YES(O(1),O(n^2)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { double(x) -> +(x, x) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , +(s(x), y) -> s(+(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add following weak dependency pairs: Strict DPs: { double^#(x) -> c_1(+^#(x, x)) , double^#(0()) -> c_2() , double^#(s(x)) -> c_3(double^#(x)) , +^#(x, 0()) -> c_4() , +^#(x, s(y)) -> c_5(+^#(x, y)) , +^#(s(x), y) -> c_6(+^#(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^2)). Strict DPs: { double^#(x) -> c_1(+^#(x, x)) , double^#(0()) -> c_2() , double^#(s(x)) -> c_3(double^#(x)) , +^#(x, 0()) -> c_4() , +^#(x, s(y)) -> c_5(+^#(x, y)) , +^#(s(x), y) -> c_6(+^#(x, y)) } Strict Trs: { double(x) -> +(x, x) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , +(s(x), y) -> s(+(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) 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^2)). Strict DPs: { double^#(x) -> c_1(+^#(x, x)) , double^#(0()) -> c_2() , double^#(s(x)) -> c_3(double^#(x)) , +^#(x, 0()) -> c_4() , +^#(x, s(y)) -> c_5(+^#(x, y)) , +^#(s(x), y) -> c_6(+^#(x, y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [0] [s](x1) = [1] x1 + [0] [double^#](x1) = [1] [c_1](x1) = [1] x1 + [2] [+^#](x1, x2) = [0] [c_2] = [0] [c_3](x1) = [1] x1 + [0] [c_4] = [1] [c_5](x1) = [1] x1 + [1] [c_6](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^2)). Strict DPs: { double^#(x) -> c_1(+^#(x, x)) , double^#(s(x)) -> c_3(double^#(x)) , +^#(x, 0()) -> c_4() , +^#(x, s(y)) -> c_5(+^#(x, y)) , +^#(s(x), y) -> c_6(+^#(x, y)) } Weak DPs: { double^#(0()) -> c_2() } Obligation: innermost runtime complexity Answer: YES(?,O(n^2)) We estimate the number of application of {3} by applications of Pre({3}) = {1,4,5}. Here rules are labeled as follows: DPs: { 1: double^#(x) -> c_1(+^#(x, x)) , 2: double^#(s(x)) -> c_3(double^#(x)) , 3: +^#(x, 0()) -> c_4() , 4: +^#(x, s(y)) -> c_5(+^#(x, y)) , 5: +^#(s(x), y) -> c_6(+^#(x, y)) , 6: double^#(0()) -> c_2() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^2)). Strict DPs: { double^#(x) -> c_1(+^#(x, x)) , double^#(s(x)) -> c_3(double^#(x)) , +^#(x, s(y)) -> c_5(+^#(x, y)) , +^#(s(x), y) -> c_6(+^#(x, y)) } Weak DPs: { double^#(0()) -> c_2() , +^#(x, 0()) -> c_4() } Obligation: innermost runtime complexity Answer: YES(?,O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { double^#(0()) -> c_2() , +^#(x, 0()) -> c_4() } We are left with following problem, upon which TcT provides the certificate YES(?,O(n^2)). Strict DPs: { double^#(x) -> c_1(+^#(x, x)) , double^#(s(x)) -> c_3(double^#(x)) , +^#(x, s(y)) -> c_5(+^#(x, y)) , +^#(s(x), y) -> c_6(+^#(x, y)) } Obligation: innermost runtime complexity Answer: YES(?,O(n^2)) The input was oriented with the instance of 'Small Polynomial Path Order (PS)' as induced by the safe mapping safe(s) = {1}, safe(double^#) = {}, safe(c_1) = {}, safe(+^#) = {}, safe(c_3) = {}, safe(c_5) = {}, safe(c_6) = {} and precedence double^# > +^# . Following symbols are considered recursive: {double^#, +^#} The recursion depth is 2. Further, following argument filtering is employed: pi(s) = [1], pi(double^#) = [1], pi(c_1) = [1], pi(+^#) = [1, 2], pi(c_3) = [1], pi(c_5) = [1], pi(c_6) = [1] Usable defined function symbols are a subset of: {double^#, +^#} For your convenience, here are the satisfied ordering constraints: pi(double^#(x)) = double^#(x;) > c_1(+^#(x, x;);) = pi(c_1(+^#(x, x))) pi(double^#(s(x))) = double^#(s(; x);) > c_3(double^#(x;);) = pi(c_3(double^#(x))) pi(+^#(x, s(y))) = +^#(x, s(; y);) > c_5(+^#(x, y;);) = pi(c_5(+^#(x, y))) pi(+^#(s(x), y)) = +^#(s(; x), y;) > c_6(+^#(x, y;);) = pi(c_6(+^#(x, y))) Hurray, we answered YES(O(1),O(n^2))