Problem Maude 06 LISTUTILITIES nosorts

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 LISTUTILITIES nosorts

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 LISTUTILITIES nosorts

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
     , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
     , afterNth(N, XS) -> snd(splitAt(N, XS))
     , and(tt(), X) -> X
     , fst(pair(X, Y)) -> X
     , head(cons(N, XS)) -> N
     , natsFrom(N) -> cons(N, natsFrom(s(N)))
     , sel(N, XS) -> head(afterNth(N, XS))
     , snd(pair(X, Y)) -> Y
     , splitAt(0(), XS) -> pair(nil(), XS)
     , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
     , tail(cons(N, XS)) -> XS
     , take(N, XS) -> fst(splitAt(N, XS))}

Proof Output:
  None of the processors succeeded.

  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: U11^#(tt(), N, X, XS) -> c_0(U12^#(splitAt(N, XS), X))
              , 2: U12^#(pair(YS, ZS), X) -> c_1()
              , 3: afterNth^#(N, XS) -> c_2(snd^#(splitAt(N, XS)))
              , 4: and^#(tt(), X) -> c_3()
              , 5: fst^#(pair(X, Y)) -> c_4()
              , 6: head^#(cons(N, XS)) -> c_5()
              , 7: natsFrom^#(N) -> c_6(natsFrom^#(s(N)))
              , 8: sel^#(N, XS) -> c_7(head^#(afterNth(N, XS)))
              , 9: snd^#(pair(X, Y)) -> c_8()
              , 10: splitAt^#(0(), XS) -> c_9()
              , 11: splitAt^#(s(N), cons(X, XS)) -> c_10(U11^#(tt(), N, X, XS))
              , 12: tail^#(cons(N, XS)) -> c_11()
              , 13: take^#(N, XS) -> c_12(fst^#(splitAt(N, XS)))}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{13}                                                      [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{12}                                                      [    YES(?,O(1))     ]

             ->{11}                                                      [         NA         ]
                |
                `->{1}                                                   [         NA         ]
                    |
                    `->{2}                                               [         NA         ]

             ->{10}                                                      [    YES(?,O(1))     ]

             ->{8}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{7}                                                       [       MAYBE        ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [         NA         ]
                |
                `->{9}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {3}: NA
             ------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {1}, Uargs(snd^#) = {1},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [2 0 1] x1 + [1 1 0] x2 + [0 0 0] x3 + [0 1 0] x4 + [0]
                                      [0 0 0]      [2 0 0]      [1 0 1]      [3 1 3]      [2]
                                      [0 0 0]      [2 1 0]      [1 0 1]      [1 3 3]      [1]
                tt() = [3]
                       [0]
                       [2]
                U12(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
                              [0 0 0]      [1 0 1]      [1]
                              [0 0 1]      [1 0 1]      [0]
                splitAt(x1, x2) = [1 1 0] x1 + [0 1 0] x2 + [2]
                                  [2 0 0]      [3 1 2]      [0]
                                  [2 1 0]      [1 3 1]      [1]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 1]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [1]
                cons(x1, x2) = [1 0 0] x1 + [1 2 3] x2 + [1]
                               [0 0 0]      [0 1 0]      [3]
                               [0 0 1]      [0 0 0]      [1]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [1]
                        [0 1 0]      [3]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                snd^#(x1) = [1 1 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {3}->{9}: NA
             -----------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {1}, Uargs(snd^#) = {1},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 1 0] x3 + [0 1 2] x4 + [3]
                                      [0 0 0]      [0 0 0]      [1 0 0]      [3 3 0]      [0]
                                      [0 0 0]      [0 1 0]      [1 0 0]      [1 3 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [1 0 0] x1 + [0 1 0] x2 + [1]
                              [0 0 0]      [1 0 0]      [0]
                              [0 0 0]      [1 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 1 2] x2 + [1]
                                  [0 0 0]      [3 0 0]      [0]
                                  [0 1 0]      [1 2 0]      [0]
                pair(x1, x2) = [0 1 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
                               [0 1 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                snd^#(x1) = [3 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(tt(), X) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                tt() = [2]
                       [2]
                       [2]
                and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                [2 2 0]      [0 0 0]      [3]
                                [2 2 2]      [0 0 0]      [3]
                c_3() = [0]
                        [1]
                        [1]

           * Path {7}: MAYBE
             ---------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {1}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [3 3 3] x1 + [0]
                                 [3 3 3]      [0]
                                 [3 3 3]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {natsFrom^#(N) -> c_6(natsFrom^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {8}: NA
             ------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {1},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {1}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [1 3 2] x2 + [1 0 0] x3 + [1 3 2] x4 + [3]
                                      [0 0 2]      [0 1 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 1 0]      [0]
                tt() = [0]
                       [0]
                       [2]
                U12(x1, x2) = [1 2 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [1]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [1 1 0] x1 + [1 2 2] x2 + [0]
                                  [0 1 1]      [0 0 0]      [0]
                                  [0 0 0]      [0 1 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [1 1 1] x2 + [0]
                               [0 0 0]      [0 0 0]      [1]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 1 2] x2 + [2]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [3]
                afterNth(x1, x2) = [1 1 0] x1 + [2 2 2] x2 + [3]
                                   [2 2 2]      [2 2 2]      [3]
                                   [2 2 2]      [2 2 2]      [3]
                snd(x1) = [1 0 0] x1 + [2]
                          [1 0 0]      [3]
                          [1 0 0]      [3]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 3 2] x1 + [0]
                        [0 1 0]      [1]
                        [0 0 0]      [3]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [2]
                      [2]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [1 0 0] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {8}->{6}: NA
             -----------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {1},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {1}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [1 2 3] x2 + [0 0 0] x3 + [2 1 0] x4 + [3]
                                      [0 2 0]      [1 2 3]      [0 0 0]      [2 1 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 1]      [0]
                tt() = [0]
                       [2]
                       [0]
                U12(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
                              [1 0 0]      [0 0 0]      [0]
                              [0 0 1]      [0 0 0]      [0]
                splitAt(x1, x2) = [1 2 2] x1 + [2 1 0] x2 + [1]
                                  [3 0 2]      [2 1 0]      [0]
                                  [0 0 0]      [0 0 1]      [0]
                pair(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [0]
                afterNth(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [3]
                                   [3 3 3]      [3 3 3]      [3]
                                   [3 3 3]      [3 3 3]      [3]
                snd(x1) = [1 0 0] x1 + [1]
                          [0 1 0]      [3]
                          [0 0 2]      [3]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 2 3] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [2]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [3 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {10}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {splitAt^#(0(), XS) -> c_9()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(splitAt^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                splitAt^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                    [2 2 0]      [0 0 0]      [3]
                                    [2 2 2]      [0 0 0]      [3]
                c_9() = [0]
                        [1]
                        [1]

