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: { rec[map_0][2](plus_x(x6), Nil()) -> Nil() , rec[map_0][2](plus_x(x14), Cons(x10, x6)) -> Cons(plus_x[1](x14, x10), rec[map_0][2](plus_x(x14), x6)) , plus_x[1](0(), x13) -> x13 , plus_x[1](S(x18), x4) -> S(plus_x[1](x18, x4)) , main(x11, x18) -> rec[map_0][2](plus_x(x18), x11) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add the following innermost weak dependency pairs: Strict DPs: { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(0(), x13) -> c_3() , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) , main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) } 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: { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(0(), x13) -> c_3() , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) , main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) } Strict Trs: { rec[map_0][2](plus_x(x6), Nil()) -> Nil() , rec[map_0][2](plus_x(x14), Cons(x10, x6)) -> Cons(plus_x[1](x14, x10), rec[map_0][2](plus_x(x14), x6)) , plus_x[1](0(), x13) -> x13 , plus_x[1](S(x18), x4) -> S(plus_x[1](x18, x4)) , main(x11, x18) -> rec[map_0][2](plus_x(x18), x11) } 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: { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(0(), x13) -> c_3() , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) , main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) } 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_2) = {1, 2}, Uargs(c_4) = {1}, Uargs(c_5) = {1} TcT has computed the following constructor-restricted matrix interpretation. [plus_x](x1) = [1 0] x1 + [0] [0 0] [0] [Nil] = [0] [0] [Cons](x1, x2) = [1 2] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] [0] = [2] [0] [S](x1) = [1 1] x1 + [2] [0 1] [0] [rec[map_0][2]^#](x1, x2) = [0] [0] [c_1] = [1] [0] [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [2] [plus_x[1]^#](x1, x2) = [0 0] x1 + [0 0] x2 + [0] [2 0] [2 2] [0] [c_3] = [2] [0] [c_4](x1) = [1 0] x1 + [2] [0 1] [0] [main^#](x1, x2) = [2 2] x1 + [2 2] x2 + [2] [2 2] [2 2] [2] [c_5](x1) = [1 0] x1 + [1] [0 1] [2] The following symbols are considered usable {rec[map_0][2]^#, plus_x[1]^#, main^#} The order satisfies the following ordering constraints: [rec[map_0][2]^#(plus_x(x6), Nil())] = [0] [0] ? [1] [0] = [c_1()] [rec[map_0][2]^#(plus_x(x14), Cons(x10, x6))] = [0] [0] ? [0 0] x14 + [0 0] x10 + [1] [2 0] [2 2] [2] = [c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6))] [plus_x[1]^#(0(), x13)] = [0 0] x13 + [0] [2 2] [4] ? [2] [0] = [c_3()] [plus_x[1]^#(S(x18), x4)] = [0 0] x18 + [0 0] x4 + [0] [2 2] [2 2] [4] ? [0 0] x18 + [0 0] x4 + [2] [2 0] [2 2] [0] = [c_4(plus_x[1]^#(x18, x4))] [main^#(x11, x18)] = [2 2] x18 + [2 2] x11 + [2] [2 2] [2 2] [2] > [1] [2] = [c_5(rec[map_0][2]^#(plus_x(x18), x11))] 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(1),O(n^2)). Strict DPs: { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(0(), x13) -> c_3() , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } Weak DPs: { main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {3} by applications of Pre({3}) = {2,4}. Here rules are labeled as follows: DPs: { 1: rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , 2: rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , 3: plus_x[1]^#(0(), x13) -> c_3() , 4: plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) , 5: main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } Weak DPs: { plus_x[1]^#(0(), x13) -> c_3() , main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { plus_x[1]^#(0(), x13) -> c_3() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } Weak DPs: { main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) Consider the dependency graph 1: rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() 2: rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) -->_1 plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) :3 -->_2 rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) :2 -->_2 rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() :1 3: plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) -->_1 plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) :3 4: main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) -->_1 rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) :2 -->_1 rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() :1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). { main^#(x11, x18) -> c_5(rec[map_0][2]^#(plus_x(x18), x11)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {1} by applications of Pre({1}) = {2}. Here rules are labeled as follows: DPs: { 1: rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() , 2: rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , 3: plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } Weak DPs: { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { rec[map_0][2]^#(plus_x(x6), Nil()) -> c_1() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'Small Polynomial Path Order (PS,2-bounded)' to orient following rules strictly. DPs: { 1: rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , 2: plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } Sub-proof: ---------- The input was oriented with the instance of 'Small Polynomial Path Order (PS,2-bounded)' as induced by the safe mapping safe(plus_x) = {1}, safe(Nil) = {}, safe(rec[map_0][2]) = {}, safe(Cons) = {1, 2}, safe(plus_x[1]) = {}, safe(0) = {}, safe(S) = {1}, safe(main) = {}, safe(rec[map_0][2]^#) = {}, safe(c_1) = {}, safe(c_2) = {}, safe(plus_x[1]^#) = {2}, safe(c_3) = {}, safe(c_4) = {}, safe(main^#) = {}, safe(c_5) = {} and precedence rec[map_0][2]^# > plus_x[1]^# . Following symbols are considered recursive: {rec[map_0][2]^#, plus_x[1]^#} The recursion depth is 2. Further, following argument filtering is employed: pi(plus_x) = [1], pi(Nil) = [], pi(rec[map_0][2]) = [], pi(Cons) = [2], pi(plus_x[1]) = [], pi(0) = [], pi(S) = [1], pi(main) = [], pi(rec[map_0][2]^#) = [1, 2], pi(c_1) = [], pi(c_2) = [1, 2], pi(plus_x[1]^#) = [1], pi(c_3) = [], pi(c_4) = [1], pi(main^#) = [], pi(c_5) = [] Usable defined function symbols are a subset of: {rec[map_0][2]^#, plus_x[1]^#, main^#} For your convenience, here are the satisfied ordering constraints: pi(rec[map_0][2]^#(plus_x(x14), Cons(x10, x6))) = rec[map_0][2]^#(plus_x(; x14), Cons(; x6);) > c_2(plus_x[1]^#(x14;), rec[map_0][2]^#(plus_x(; x14), x6;);) = pi(c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6))) pi(plus_x[1]^#(S(x18), x4)) = plus_x[1]^#(S(; x18);) > c_4(plus_x[1]^#(x18;);) = pi(c_4(plus_x[1]^#(x18, x4))) The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { rec[map_0][2]^#(plus_x(x14), Cons(x10, x6)) -> c_2(plus_x[1]^#(x14, x10), rec[map_0][2]^#(plus_x(x14), x6)) , plus_x[1]^#(S(x18), x4) -> c_4(plus_x[1]^#(x18, x4)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^2))