Problem Strategy outermost added 08 Ex7 BLR02

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex7 BLR02

stdout:

MAYBE

Problem:
from(X) -> cons(X,from(s(X)))
head(cons(X,XS)) -> X
2nd(cons(X,XS)) -> head(XS)
take(0(),XS) -> nil()
take(s(N),cons(X,XS)) -> cons(X,take(N,XS))
sel(0(),cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,XS)

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex7 BLR02

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex7 BLR02

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, from(s(X)))
     , head(cons(X, XS)) -> X
     , 2nd(cons(X, XS)) -> head(XS)
     , take(0(), XS) -> nil()
     , take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
     , sel(0(), cons(X, XS)) -> X
     , sel(s(N), cons(X, XS)) -> sel(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: from^#(X) -> c_0(from^#(s(X)))
              , 2: head^#(cons(X, XS)) -> c_1()
              , 3: 2nd^#(cons(X, XS)) -> c_2(head^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
              , 6: sel^#(0(), cons(X, XS)) -> c_5()
              , 7: sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS))}

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

             ->{7}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{5}                                                       [   YES(?,O(n^3))    ]
                |
                `->{4}                                                   [         NA         ]

             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [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]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [3 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(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_5() = [0]
                        [0]
                        [0]
                c_6(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: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [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]
                head^#(x1) = [3 0 0] x1 + [0]
                             [3 0 0]      [0]
                             [3 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(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_5() = [0]
                        [0]
                        [0]
                c_6(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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}
               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 1 2]      [2]
                               [0 0 0]      [0 0 0]      [2]
                head^#(x1) = [0 2 0] x1 + [2]
                             [0 2 2]      [2]
                             [0 0 2]      [2]
                2nd^#(x1) = [3 2 0] x1 + [5]
                            [2 0 2]      [6]
                            [2 3 2]      [0]
                c_2(x1) = [2 2 2] x1 + [1]
                          [2 0 0]      [6]
                          [0 0 0]      [7]

           * Path {3}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {1},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [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]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [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]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(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_5() = [0]
                        [0]
                        [0]
                c_6(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {head^#(cons(X, XS)) -> c_1()}
               Weak Rules: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {1}
               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 0]      [2]
                               [0 0 0]      [0 0 0]      [2]
                head^#(x1) = [2 2 2] x1 + [2]
                             [2 0 2]      [0]
                             [2 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]
                2nd^#(x1) = [2 2 0] x1 + [7]
                            [4 2 0]      [3]
                            [6 0 0]      [3]
                c_2(x1) = [1 0 0] x1 + [5]
                          [2 0 0]      [7]
                          [0 2 0]      [7]

           * Path {5}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
                               [0 1 0]      [0 1 3]      [0]
                               [0 0 0]      [0 0 1]      [0]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [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]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [3 3 3]      [3 3 3]      [0]
                                 [3 3 3]      [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(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_5() = [0]
                        [0]
                        [0]
                c_6(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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [0]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 0]      [2]
                        [0 0 0]      [0]
                take^#(x1, x2) = [0 4 0] x1 + [4 1 0] x2 + [0]
                                 [0 0 0]      [2 0 0]      [0]
                                 [4 0 0]      [0 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

           * Path {5}->{4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [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]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [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]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(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_5() = [0]
                        [0]
                        [0]
                c_6(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 {7}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
                               [0 1 0]      [0 1 3]      [0]
                               [0 0 0]      [0 0 1]      [0]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [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]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(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]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]

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

           * Path {7}->{6}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [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]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [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]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(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_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]

             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: from^#(X) -> c_0(from^#(s(X)))
              , 2: head^#(cons(X, XS)) -> c_1()
              , 3: 2nd^#(cons(X, XS)) -> c_2(head^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
              , 6: sel^#(0(), cons(X, XS)) -> c_5()
              , 7: sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS))}

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

             ->{7}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]

             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                `->{2}                                                   [   YES(?,O(n^2))    ]

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [3 3] x1 + [0]
                             [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1() = [0]
                        [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [3 0] x1 + [0]
                             [3 0]      [0]
                c_1() = [0]
                        [0]
                2nd^#(x1) = [1 3] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [0]
                head^#(x1) = [0 2] x1 + [2]
                             [0 0]      [2]
                2nd^#(x1) = [6 1] x1 + [3]
                            [6 1]      [2]
                c_2(x1) = [2 0] x1 + [7]
                          [2 2]      [3]