           * Path {11}: NA
             -------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [3 0 0] x2 + [3 0 0] x3 + [3 0 0] x4 + [0]
                                        [0 0 0]      [3 0 0]      [3 0 0]      [3 0 0]      [0]
                                        [0 0 0]      [3 0 0]      [3 0 0]      [3 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [1 3 1] x3 + [3 3 3] x4 + [3]
                                      [0 0 0]      [0 0 0]      [1 3 2]      [3 3 3]      [0]
                                      [0 0 0]      [0 0 0]      [1 1 0]      [3 3 3]      [2]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [1 0 0] x1 + [1 3 1] x2 + [1]
                              [0 0 0]      [1 1 2]      [0]
                              [0 0 0]      [1 1 0]      [2]
                splitAt(x1, x2) = [0 0 0] x1 + [3 0 3] x2 + [1]
                                  [0 0 0]      [3 2 2]      [0]
                                  [0 0 0]      [3 0 0]      [2]
                pair(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [1]
                cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
                               [0 0 3]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [1]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [3 3 3] x2 + [3 3 3] x3 + [3 3 3] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                U12^#(x1, x2) = [1 0 0] x1 + [3 3 3] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}->{2}: NA
             -----------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0 1 1] x4 + [3]
                                      [0 0 0]      [0 1 0]      [0 0 0]      [0 0 3]      [3]
                                      [0 0 0]      [0 0 0]      [1 0 0]      [1 3 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [1 0 0] x1 + [0 1 0] x2 + [1]
                              [0 1 0]      [0 0 0]      [3]
                              [0 0 0]      [1 0 0]      [0]
                splitAt(x1, x2) = [0 1 0] x1 + [0 1 1] x2 + [1]
                                  [0 1 0]      [0 0 3]      [0]
                                  [0 0 0]      [1 2 0]      [0]
                pair(x1, x2) = [0 1 0] x1 + [0 0 0] x2 + [0]
                               [0 0 1]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
                               [0 1 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [3]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                U12^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {12}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(N, XS)) -> c_11()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
                               [0 0 0]      [0 0 0]      [2]
                               [0 0 0]      [0 0 0]      [2]
                tail^#(x1) = [0 2 0] x1 + [7]
                             [2 2 0]      [3]
                             [2 2 2]      [3]
                c_11() = [0]
                         [1]
                         [1]

           * Path {13}: NA
             -------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {1}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 0 0] x1 + [1 1 0] x2 + [1 0 0] x3 + [1 0 3] x4 + [3]
                                      [0 0 0]      [0 1 0]      [0 0 0]      [0 0 1]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [1]
                tt() = [3]
                       [0]
                       [0]
                U12(x1, x2) = [1 0 3] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [1]
                splitAt(x1, x2) = [1 1 0] x1 + [1 0 3] x2 + [1]
                                  [0 1 1]      [0 0 1]      [0]
                                  [0 0 0]      [0 0 0]      [3]
                pair(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [1]
                cons(x1, x2) = [1 0 0] x1 + [1 0 2] x2 + [1]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [3]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [2]
                        [0 1 0]      [3]
                        [0 0 0]      [2]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [1 0 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {13}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {1}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 1 0] x3 + [0 1 2] x4 + [3]
                                      [0 0 0]      [0 0 0]      [1 0 0]      [3 3 0]      [0]
                                      [0 0 0]      [0 1 0]      [1 0 0]      [1 3 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [1 0 0] x1 + [0 1 0] x2 + [1]
                              [0 0 0]      [1 0 0]      [0]
                              [0 0 0]      [1 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 1 2] x2 + [1]
                                  [0 0 0]      [3 0 0]      [0]
                                  [0 1 0]      [1 2 0]      [0]
                pair(x1, x2) = [0 1 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
                               [0 1 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                fst^#(x1) = [3 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: U11^#(tt(), N, X, XS) -> c_0(U12^#(splitAt(N, XS), X))
              , 2: U12^#(pair(YS, ZS), X) -> c_1()
              , 3: afterNth^#(N, XS) -> c_2(snd^#(splitAt(N, XS)))
              , 4: and^#(tt(), X) -> c_3()
              , 5: fst^#(pair(X, Y)) -> c_4()
              , 6: head^#(cons(N, XS)) -> c_5()
              , 7: natsFrom^#(N) -> c_6(natsFrom^#(s(N)))
              , 8: sel^#(N, XS) -> c_7(head^#(afterNth(N, XS)))
              , 9: snd^#(pair(X, Y)) -> c_8()
              , 10: splitAt^#(0(), XS) -> c_9()
              , 11: splitAt^#(s(N), cons(X, XS)) -> c_10(U11^#(tt(), N, X, XS))
              , 12: tail^#(cons(N, XS)) -> c_11()
              , 13: take^#(N, XS) -> c_12(fst^#(splitAt(N, XS)))}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{13}                                                      [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{12}                                                      [    YES(?,O(1))     ]

             ->{11}                                                      [         NA         ]
                |
                `->{1}                                                   [         NA         ]
                    |
                    `->{2}                                               [         NA         ]

             ->{10}                                                      [    YES(?,O(1))     ]

