0] = [c(a())] [h(d())] = [2] > [0] = [c(b())] 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: { g^#(X) -> c_1(u^#(h(X), h(X), X)) , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) } Weak DPs: { u^#(d(), c(Y), X) -> c_2() , h^#(d()) -> c_3() , h^#(d()) -> c_4() } Weak Trs: { h(d()) -> c(a()) , h(d()) -> c(b()) } 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: g^#(X) -> c_1(u^#(h(X), h(X), X)) , 2: f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) , 3: u^#(d(), c(Y), X) -> c_2() , 4: h^#(d()) -> c_3() , 5: h^#(d()) -> c_4() } We are left with following problem, upon which TcT provides the certificate YES(?,O(1)). Weak DPs: { g^#(X) -> c_1(u^#(h(X), h(X), X)) , u^#(d(), c(Y), X) -> c_2() , h^#(d()) -> c_3() , h^#(d()) -> c_4() , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) } Weak Trs: { h(d()) -> c(a()) , h(d()) -> c(b()) } 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. { g^#(X) -> c_1(u^#(h(X), h(X), X)) , u^#(d(), c(Y), X) -> c_2() , h^#(d()) -> c_3() , h^#(d()) -> c_4() , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) } We are left with following problem, upon which TcT provides the certificate YES(?,O(1)). Weak Trs: { h(d()) -> c(a()) , h(d()) -> c(b()) } 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))