           * Path {3}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {1},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1() = [0]
                        [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {head^#(cons(X, XS)) -> c_1()}
               Weak Rules: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [2]
                head^#(x1) = [2 2] x1 + [2]
                             [2 2]      [2]
                c_1() = [1]
                        [0]
                2nd^#(x1) = [6 1] x1 + [0]
                            [4 2]      [3]
                c_2(x1) = [2 0] x1 + [7]
                          [0 2]      [7]

           * Path {5}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
                               [0 0]      [0 1]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1() = [0]
                        [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {5}->{4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1() = [0]
                        [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {7}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
                               [0 0]      [0 1]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1() = [0]
                        [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]

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

           * Path {7}->{6}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1() = [0]
                        [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]

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

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

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

             {  1: from^#(X) -> c_0(from^#(s(X)))
              , 2: head^#(cons(X, XS)) -> c_1()
              , 3: 2nd^#(cons(X, XS)) -> c_2(head^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
              , 6: sel^#(0(), cons(X, XS)) -> c_5()
              , 7: sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS))}

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

             ->{7}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]

             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1() = [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(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: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                head^#(x1) = [1] x1 + [0]
                c_1() = [0]
                2nd^#(x1) = [3] x1 + [0]
                c_2(x1) = [3] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]

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

           * Path {3}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {1},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1() = [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]

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

           * Path {5}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1() = [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [1] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]

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

           * Path {5}->{4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(sel^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1() = [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]

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

           * Path {7}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1() = [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [1] x1 + [3] x2 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]

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

           * Path {7}->{6}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(2nd^#) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1() = [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]

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

    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	Strategy outermost added 08 Ex7 BLR02

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex7 BLR02

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, from(s(X)))
     , head(cons(X, XS)) -> X
     , 2nd(cons(X, XS)) -> head(XS)
     , take(0(), XS) -> nil()
     , take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
     , sel(0(), cons(X, XS)) -> X
     , sel(s(N), cons(X, XS)) -> sel(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: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: head^#(cons(X, XS)) -> c_1(X)
              , 3: 2nd^#(cons(X, XS)) -> c_2(head^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
              , 6: sel^#(0(), cons(X, XS)) -> c_5(X)
              , 7: sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS))}

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

             ->{7}                                                       [   YES(?,O(n^3))    ]
                |
                `->{6}                                                   [         NA         ]

             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]

             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                `->{2}                                                   [   YES(?,O(n^3))    ]

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [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]
                s(x1) = [0 1 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [1 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_0(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]
                head^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1, 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_5(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]

             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: {from^#(X) -> c_0(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                head^#(x1) = [3 0 0] x1 + [0]
                             [3 0 0]      [0]
                             [3 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1, 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_5(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]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}
               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 1 2]      [2]
                               [0 0 0]      [0 0 0]      [2]
                head^#(x1) = [0 2 0] x1 + [2]
                             [0 2 2]      [2]
                             [0 0 2]      [2]
                2nd^#(x1) = [3 2 0] x1 + [5]
                            [2 0 2]      [6]
                            [2 3 2]      [0]
                c_2(x1) = [2 2 2] x1 + [1]
                          [2 0 0]      [6]
                          [0 0 0]      [7]

           * Path {3}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {1}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 1 1] x1 + [0 0 0] x2 + [0]
                               [0 1 3]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 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_1(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1, 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_5(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]

             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: {head^#(cons(X, XS)) -> c_1(X)}
               Weak Rules: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_1) = {},
                 Uargs(2nd^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 1 3] x2 + [3]
                               [0 0 2]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                head^#(x1) = [2 2 0] x1 + [2]
                             [2 2 0]      [2]
                             [2 0 0]      [2]
                c_1(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2nd^#(x1) = [3 0 0] x1 + [5]
                            [4 0 0]      [3]
                            [4 0 0]      [3]
                c_2(x1) = [1 0 0] x1 + [5]
                          [0 0 2]      [7]
                          [0 2 0]      [7]