             ->{8}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{7}                                                       [       MAYBE        ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [         NA         ]
                |
                `->{9}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {3}: NA
             ------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {1}, Uargs(snd^#) = {1},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 3] x1 + [0 3] x2 + [1 0] x3 + [3 3] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [1]
                tt() = [0]
                       [3]
                U12(x1, x2) = [1 3] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [1]
                splitAt(x1, x2) = [0 3] x1 + [3 1] x2 + [2]
                                  [0 0]      [0 0]      [1]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [2]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [3]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                snd^#(x1) = [1 3] x1 + [0]
                            [3 3]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {3}->{9}: NA
             -----------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {1}, Uargs(snd^#) = {1},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [3 0] x4 + [3]
                                      [0 0]      [0 0]      [0 0]      [0 1]      [1]
                tt() = [0]
                       [0]
                U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 0]      [1]
                splitAt(x1, x2) = [1 1] x1 + [3 0] x2 + [1]
                                  [0 0]      [0 1]      [1]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 1]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [1]
                        [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                snd^#(x1) = [3 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(tt(), X) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                tt() = [2]
                       [2]
                and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                [2 2]      [0 0]      [7]
                c_3() = [0]
                        [1]

           * Path {7}: MAYBE
             ---------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {1}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [3 3] x1 + [0]
                                 [3 3]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {natsFrom^#(N) -> c_6(natsFrom^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {8}: NA
             ------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {1},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {1}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 2] x1 + [0 1] x2 + [0 0] x3 + [2 3] x4 + [3]
                                      [0 0]      [0 0]      [0 0]      [0 1]      [1]
                tt() = [0]
                       [3]
                U12(x1, x2) = [1 2] x1 + [0 0] x2 + [2]
                              [0 1]      [0 0]      [0]
                splitAt(x1, x2) = [0 1] x1 + [2 1] x2 + [1]
                                  [0 0]      [0 1]      [1]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [1]
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [3]
                               [0 0]      [0 1]      [2]
                afterNth(x1, x2) = [2 2] x1 + [2 1] x2 + [3]
                                   [2 2]      [2 2]      [3]
                snd(x1) = [1 0] x1 + [1]
                          [0 1]      [2]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [3]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [1 0] x1 + [0]
                             [3 3]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {8}->{6}: NA
             -----------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {1},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {1}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 1] x2 + [0 0] x3 + [2 0] x4 + [3]
                                      [0 0]      [0 1]      [0 0]      [0 1]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                              [0 1]      [0 0]      [0]
                splitAt(x1, x2) = [0 1] x1 + [2 0] x2 + [1]
                                  [0 1]      [0 1]      [0]
                pair(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                               [0 0]      [0 1]      [0]
                afterNth(x1, x2) = [3 3] x1 + [3 3] x2 + [3]
                                   [3 3]      [3 3]      [3]
                snd(x1) = [1 0] x1 + [1]
                          [0 2]      [3]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [3 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {10}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {splitAt^#(0(), XS) -> c_9()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(splitAt^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                splitAt^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                    [2 2]      [0 0]      [7]
                c_9() = [0]
                        [1]

           * Path {11}: NA
             -------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [3 0] x2 + [3 0] x3 + [3 0] x4 + [0]
                                        [0 0]      [3 0]      [3 0]      [3 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [1 3] x1 + [0 1] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 0] x1 + [3 0] x2 + [0 1] x3 + [0 1] x4 + [0]
                                      [2 0]      [2 0]      [0 0]      [0 0]      [2]
                tt() = [3]
                       [0]
                U12(x1, x2) = [1 0] x1 + [0 1] x2 + [1]
                              [0 0]      [0 0]      [2]
                splitAt(x1, x2) = [3 0] x1 + [0 1] x2 + [1]
                                  [2 0]      [0 0]      [2]
                pair(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [3]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [3 3] x2 + [3 3] x3 + [3 3] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                U12^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}->{2}: NA
             -----------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 1] x2 + [0 0] x3 + [2 3] x4 + [3]
                                      [0 0]      [0 1]      [0 0]      [2 0]      [1]
                tt() = [0]
                       [0]
                U12(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                              [0 0]      [0 0]      [1]
                splitAt(x1, x2) = [0 1] x1 + [2 3] x2 + [1]
                                  [0 1]      [2 0]      [1]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
                               [0 0]      [0 1]      [2]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                U12^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {12}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(N, XS)) -> c_11()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
                               [0 0]      [0 0]      [2]
                tail^#(x1) = [2 0] x1 + [7]
                             [2 2]      [7]
                c_11() = [0]
                         [1]

           * Path {13}: NA
             -------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {1}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [2 3] x1 + [3 3] x2 + [1 0] x3 + [3 3] x4 + [3]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [2]
                       [3]
                U12(x1, x2) = [1 2] x1 + [1 0] x2 + [1]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [3 2] x1 + [3 3] x2 + [2]
                                  [0 0]      [0 0]      [3]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
                               [0 0]      [0 1]      [3]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 3] x1 + [3]
                        [0 0]      [3]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [1 3] x1 + [0]
                            [3 3]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {13}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {1}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [3 0] x4 + [3]
                                      [0 0]      [0 0]      [0 0]      [0 1]      [1]
                tt() = [0]
                       [0]
                U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 0]      [1]
                splitAt(x1, x2) = [1 1] x1 + [3 0] x2 + [1]
                                  [0 0]      [0 1]      [1]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 1]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [1]
                        [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                fst^#(x1) = [3 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5() = [0]
                        [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8() = [0]
                        [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11() = [0]
                         [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: U11^#(tt(), N, X, XS) -> c_0(U12^#(splitAt(N, XS), X))
              , 2: U12^#(pair(YS, ZS), X) -> c_1()
              , 3: afterNth^#(N, XS) -> c_2(snd^#(splitAt(N, XS)))
              , 4: and^#(tt(), X) -> c_3()
              , 5: fst^#(pair(X, Y)) -> c_4()
              , 6: head^#(cons(N, XS)) -> c_5()
              , 7: natsFrom^#(N) -> c_6(natsFrom^#(s(N)))
              , 8: sel^#(N, XS) -> c_7(head^#(afterNth(N, XS)))
              , 9: snd^#(pair(X, Y)) -> c_8()
              , 10: splitAt^#(0(), XS) -> c_9()
              , 11: splitAt^#(s(N), cons(X, XS)) -> c_10(U11^#(tt(), N, X, XS))
              , 12: tail^#(cons(N, XS)) -> c_11()
              , 13: take^#(N, XS) -> c_12(fst^#(splitAt(N, XS)))}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{13}                                                      [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{12}                                                      [    YES(?,O(1))     ]

