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: { f(g(i(a(), b(), b'()), c()), d()) -> if(e(), f(.(b(), c()), d'()), f(.(b'(), c()), d'())) , f(g(h(a(), b()), c()), d()) -> if(e(), f(.(b(), g(h(a(), b()), c())), d()), f(c(), d'())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { f^#(g(i(a(), b(), b'()), c()), d()) -> c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'())) , f^#(g(h(a(), b()), c()), d()) -> c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) } 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: { f^#(g(i(a(), b(), b'()), c()), d()) -> c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'())) , f^#(g(h(a(), b()), c()), d()) -> c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) } Strict Trs: { f(g(i(a(), b(), b'()), c()), d()) -> if(e(), f(.(b(), c()), d'()), f(.(b'(), c()), d'())) , f(g(h(a(), b()), c()), d()) -> if(e(), f(.(b(), g(h(a(), b()), c())), d()), f(c(), d'())) } 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: { f^#(g(i(a(), b(), b'()), c()), d()) -> c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'())) , f^#(g(h(a(), b()), c()), d()) -> c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) } 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: none TcT has computed following constructor-restricted matrix interpretation. [g](x1, x2) = [0] [i](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2] [a] = [0] [b] = [0] [b'] = [0] [c] = [0] [d] = [0] [.](x1, x2) = [0] [d'] = [0] [h](x1, x2) = [0] [f^#](x1, x2) = [1] [c_1](x1, x2) = [0] [c_2](x1, x2) = [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),O(1)). Strict DPs: { f^#(g(h(a(), b()), c()), d()) -> c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) } Weak DPs: { f^#(g(i(a(), b(), b'()), c()), d()) -> c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) We estimate the number of application of {1} by applications of Pre({1}) = {}. Here rules are labeled as follows: DPs: { 1: f^#(g(h(a(), b()), c()), d()) -> c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) , 2: f^#(g(i(a(), b(), b'()), c()), d()) -> c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'())) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { f^#(g(i(a(), b(), b'()), c()), d()) -> c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'())) , f^#(g(h(a(), b()), c()), d()) -> c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) } 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. { f^#(g(i(a(), b(), b'()), c()), d()) -> c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'())) , f^#(g(h(a(), b()), c()), d()) -> c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) } 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))