Problem Strategy outermost added 08 Ex3 2 Luc97

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex3 2 Luc97

stdout:

MAYBE

Problem:
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
dbls(nil()) -> nil()
dbls(cons(X,Y)) -> cons(dbl(X),dbls(Y))
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,Z)
indx(nil(),X) -> nil()
indx(cons(X,Y),Z) -> cons(sel(X,Z),indx(Y,Z))
from(X) -> cons(X,from(s(X)))

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex3 2 Luc97

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex3 2 Luc97

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  dbl(0()) -> 0()
     , dbl(s(X)) -> s(s(dbl(X)))
     , dbls(nil()) -> nil()
     , dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
     , sel(0(), cons(X, Y)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, Z)
     , indx(nil(), X) -> nil()
     , indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
     , from(X) -> cons(X, from(s(X)))}

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: dbl^#(0()) -> c_0()
              , 2: dbl^#(s(X)) -> c_1(dbl^#(X))
              , 3: dbls^#(nil()) -> c_2()
              , 4: dbls^#(cons(X, Y)) -> c_3(dbl^#(X), dbls^#(Y))
              , 5: sel^#(0(), cons(X, Y)) -> c_4()
              , 6: sel^#(s(X), cons(Y, Z)) -> c_5(sel^#(X, Z))
              , 7: indx^#(nil(), X) -> c_6()
              , 8: indx^#(cons(X, Y), Z) -> c_7(sel^#(X, Z), indx^#(Y, Z))
              , 9: from^#(X) -> c_8(from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                |->{5}                                                   [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |   |
                |   `->{5}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{4}                                                       [   YES(?,O(n^3))    ]
                |
                |->{1}                                                   [         NA         ]
                |
                |->{2}                                                   [         NA         ]
                |   |
                |   `->{1}                                               [         NA         ]
                |
                `->{3}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {4}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
                               [0 1 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 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]
                indx(x1, 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]
                dbl^#(x1) = [3 0 0] x1 + [0]
                            [3 0 0]      [0]
                            [3 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 1 0] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(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^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbls^#(cons(X, Y)) -> c_3(dbl^#(X), dbls^#(Y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(dbl^#) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 3] x1 + [1 4 3] x2 + [2]
                               [0 1 0]      [0 0 1]      [0]
                               [0 0 1]      [0 0 1]      [2]
                dbl^#(x1) = [0 0 2] x1 + [0]
                            [0 0 0]      [2]
                            [0 0 2]      [2]
                dbls^#(x1) = [2 4 0] x1 + [0]
                             [2 0 2]      [0]
                             [0 0 4]      [2]
                c_3(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 2]      [0 0 2]      [0]
                              [2 2 0]      [0 0 0]      [3]

           * Path {4}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(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 {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(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 {4}->{2}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(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 {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(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 {8}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
                               [0 0 0]      [0 1 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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [3 0 0] x1 + [3 3 3] x2 + [0]
                                [3 0 0]      [3 3 3]      [0]
                                [3 0 0]      [3 3 3]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, x2) = [0 0 0] x1 + [0 0 3] x2 + [0]
                                 [3 3 3]      [3 3 3]      [0]
                                 [3 3 3]      [3 3 3]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {1},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

           * Path {8}->{6}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {1},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