           * Path {5}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {2},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
                               [0 1 3]      [0 1 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                head(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [3 3 3]      [3 3 3]      [0]
                                 [3 3 3]      [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 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_5(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]

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

           * Path {5}->{4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {2},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [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]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 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_5(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]

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

           * Path {7}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
                               [0 1 0]      [0 1 3]      [0]
                               [0 0 0]      [0 0 1]      [0]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1, 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]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [0]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 0]      [2]
                        [0 0 0]      [0]
                sel^#(x1, x2) = [0 4 0] x1 + [4 1 0] x2 + [0]
                                [0 0 0]      [2 0 0]      [0]
                                [4 0 0]      [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

           * Path {7}->{6}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 1 1] x1 + [0 0 0] x2 + [0]
                               [0 1 3]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                s(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]
                2nd(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]
                0() = [0]
                      [0]
                      [0]
                nil() = [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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2nd^#(x1) = [0 0 0] x1 + [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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1, 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 + [3 1 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_5(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]

             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: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: head^#(cons(X, XS)) -> c_1(X)
              , 3: 2nd^#(cons(X, XS)) -> c_2(head^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
              , 6: sel^#(0(), cons(X, XS)) -> c_5(X)
              , 7: sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS))}

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

             ->{7}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]

             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                `->{2}                                                   [   YES(?,O(n^2))    ]

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_0(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(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_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(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: {from^#(X) -> c_0(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [3 0] x1 + [0]
                             [3 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                2nd^#(x1) = [1 3] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(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_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(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^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [0]
                head^#(x1) = [0 2] x1 + [2]
                             [0 0]      [2]
                2nd^#(x1) = [6 1] x1 + [3]
                            [6 1]      [2]
                c_2(x1) = [2 0] x1 + [7]
                          [2 2]      [3]

           * Path {3}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {1}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [1 3] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(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_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(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^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {head^#(cons(X, XS)) -> c_1(X)}
               Weak Rules: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_1) = {},
                 Uargs(2nd^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [2]
                head^#(x1) = [2 2] x1 + [0]
                             [2 2]      [0]
                c_1(x1) = [0 0] x1 + [1]
                          [0 0]      [0]
                2nd^#(x1) = [2 3] x1 + [4]
                            [6 1]      [1]
                c_2(x1) = [1 0] x1 + [7]
                          [2 0]      [0]

           * Path {5}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {2},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1, x2) = [1 0] 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_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {5}->{4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {2},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] 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_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {7}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
                               [0 0]      [0 1]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]

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

           * Path {7}->{6}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                nil() = [0]
                        [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]

             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: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: head^#(cons(X, XS)) -> c_1(X)
              , 3: 2nd^#(cons(X, XS)) -> c_2(head^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
              , 6: sel^#(0(), cons(X, XS)) -> c_5(X)
              , 7: sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS))}

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

             ->{7}                                                       [         NA         ]
                |
                `->{6}                                                   [         NA         ]

             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]

             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [         NA         ]

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1, x2) = [2] x1 + [1] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(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: {from^#(X) -> c_0(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [1] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                2nd^#(x1) = [3] x1 + [0]
                c_2(x1) = [3] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(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: {2nd^#(cons(X, XS)) -> c_2(head^#(XS))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                head^#(x1) = [7] x1 + [0]
                2nd^#(x1) = [0] x1 + [7]
                c_2(x1) = [0] x1 + [5]

           * Path {3}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {1}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [3] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]

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

           * Path {5}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {2},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [1] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [1] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]

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

           * Path {5}->{4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {2},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]

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

           * Path {7}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [1] x1 + [3] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]

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

           * Path {7}->{6}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(head) = {}, Uargs(2nd) = {}, Uargs(take) = {},
                 Uargs(sel) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(head^#) = {}, Uargs(c_1) = {}, Uargs(2nd^#) = {},
                 Uargs(c_2) = {}, Uargs(take^#) = {}, Uargs(c_4) = {},
                 Uargs(sel^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                head(x1) = [0] x1 + [0]
                2nd(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                nil() = [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                2nd^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_5(x1) = [1] x1 + [0]
                c_6(x1) = [1] x1 + [0]

             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.