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: { +(0(), y) -> y , +(s(x), y) -> +(x, s(y)) , +(s(x), y) -> s(+(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We add following weak dependency pairs: Strict DPs: { +^#(0(), y) -> c_1() , +^#(s(x), y) -> c_2(+^#(x, s(y))) , +^#(s(x), y) -> c_3(+^#(x, y)) } 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: { +^#(0(), y) -> c_1() , +^#(s(x), y) -> c_2(+^#(x, s(y))) , +^#(s(x), y) -> c_3(+^#(x, y)) } Strict Trs: { +(0(), y) -> y , +(s(x), y) -> +(x, s(y)) , +(s(x), y) -> s(+(x, y)) } 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: { +^#(0(), y) -> c_1() , +^#(s(x), y) -> c_2(+^#(x, s(y))) , +^#(s(x), y) -> c_3(+^#(x, y)) } 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_3) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [2] [s](x1) = [1] x1 + [2] [+^#](x1, x2) = [2] x1 + [1] x2 + [2] [c_1] = [1] [c_2](x1) = [1] x1 + [2] [c_3](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: { +^#(s(x), y) -> c_2(+^#(x, s(y))) } Weak DPs: { +^#(0(), y) -> c_1() , +^#(s(x), y) -> c_3(+^#(x, y)) } 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. { +^#(0(), y) -> c_1() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { +^#(s(x), y) -> c_2(+^#(x, s(y))) } Weak DPs: { +^#(s(x), y) -> c_3(+^#(x, y)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_3) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [s](x1) = [1] x1 + [2] [+^#](x1, x2) = [2] x1 + [0] [c_2](x1) = [1] x1 + [3] [c_3](x1) = [1] x1 + [3] This order satisfies following ordering constraints: [+^#(s(x), y)] = [2] x + [4] > [2] x + [3] = [c_2(+^#(x, s(y)))] [+^#(s(x), y)] = [2] x + [4] > [2] x + [3] = [c_3(+^#(x, y))] Hurray, we answered YES(O(1),O(n^1))