             ->{11}                                                      [         NA         ]
                |
                `->{1}                                                   [         NA         ]
                    |
                    `->{2}                                               [         NA         ]

             ->{10}                                                      [    YES(?,O(1))     ]

             ->{8}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{7}                                                       [       MAYBE        ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [         NA         ]
                |
                `->{9}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {3}: NA
             ------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {1}, Uargs(snd^#) = {1},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [2] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [0] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [2] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                snd^#(x1) = [1] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {3}->{9}: NA
             -----------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {1}, Uargs(snd^#) = {1},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [3]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                snd^#(x1) = [3] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(tt(), X) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                tt() = [7]
                and^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_3() = [1]

           * Path {7}: MAYBE
             ---------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {1}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [3] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {natsFrom^#(N) -> c_6(natsFrom^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {8}: NA
             ------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {1},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {1}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [2] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [0] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [2] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                afterNth(x1, x2) = [3] x1 + [3] x2 + [3]
                snd(x1) = [1] x1 + [1]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [1] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7(x1) = [1] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {8}->{6}: NA
             -----------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {1},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {1}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                afterNth(x1, x2) = [3] x1 + [3] x2 + [3]
                snd(x1) = [1] x1 + [1]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [3] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [1] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {10}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {splitAt^#(0(), XS) -> c_9()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(splitAt^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                splitAt^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_9() = [1]

           * Path {11}: NA
             -------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [3] x2 + [3] x3 + [3] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [2] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [0] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [2] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [3] x2 + [3] x3 + [3] x4 + [0]
                c_0(x1) = [1] x1 + [0]
                U12^#(x1, x2) = [1] x1 + [3] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}->{2}: NA
             -----------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [1] x1 + [0]
                U12^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {12}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [0] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(N, XS)) -> c_11()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [0] x2 + [7]
                tail^#(x1) = [1] x1 + [7]
                c_11() = [1]

           * Path {13}: NA
             -------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {1}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [1] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_12(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {13}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(afterNth^#) = {}, Uargs(c_2) = {}, Uargs(snd^#) = {},
                 Uargs(and^#) = {}, Uargs(fst^#) = {1}, Uargs(head^#) = {},
                 Uargs(natsFrom^#) = {}, Uargs(c_6) = {}, Uargs(sel^#) = {},
                 Uargs(c_7) = {}, Uargs(splitAt^#) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(take^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [3]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                fst^#(x1) = [3] x1 + [0]
                c_4() = [0]
                head^#(x1) = [0] x1 + [0]
                c_5() = [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8() = [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11() = [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible

    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.

    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 LISTUTILITIES nosorts

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 LISTUTILITIES nosorts

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
     , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
     , afterNth(N, XS) -> snd(splitAt(N, XS))
     , and(tt(), X) -> X
     , fst(pair(X, Y)) -> X
     , head(cons(N, XS)) -> N
     , natsFrom(N) -> cons(N, natsFrom(s(N)))
     , sel(N, XS) -> head(afterNth(N, XS))
     , snd(pair(X, Y)) -> Y
     , splitAt(0(), XS) -> pair(nil(), XS)
     , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
     , tail(cons(N, XS)) -> XS
     , take(N, XS) -> fst(splitAt(N, XS))}

Proof Output:
  None of the processors succeeded.

  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: U11^#(tt(), N, X, XS) -> c_0(U12^#(splitAt(N, XS), X))
              , 2: U12^#(pair(YS, ZS), X) -> c_1(X, YS, ZS)
              , 3: afterNth^#(N, XS) -> c_2(snd^#(splitAt(N, XS)))
              , 4: and^#(tt(), X) -> c_3(X)
              , 5: fst^#(pair(X, Y)) -> c_4(X)
              , 6: head^#(cons(N, XS)) -> c_5(N)
              , 7: natsFrom^#(N) -> c_6(N, natsFrom^#(s(N)))
              , 8: sel^#(N, XS) -> c_7(head^#(afterNth(N, XS)))
              , 9: snd^#(pair(X, Y)) -> c_8(Y)
              , 10: splitAt^#(0(), XS) -> c_9(XS)
              , 11: splitAt^#(s(N), cons(X, XS)) -> c_10(U11^#(tt(), N, X, XS))
              , 12: tail^#(cons(N, XS)) -> c_11(XS)
              , 13: take^#(N, XS) -> c_12(fst^#(splitAt(N, XS)))}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{13}                                                      [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{12}                                                      [   YES(?,O(n^3))    ]

             ->{11}                                                      [         NA         ]
                |
                `->{1}                                                   [         NA         ]
                    |
                    `->{2}                                               [         NA         ]

             ->{10}                                                      [    YES(?,O(1))     ]

