We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add the following dependency tuples: Strict DPs: { h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(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: { h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) } Weak Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. DPs: { 1: h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(1)). [h](x1) = [0 1] x1 + [0] [4 0] [2] [f](x1, x2) = [1 1] x1 + [0 0] x2 + [2] [0 0] [1 1] [2] [a] = [0] [0] [h^#](x1) = [3 0] x1 + [0] [0 1] [0] [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 0] [0 0] [0] The order satisfies the following ordering constraints: [h(f(x, y))] = [0 0] x + [1 1] y + [2] [4 4] [0 0] [10] >= [0 0] x + [1 1] y + [2] [4 4] [0 0] [10] = [f(y, f(h(h(x)), a()))] [h^#(f(x, y))] = [3 3] x + [0 0] y + [6] [0 0] [1 1] [2] > [3 3] x + [1] [0 0] [0] = [c_1(h^#(h(x)), h^#(x))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) } Weak Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) } 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. { h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) } Obligation: innermost runtime complexity Answer: YES(O(1),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),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))