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) -> s(+(x, y)) , -(x, 0()) -> x , -(0(), y) -> 0() , -(s(x), s(y)) -> -(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, y)) , -^#(x, 0()) -> c_3() , -^#(0(), y) -> c_4() , -^#(s(x), s(y)) -> c_5(-^#(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, y)) , -^#(x, 0()) -> c_3() , -^#(0(), y) -> c_4() , -^#(s(x), s(y)) -> c_5(-^#(x, y)) } Strict Trs: { +(0(), y) -> y , +(s(x), y) -> s(+(x, y)) , -(x, 0()) -> x , -(0(), y) -> 0() , -(s(x), s(y)) -> -(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, y)) , -^#(x, 0()) -> c_3() , -^#(0(), y) -> c_4() , -^#(s(x), s(y)) -> c_5(-^#(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_5) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [0] [s](x1) = [1] x1 + [0] [+^#](x1, x2) = [2] x2 + [0] [c_1] = [1] [c_2](x1) = [1] x1 + [1] [-^#](x1, x2) = [1] [c_3] = [0] [c_4] = [0] [c_5](x1) = [1] x1 + [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(n^1)). Strict DPs: { +^#(0(), y) -> c_1() , +^#(s(x), y) -> c_2(+^#(x, y)) , -^#(s(x), s(y)) -> c_5(-^#(x, y)) } Weak DPs: { -^#(x, 0()) -> c_3() , -^#(0(), y) -> c_4() } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) We estimate the number of application of {1} by applications of Pre({1}) = {2}. Here rules are labeled as follows: DPs: { 1: +^#(0(), y) -> c_1() , 2: +^#(s(x), y) -> c_2(+^#(x, y)) , 3: -^#(s(x), s(y)) -> c_5(-^#(x, y)) , 4: -^#(x, 0()) -> c_3() , 5: -^#(0(), y) -> c_4() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { +^#(s(x), y) -> c_2(+^#(x, y)) , -^#(s(x), s(y)) -> c_5(-^#(x, y)) } Weak DPs: { +^#(0(), y) -> c_1() , -^#(x, 0()) -> c_3() , -^#(0(), y) -> c_4() } 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() , -^#(x, 0()) -> c_3() , -^#(0(), y) -> c_4() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^1)). Strict DPs: { +^#(s(x), y) -> c_2(+^#(x, y)) , -^#(s(x), s(y)) -> c_5(-^#(x, y)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^1)) The input was oriented with the instance of 'Small Polynomial Path Order (PS)' as induced by the safe mapping safe(s) = {1}, safe(+^#) = {2}, safe(c_2) = {}, safe(-^#) = {}, safe(c_5) = {} and precedence +^# ~ -^# . Following symbols are considered recursive: {+^#, -^#} The recursion depth is 1. Further, following argument filtering is employed: pi(s) = [1], pi(+^#) = [1, 2], pi(c_2) = [1], pi(-^#) = [1, 2], pi(c_5) = [1] Usable defined function symbols are a subset of: {+^#, -^#} For your convenience, here are the satisfied ordering constraints: pi(+^#(s(x), y)) = +^#(s(; x); y) > c_2(+^#(x; y);) = pi(c_2(+^#(x, y))) pi(-^#(s(x), s(y))) = -^#(s(; x), s(; y);) > c_5(-^#(x, y;);) = pi(c_5(-^#(x, y))) Hurray, we answered YES(O(1),O(n^1))