             ->{8}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{7}                                                       [       MAYBE        ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [         NA         ]
                |
                `->{9}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {3}: NA
             ------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {1},
                 Uargs(snd^#) = {1}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [2 0 1] x1 + [1 1 0] x2 + [0 0 0] x3 + [0 1 0] x4 + [0]
                                      [0 0 0]      [2 0 0]      [1 0 1]      [3 1 3]      [2]
                                      [0 0 0]      [2 1 0]      [1 0 1]      [1 3 3]      [1]
                tt() = [3]
                       [0]
                       [2]
                U12(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
                              [0 0 0]      [1 0 1]      [1]
                              [0 0 1]      [1 0 1]      [0]
                splitAt(x1, x2) = [1 1 0] x1 + [0 1 0] x2 + [2]
                                  [2 0 0]      [3 1 2]      [0]
                                  [2 1 0]      [1 3 1]      [1]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 1]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [1]
                cons(x1, x2) = [1 0 0] x1 + [1 2 3] x2 + [1]
                               [0 0 0]      [0 1 0]      [3]
                               [0 0 1]      [0 0 0]      [1]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [1]
                        [0 1 0]      [3]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                snd^#(x1) = [1 1 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {3}->{9}: NA
             -----------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {1},
                 Uargs(snd^#) = {1}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [2 2 3] x1 + [3 3 3] x2 + [0 0 0] x3 + [0 0 1] x4 + [3]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 1 1]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [1]
                tt() = [2]
                       [2]
                       [2]
                U12(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [3]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 1]      [0 0 0]      [0]
                splitAt(x1, x2) = [3 3 3] x1 + [0 0 1] x2 + [3]
                                  [0 0 0]      [2 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [1]
                pair(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [1]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [1]
                cons(x1, x2) = [1 0 0] x1 + [1 0 1] x2 + [0]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [3]
                        [0 1 2]      [3]
                        [0 0 1]      [3]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                snd^#(x1) = [3 3 3] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(tt(), X) -> c_3(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                tt() = [2]
                       [0]
                       [2]
                and^#(x1, x2) = [2 0 2] x1 + [7 7 7] x2 + [7]
                                [2 0 2]      [7 7 7]      [7]
                                [2 0 2]      [7 7 7]      [7]
                c_3(x1) = [1 3 3] x1 + [0]
                          [1 1 1]      [1]
                          [1 1 1]      [1]

           * Path {7}: MAYBE
             ---------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {2},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 1 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [1 3 3] x1 + [0]
                                 [3 3 3]      [0]
                                 [3 3 3]      [0]
                c_6(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {natsFrom^#(N) -> c_6(N, natsFrom^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {8}: NA
             ------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {1},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {1}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [1 3 2] x2 + [1 0 0] x3 + [1 3 2] x4 + [3]
                                      [0 0 2]      [0 1 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 1 0]      [0]
                tt() = [0]
                       [0]
                       [2]
                U12(x1, x2) = [1 2 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [1]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [1 1 0] x1 + [1 2 2] x2 + [0]
                                  [0 1 1]      [0 0 0]      [0]
                                  [0 0 0]      [0 1 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [1 1 1] x2 + [0]
                               [0 0 0]      [0 0 0]      [1]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 1 2] x2 + [2]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [3]
                afterNth(x1, x2) = [1 1 0] x1 + [2 2 2] x2 + [3]
                                   [2 2 2]      [2 2 2]      [3]
                                   [2 2 2]      [2 2 2]      [3]
                snd(x1) = [1 0 0] x1 + [2]
                          [1 0 0]      [3]
                          [1 0 0]      [3]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 3 2] x1 + [0]
                        [0 1 0]      [1]
                        [0 0 0]      [3]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [2]
                      [2]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [1 0 0] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {8}->{6}: NA
             -----------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {1},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {1}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 2 2] x1 + [1 3 3] x2 + [1 2 0] x3 + [1 1 0] x4 + [0]
                                      [0 0 2]      [2 3 1]      [0 1 0]      [2 3 1]      [0]
                                      [0 0 0]      [0 0 0]      [0 1 0]      [0 1 1]      [0]
                tt() = [3]
                       [2]
                       [2]
                U12(x1, x2) = [1 0 0] x1 + [1 2 0] x2 + [1]
                              [0 1 0]      [0 1 0]      [3]
                              [0 0 1]      [0 1 0]      [0]
                splitAt(x1, x2) = [1 3 3] x1 + [1 0 0] x2 + [1]
                                  [2 1 0]      [2 3 0]      [1]
                                  [0 0 0]      [0 1 1]      [0]
                pair(x1, x2) = [0 2 2] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 1 0]      [1]
                               [0 0 0]      [0 0 1]      [0]
                cons(x1, x2) = [1 2 0] x1 + [1 1 0] x2 + [0]
                               [0 1 0]      [0 1 1]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [3]
                                   [3 3 3]      [3 3 3]      [3]
                                   [3 3 3]      [3 3 3]      [3]
                snd(x1) = [1 0 0] x1 + [1]
                          [0 1 0]      [2]
                          [0 0 1]      [3]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 2 0] x1 + [0]
                        [0 1 1]      [3]
                        [0 0 0]      [3]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [3 1 3] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [1 1 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {10}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {splitAt^#(0(), XS) -> c_9(XS)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                      [2]
                splitAt^#(x1, x2) = [2 0 2] x1 + [7 7 7] x2 + [7]
                                    [2 0 2]      [7 7 7]      [7]
                                    [2 0 2]      [7 7 7]      [7]
                c_9(x1) = [1 3 3] x1 + [0]
                          [1 1 1]      [1]
                          [1 1 1]      [1]

