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: { p(m, n, s(r)) -> p(m, r, n) , p(m, s(n), 0()) -> p(0(), n, m) , p(m, 0(), 0()) -> m } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { p^#(m, n, s(r)) -> c_1(p^#(m, r, n)) , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m)) , p^#(m, 0(), 0()) -> c_3() } 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: { p^#(m, n, s(r)) -> c_1(p^#(m, r, n)) , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m)) , p^#(m, 0(), 0()) -> c_3() } Strict Trs: { p(m, n, s(r)) -> p(m, r, n) , p(m, s(n), 0()) -> p(0(), n, m) , p(m, 0(), 0()) -> m } 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: { p^#(m, n, s(r)) -> c_1(p^#(m, r, n)) , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m)) , p^#(m, 0(), 0()) -> c_3() } 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(c_2) = {1} TcT has computed following constructor-restricted matrix interpretation. [s](x1) = [1] x1 + [0] [0] = [0] [p^#](x1, x2, x3) = [1] [c_1](x1) = [1] x1 + [1] [c_2](x1) = [1] x1 + [0] [c_3] = [0] 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(n^1)). Strict DPs: { p^#(m, n, s(r)) -> c_1(p^#(m, r, n)) , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m)) } Weak DPs: { p^#(m, 0(), 0()) -> c_3() } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { p^#(m, 0(), 0()) -> c_3() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { p^#(m, n, s(r)) -> c_1(p^#(m, r, n)) , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [s](x1) = [1] x1 + [2] [0] = [0] [p^#](x1, x2, x3) = [2] x1 + [2] x2 + [2] x3 + [0] [c_1](x1) = [1] x1 + [1] [c_2](x1) = [1] x1 + [3] This order satisfies following ordering constraints: [p^#(m, n, s(r))] = [2] m + [2] n + [2] r + [4] > [2] m + [2] n + [2] r + [1] = [c_1(p^#(m, r, n))] [p^#(m, s(n), 0())] = [2] m + [2] n + [4] > [2] m + [2] n + [3] = [c_2(p^#(0(), n, m))] Hurray, we answered YES(O(1),O(n^1))