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: { norm(nil()) -> 0() , norm(g(x, y)) -> s(norm(x)) , f(x, nil()) -> g(nil(), x) , f(x, g(y, z)) -> g(f(x, y), z) , rem(nil(), y) -> nil() , rem(g(x, y), 0()) -> g(x, y) , rem(g(x, y), s(z)) -> rem(x, z) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { norm^#(nil()) -> c_1() , norm^#(g(x, y)) -> c_2(norm^#(x)) , f^#(x, nil()) -> c_3() , f^#(x, g(y, z)) -> c_4(f^#(x, y)) , rem^#(nil(), y) -> c_5() , rem^#(g(x, y), 0()) -> c_6() , rem^#(g(x, y), s(z)) -> c_7(rem^#(x, z)) } 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: { norm^#(nil()) -> c_1() , norm^#(g(x, y)) -> c_2(norm^#(x)) , f^#(x, nil()) -> c_3() , f^#(x, g(y, z)) -> c_4(f^#(x, y)) , rem^#(nil(), y) -> c_5() , rem^#(g(x, y), 0()) -> c_6() , rem^#(g(x, y), s(z)) -> c_7(rem^#(x, z)) } Strict Trs: { norm(nil()) -> 0() , norm(g(x, y)) -> s(norm(x)) , f(x, nil()) -> g(nil(), x) , f(x, g(y, z)) -> g(f(x, y), z) , rem(nil(), y) -> nil() , rem(g(x, y), 0()) -> g(x, y) , rem(g(x, y), s(z)) -> rem(x, z) } 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: { norm^#(nil()) -> c_1() , norm^#(g(x, y)) -> c_2(norm^#(x)) , f^#(x, nil()) -> c_3() , f^#(x, g(y, z)) -> c_4(f^#(x, y)) , rem^#(nil(), y) -> c_5() , rem^#(g(x, y), 0()) -> c_6() , rem^#(g(x, y), s(z)) -> c_7(rem^#(x, z)) } 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_2) = {1}, Uargs(c_4) = {1}, Uargs(c_7) = {1} TcT has computed following constructor-restricted matrix interpretation. [nil] = [0] [0] = [1] [g](x1, x2) = [1] x1 + [0] [s](x1) = [1] x1 + [2] [norm^#](x1) = [1] [c_1] = [0] [c_2](x1) = [1] x1 + [0] [f^#](x1, x2) = [2] x1 + [0] [c_3] = [1] [c_4](x1) = [1] x1 + [2] [rem^#](x1, x2) = [0] [c_5] = [1] [c_6] = [1] [c_7](x1) = [1] x1 + [2] 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: { norm^#(g(x, y)) -> c_2(norm^#(x)) , f^#(x, nil()) -> c_3() , f^#(x, g(y, z)) -> c_4(f^#(x, y)) , rem^#(nil(), y) -> c_5() , rem^#(g(x, y), 0()) -> c_6() , rem^#(g(x, y), s(z)) -> c_7(rem^#(x, z)) } Weak DPs: { norm^#(nil()) -> c_1() } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) We estimate the number of application of {2,4,5} by applications of Pre({2,4,5}) = {3,6}. Here rules are labeled as follows: DPs: { 1: norm^#(g(x, y)) -> c_2(norm^#(x)) , 2: f^#(x, nil()) -> c_3() , 3: f^#(x, g(y, z)) -> c_4(f^#(x, y)) , 4: rem^#(nil(), y) -> c_5() , 5: rem^#(g(x, y), 0()) -> c_6() , 6: rem^#(g(x, y), s(z)) -> c_7(rem^#(x, z)) , 7: norm^#(nil()) -> c_1() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { norm^#(g(x, y)) -> c_2(norm^#(x)) , f^#(x, g(y, z)) -> c_4(f^#(x, y)) , rem^#(g(x, y), s(z)) -> c_7(rem^#(x, z)) } Weak DPs: { norm^#(nil()) -> c_1() , f^#(x, nil()) -> c_3() , rem^#(nil(), y) -> c_5() , rem^#(g(x, y), 0()) -> c_6() } 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. { norm^#(nil()) -> c_1() , f^#(x, nil()) -> c_3() , rem^#(nil(), y) -> c_5() , rem^#(g(x, y), 0()) -> c_6() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { norm^#(g(x, y)) -> c_2(norm^#(x)) , f^#(x, g(y, z)) -> c_4(f^#(x, y)) , rem^#(g(x, y), s(z)) -> c_7(rem^#(x, z)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_4) = {1}, Uargs(c_7) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [g](x1, x2) = [1] x1 + [1] x2 + [2] [s](x1) = [1] x1 + [2] [norm^#](x1) = [2] x1 + [0] [c_2](x1) = [1] x1 + [1] [f^#](x1, x2) = [3] x1 + [2] x2 + [0] [c_4](x1) = [1] x1 + [1] [rem^#](x1, x2) = [2] x2 + [0] [c_7](x1) = [1] x1 + [1] This order satisfies following ordering constraints: [norm^#(g(x, y))] = [2] x + [2] y + [4] > [2] x + [1] = [c_2(norm^#(x))] [f^#(x, g(y, z))] = [3] x + [2] y + [2] z + [4] > [3] x + [2] y + [1] = [c_4(f^#(x, y))] [rem^#(g(x, y), s(z))] = [2] z + [4] > [2] z + [1] = [c_7(rem^#(x, z))] Hurray, we answered YES(O(1),O(n^1))