           * Path {11}: NA
             -------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [3 0 0] x2 + [3 0 0] x3 + [3 0 0] x4 + [0]
                                        [0 0 0]      [3 0 0]      [3 0 0]      [3 0 0]      [0]
                                        [0 0 0]      [3 0 0]      [3 0 0]      [3 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [1 3 1] x3 + [3 3 3] x4 + [3]
                                      [0 0 0]      [0 0 0]      [1 3 2]      [3 3 3]      [0]
                                      [0 0 0]      [0 0 0]      [1 1 0]      [3 3 3]      [2]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [1 0 0] x1 + [1 3 1] x2 + [1]
                              [0 0 0]      [1 1 2]      [0]
                              [0 0 0]      [1 1 0]      [2]
                splitAt(x1, x2) = [0 0 0] x1 + [3 0 3] x2 + [1]
                                  [0 0 0]      [3 2 2]      [0]
                                  [0 0 0]      [3 0 0]      [2]
                pair(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [1]
                cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
                               [0 0 3]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [1]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [3 3 3] x2 + [3 3 3] x3 + [3 3 3] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                U12^#(x1, x2) = [1 0 0] x1 + [3 3 3] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}->{2}: NA
             -----------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 1 2] x1 + [3 1 0] x2 + [1 0 0] x3 + [3 0 3] x4 + [3]
                                      [0 0 0]      [0 1 0]      [1 0 0]      [3 0 0]      [0]
                                      [1 0 0]      [0 0 0]      [0 0 0]      [0 0 1]      [2]
                tt() = [3]
                       [3]
                       [1]
                U12(x1, x2) = [1 0 3] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [1 0 0]      [0]
                              [0 0 1]      [0 0 0]      [2]
                splitAt(x1, x2) = [3 1 0] x1 + [3 0 0] x2 + [3]
                                  [0 1 2]      [3 0 3]      [0]
                                  [0 0 0]      [0 0 1]      [3]
                pair(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [3]
                cons(x1, x2) = [1 0 0] x1 + [1 0 1] x2 + [1]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [3]
                        [0 1 0]      [3]
                        [0 0 0]      [2]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                U12^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [1 1 1] x1 + [0 1 1] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {12}: YES(?,O(n^3))
             ------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tt() = [0]
                       [0]
                       [0]
                U12(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                splitAt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
                               [0 1 1]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [0]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [1 3 3] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [1 0 1] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(N, XS)) -> c_11(XS)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
                               [0 0 0]      [0 0 2]      [2]
                               [0 0 0]      [0 0 0]      [2]
                tail^#(x1) = [2 2 2] x1 + [3]
                             [2 2 2]      [3]
                             [2 2 2]      [3]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [1]
                           [0 0 0]      [1]

           * Path {13}: NA
             -------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {1}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 0 0] x1 + [1 1 0] x2 + [1 0 0] x3 + [1 0 3] x4 + [3]
                                      [0 0 0]      [0 1 0]      [0 0 0]      [0 0 1]      [0]
                                      [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [1]
                tt() = [3]
                       [0]
                       [0]
                U12(x1, x2) = [1 0 3] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [1]
                splitAt(x1, x2) = [1 1 0] x1 + [1 0 3] x2 + [1]
                                  [0 1 1]      [0 0 1]      [0]
                                  [0 0 0]      [0 0 0]      [3]
                pair(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [1]
                cons(x1, x2) = [1 0 0] x1 + [1 0 2] x2 + [1]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [3]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [2]
                        [0 1 0]      [3]
                        [0 0 0]      [2]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [1 0 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

           * Path {13}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {1}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 2 3] x1 + [2 1 3] x2 + [0 2 0] x3 + [2 2 3] x4 + [1]
                                      [0 0 0]      [0 1 0]      [1 0 0]      [1 1 1]      [0]
                                      [0 0 0]      [0 1 0]      [1 0 0]      [1 1 2]      [2]
                tt() = [1]
                       [2]
                       [3]
                U12(x1, x2) = [1 0 0] x1 + [0 2 0] x2 + [1]
                              [0 0 0]      [1 0 0]      [0]
                              [0 0 0]      [1 0 0]      [1]
                splitAt(x1, x2) = [2 1 3] x1 + [2 2 3] x2 + [1]
                                  [0 3 0]      [1 0 3]      [0]
                                  [0 3 0]      [1 0 0]      [2]
                pair(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [1]
                cons(x1, x2) = [1 0 0] x1 + [1 1 2] x2 + [2]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [3]
                afterNth(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                snd(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                fst(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                natsFrom(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 0 1] x1 + [3]
                        [0 1 0]      [3]
                        [0 0 1]      [3]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                nil() = [3]
                        [0]
                        [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U12^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                afterNth^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                     [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                snd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                fst^#(x1) = [3 3 3] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_4(x1) = [0 1 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                natsFrom^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                splitAt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                    [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

             We have not generated a proof for the resulting sub-problem.

    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: U11^#(tt(), N, X, XS) -> c_0(U12^#(splitAt(N, XS), X))
              , 2: U12^#(pair(YS, ZS), X) -> c_1(X, YS, ZS)
              , 3: afterNth^#(N, XS) -> c_2(snd^#(splitAt(N, XS)))
              , 4: and^#(tt(), X) -> c_3(X)
              , 5: fst^#(pair(X, Y)) -> c_4(X)
              , 6: head^#(cons(N, XS)) -> c_5(N)
              , 7: natsFrom^#(N) -> c_6(N, natsFrom^#(s(N)))
              , 8: sel^#(N, XS) -> c_7(head^#(afterNth(N, XS)))
              , 9: snd^#(pair(X, Y)) -> c_8(Y)
              , 10: splitAt^#(0(), XS) -> c_9(XS)
              , 11: splitAt^#(s(N), cons(X, XS)) -> c_10(U11^#(tt(), N, X, XS))
              , 12: tail^#(cons(N, XS)) -> c_11(XS)
              , 13: take^#(N, XS) -> c_12(fst^#(splitAt(N, XS)))}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{13}                                                      [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{12}                                                      [   YES(?,O(n^2))    ]

             ->{11}                                                      [         NA         ]
                |
                `->{1}                                                   [         NA         ]
                    |
                    `->{2}                                               [         NA         ]

             ->{10}                                                      [    YES(?,O(1))     ]

             ->{8}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{7}                                                       [       MAYBE        ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [         NA         ]
                |
                `->{9}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {3}: NA
             ------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {1},
                 Uargs(snd^#) = {1}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 3] x1 + [0 3] x2 + [1 0] x3 + [3 3] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [1]
                tt() = [0]
                       [3]
                U12(x1, x2) = [1 3] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [1]
                splitAt(x1, x2) = [0 3] x1 + [3 1] x2 + [2]
                                  [0 0]      [0 0]      [1]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [2]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [3]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                snd^#(x1) = [1 3] x1 + [0]
                            [3 3]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {3}->{9}: NA
             -----------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {1},
                 Uargs(snd^#) = {1}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 1] x1 + [3 0] x2 + [0 0] x3 + [2 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [3]
                tt() = [3]
                       [3]
                U12(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [3]
                splitAt(x1, x2) = [3 0] x1 + [2 0] x2 + [2]
                                  [0 0]      [0 0]      [3]
                pair(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [3]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 1]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [3]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                snd^#(x1) = [3 3] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 1] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(tt(), X) -> c_3(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                tt() = [2]
                       [2]
                and^#(x1, x2) = [2 2] x1 + [7 7] x2 + [7]
                                [2 2]      [7 7]      [3]
                c_3(x1) = [1 3] x1 + [0]
                          [1 1]      [1]

           * Path {7}: MAYBE
             ---------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {2},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [1 3] x1 + [0]
                                 [3 3]      [0]
                c_6(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {natsFrom^#(N) -> c_6(N, natsFrom^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {8}: NA
             ------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {1},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {1}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 2] x1 + [0 1] x2 + [0 0] x3 + [2 3] x4 + [3]
                                      [0 0]      [0 0]      [0 0]      [0 1]      [1]
                tt() = [0]
                       [3]
                U12(x1, x2) = [1 2] x1 + [0 0] x2 + [2]
                              [0 1]      [0 0]      [0]
                splitAt(x1, x2) = [0 1] x1 + [2 1] x2 + [1]
                                  [0 0]      [0 1]      [1]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [1]
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [3]
                               [0 0]      [0 1]      [2]
                afterNth(x1, x2) = [2 2] x1 + [2 1] x2 + [3]
                                   [2 2]      [2 2]      [3]
                snd(x1) = [1 0] x1 + [1]
                          [0 1]      [2]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [3]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [1 0] x1 + [0]
                             [3 3]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {8}->{6}: NA
             -----------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {1},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {1}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 0] x1 + [2 2] x2 + [0 0] x3 + [2 3] x4 + [0]
                                      [0 2]      [0 1]      [0 0]      [0 1]      [0]
                tt() = [3]
                       [2]
                U12(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
                              [0 1]      [0 0]      [2]
                splitAt(x1, x2) = [2 0] x1 + [2 0] x2 + [2]
                                  [0 1]      [0 1]      [2]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 1]      [2]
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [0]
                afterNth(x1, x2) = [3 3] x1 + [3 3] x2 + [3]
                                   [3 3]      [3 3]      [3]
                snd(x1) = [1 0] x1 + [0]
                          [0 1]      [1]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [2]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [3 3] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

             We have not generated a proof for the resulting sub-problem.

           * Path {10}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {splitAt^#(0(), XS) -> c_9(XS)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                splitAt^#(x1, x2) = [2 2] x1 + [7 7] x2 + [7]
                                    [2 2]      [7 7]      [3]
                c_9(x1) = [1 3] x1 + [0]
                          [1 1]      [1]

           * Path {11}: NA
             -------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [3 0] x2 + [3 0] x3 + [3 0] x4 + [0]
                                        [0 0]      [3 0]      [3 0]      [3 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [1 3] x1 + [0 1] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [3 0] x1 + [3 0] x2 + [0 1] x3 + [0 1] x4 + [0]
                                      [2 0]      [2 0]      [0 0]      [0 0]      [2]
                tt() = [3]
                       [0]
                U12(x1, x2) = [1 0] x1 + [0 1] x2 + [1]
                              [0 0]      [0 0]      [2]
                splitAt(x1, x2) = [3 0] x1 + [0 1] x2 + [1]
                                  [2 0]      [0 0]      [2]
                pair(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [3]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [3 3] x2 + [3 3] x3 + [3 3] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                U12^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}->{2}: NA
             -----------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [2 3] x1 + [1 2] x2 + [1 0] x3 + [3 1] x4 + [2]
                                      [3 3]      [0 1]      [1 0]      [3 0]      [0]
                tt() = [3]
                       [3]
                U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [1 0]      [1]
                splitAt(x1, x2) = [1 2] x1 + [3 1] x2 + [2]
                                  [0 3]      [3 0]      [1]
                pair(x1, x2) = [0 3] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
                               [0 0]      [0 1]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [1]
                        [0 1]      [3]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                U12^#(x1, x2) = [1 3] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [1 1] x1 + [0 1] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {12}: YES(?,O(n^2))
             ------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [3 3] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(N, XS)) -> c_11(XS)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [2]
                tail^#(x1) = [2 2] x1 + [7]
                             [2 0]      [7]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [1]

           * Path {13}: NA
             -------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {1}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [2 3] x1 + [3 3] x2 + [1 0] x3 + [3 3] x4 + [3]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                tt() = [2]
                       [3]
                U12(x1, x2) = [1 2] x1 + [1 0] x2 + [1]
                              [0 0]      [0 0]      [0]
                splitAt(x1, x2) = [3 2] x1 + [3 3] x2 + [2]
                                  [0 0]      [0 0]      [3]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
                               [0 0]      [0 1]      [3]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 3] x1 + [3]
                        [0 0]      [3]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [1 3] x1 + [0]
                            [3 3]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {13}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {1}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0 0] x1 + [0 1] x2 + [1 0] x3 + [1 1] x4 + [3]
                                      [0 0]      [0 0]      [1 0]      [1 1]      [1]
                tt() = [0]
                       [0]
                U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [1 0]      [1]
                splitAt(x1, x2) = [0 1] x1 + [1 0] x2 + [1]
                                  [0 0]      [1 0]      [1]
                pair(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [1]
                cons(x1, x2) = [1 0] x1 + [1 1] x2 + [2]
                               [0 0]      [0 0]      [0]
                afterNth(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                snd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fst(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                natsFrom(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [3]
                        [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U11^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                        [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U12^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                afterNth^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                     [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                snd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                fst^#(x1) = [3 3] x1 + [0]
                            [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                natsFrom^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                splitAt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: U11^#(tt(), N, X, XS) -> c_0(U12^#(splitAt(N, XS), X))
              , 2: U12^#(pair(YS, ZS), X) -> c_1(X, YS, ZS)
              , 3: afterNth^#(N, XS) -> c_2(snd^#(splitAt(N, XS)))
              , 4: and^#(tt(), X) -> c_3(X)
              , 5: fst^#(pair(X, Y)) -> c_4(X)
              , 6: head^#(cons(N, XS)) -> c_5(N)
              , 7: natsFrom^#(N) -> c_6(N, natsFrom^#(s(N)))
              , 8: sel^#(N, XS) -> c_7(head^#(afterNth(N, XS)))
              , 9: snd^#(pair(X, Y)) -> c_8(Y)
              , 10: splitAt^#(0(), XS) -> c_9(XS)
              , 11: splitAt^#(s(N), cons(X, XS)) -> c_10(U11^#(tt(), N, X, XS))
              , 12: tail^#(cons(N, XS)) -> c_11(XS)
              , 13: take^#(N, XS) -> c_12(fst^#(splitAt(N, XS)))}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{13}                                                      [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{12}                                                      [   YES(?,O(n^1))    ]

             ->{11}                                                      [         NA         ]
                |
                `->{1}                                                   [         NA         ]
                    |
                    `->{2}                                               [         NA         ]

             ->{10}                                                      [    YES(?,O(1))     ]

             ->{8}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{7}                                                       [       MAYBE        ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [         NA         ]
                |
                `->{9}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {3}: NA
             ------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {1},
                 Uargs(snd^#) = {1}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [2] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [0] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [2] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                snd^#(x1) = [1] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {3}->{9}: NA
             -----------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {1},
                 Uargs(snd^#) = {1}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                snd^#(x1) = [3] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_3(x1) = [1] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(tt(), X) -> c_3(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                tt() = [5]
                and^#(x1, x2) = [3] x1 + [7] x2 + [0]
                c_3(x1) = [1] x1 + [0]

           * Path {7}: MAYBE
             ---------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {2},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [3] x1 + [0]
                c_6(x1, x2) = [2] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {natsFrom^#(N) -> c_6(N, natsFrom^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {8}: NA
             ------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {1},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {1}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [2] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [0] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [2] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                afterNth(x1, x2) = [3] x1 + [3] x2 + [3]
                snd(x1) = [1] x1 + [1]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7(x1) = [1] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {8}->{6}: NA
             -----------------

             The usable rules for this path are:

               {  afterNth(N, XS) -> snd(splitAt(N, XS))
                , snd(pair(X, Y)) -> Y
                , splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {1},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {1}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [2] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [0] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [2] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                afterNth(x1, x2) = [3] x1 + [3] x2 + [3]
                snd(x1) = [1] x1 + [1]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [3] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [1] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {10}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_9(x1) = [1] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {splitAt^#(0(), XS) -> c_9(XS)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [5]
                splitAt^#(x1, x2) = [3] x1 + [7] x2 + [0]
                c_9(x1) = [1] x1 + [0]

           * Path {11}: NA
             -------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [3] x2 + [3] x3 + [3] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [2] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [0] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [2] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [3] x2 + [3] x3 + [3] x4 + [0]
                c_0(x1) = [1] x1 + [0]
                U12^#(x1, x2) = [1] x1 + [3] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {11}->{1}->{2}: NA
             -----------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {1}, Uargs(U12^#) = {1},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [1] x1 + [0]
                U12^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_1(x1, x2, x3) = [1] x1 + [0] x2 + [1] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {12}: YES(?,O(n^1))
             ------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                tt() = [0]
                U12(x1, x2) = [0] x1 + [0] x2 + [0]
                splitAt(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [3] x1 + [0]
                c_11(x1) = [1] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(N, XS)) -> c_11(XS)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [7]
                tail^#(x1) = [1] x1 + [7]
                c_11(x1) = [1] x1 + [1]

           * Path {13}: NA
             -------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {1}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [1] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_12(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

           * Path {13}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  splitAt(0(), XS) -> pair(nil(), XS)
                , splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, XS)
                , U11(tt(), N, X, XS) -> U12(splitAt(N, XS), X)
                , U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(U11) = {}, Uargs(U12) = {1}, Uargs(splitAt) = {},
                 Uargs(pair) = {}, Uargs(cons) = {}, Uargs(afterNth) = {},
                 Uargs(snd) = {}, Uargs(and) = {}, Uargs(fst) = {},
                 Uargs(head) = {}, Uargs(natsFrom) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(tail) = {}, Uargs(take) = {},
                 Uargs(U11^#) = {}, Uargs(c_0) = {}, Uargs(U12^#) = {},
                 Uargs(c_1) = {}, Uargs(afterNth^#) = {}, Uargs(c_2) = {},
                 Uargs(snd^#) = {}, Uargs(and^#) = {}, Uargs(c_3) = {},
                 Uargs(fst^#) = {1}, Uargs(c_4) = {}, Uargs(head^#) = {},
                 Uargs(c_5) = {}, Uargs(natsFrom^#) = {}, Uargs(c_6) = {},
                 Uargs(sel^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(splitAt^#) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(tail^#) = {}, Uargs(c_11) = {}, Uargs(take^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3, x4) = [0] x1 + [0] x2 + [1] x3 + [3] x4 + [3]
                tt() = [0]
                U12(x1, x2) = [1] x1 + [1] x2 + [1]
                splitAt(x1, x2) = [0] x1 + [3] x2 + [1]
                pair(x1, x2) = [0] x1 + [1] x2 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                afterNth(x1, x2) = [0] x1 + [0] x2 + [0]
                snd(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                fst(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                natsFrom(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [3]
                tail(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                U11^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                afterNth^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                snd^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                fst^#(x1) = [3] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                natsFrom^#(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                splitAt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_12(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

             We have not generated a proof for the resulting sub-problem.

    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible

    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.

    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.