           * Path {9}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(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:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_8(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: dbl^#(0()) -> c_0()
              , 2: dbl^#(s(X)) -> c_1(dbl^#(X))
              , 3: dbls^#(nil()) -> c_2()
              , 4: dbls^#(cons(X, Y)) -> c_3(dbl^#(X), dbls^#(Y))
              , 5: sel^#(0(), cons(X, Y)) -> c_4()
              , 6: sel^#(s(X), cons(Y, Z)) -> c_5(sel^#(X, Z))
              , 7: indx^#(nil(), X) -> c_6()
              , 8: indx^#(cons(X, Y), Z) -> c_7(sel^#(X, Z), indx^#(Y, Z))
              , 9: from^#(X) -> c_8(from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                |->{5}                                                   [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |   |
                |   `->{5}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{4}                                                       [         NA         ]
                |
                |->{1}                                                   [         NA         ]
                |
                |->{2}                                                   [         NA         ]
                |   |
                |   `->{1}                                               [         NA         ]
                |
                `->{3}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 1] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [3 0] x1 + [0]
                            [3 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {4}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [3 3] x1 + [0]
                            [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {4}->{2}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {8}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 1] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
                                [3 0]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [1 3] x1 + [3 3] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {1},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 2] 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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {8}->{6}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {1},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {9}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [3 3] x1 + [0]
                             [3 3]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [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_8(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: dbl^#(0()) -> c_0()
              , 2: dbl^#(s(X)) -> c_1(dbl^#(X))
              , 3: dbls^#(nil()) -> c_2()
              , 4: dbls^#(cons(X, Y)) -> c_3(dbl^#(X), dbls^#(Y))
              , 5: sel^#(0(), cons(X, Y)) -> c_4()
              , 6: sel^#(s(X), cons(Y, Z)) -> c_5(sel^#(X, Z))
              , 7: indx^#(nil(), X) -> c_6()
              , 8: indx^#(cons(X, Y), Z) -> c_7(sel^#(X, Z), indx^#(Y, Z))
              , 9: from^#(X) -> c_8(from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                |->{5}                                                   [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |   |
                |   `->{5}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{4}                                                       [         NA         ]
                |
                |->{1}                                                   [         NA         ]
                |
                |->{2}                                                   [         NA         ]
                |   |
                |   `->{1}                                               [         NA         ]
                |
                `->{3}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [1] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [3] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [3] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {4}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [1] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [3] x1 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [1] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {4}->{2}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [1] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {8}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {1},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {8}->{6}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {1},
                 Uargs(indx^#) = {}, Uargs(c_7) = {1, 2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {2}, Uargs(from^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {9}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_5) = {},
                 Uargs(indx^#) = {}, Uargs(c_7) = {}, Uargs(from^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [3] x1 + [0]
                c_8(x1) = [1] 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_8(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex3 2 Luc97

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex3 2 Luc97

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  dbl(0()) -> 0()
     , dbl(s(X)) -> s(s(dbl(X)))
     , dbls(nil()) -> nil()
     , dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
     , sel(0(), cons(X, Y)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, Z)
     , indx(nil(), X) -> nil()
     , indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
     , from(X) -> cons(X, from(s(X)))}

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: dbl^#(0()) -> c_0()
              , 2: dbl^#(s(X)) -> c_1(dbl^#(X))
              , 3: dbls^#(nil()) -> c_2()
              , 4: dbls^#(cons(X, Y)) -> c_3(dbl^#(X), dbls^#(Y))
              , 5: sel^#(0(), cons(X, Y)) -> c_4(X)
              , 6: sel^#(s(X), cons(Y, Z)) -> c_5(sel^#(X, Z))
              , 7: indx^#(nil(), X) -> c_6()
              , 8: indx^#(cons(X, Y), Z) -> c_7(sel^#(X, Z), indx^#(Y, Z))
              , 9: from^#(X) -> c_8(X, from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                |->{5}                                                   [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |   |
                |   `->{5}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{4}                                                       [   YES(?,O(n^3))    ]
                |
                |->{1}                                                   [         NA         ]
                |
                |->{2}                                                   [         NA         ]
                |   |
                |   `->{1}                                               [         NA         ]
                |
                `->{3}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {4}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
                               [0 1 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 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]
                indx(x1, 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]
                dbl^#(x1) = [3 0 0] x1 + [0]
                            [3 0 0]      [0]
                            [3 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 1 0] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 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^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbls^#(cons(X, Y)) -> c_3(dbl^#(X), dbls^#(Y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(dbl^#) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 3] x1 + [1 4 3] x2 + [2]
                               [0 1 0]      [0 0 1]      [0]
                               [0 0 1]      [0 0 1]      [2]
                dbl^#(x1) = [0 0 2] x1 + [0]
                            [0 0 0]      [2]
                            [0 0 2]      [2]
                dbls^#(x1) = [2 4 0] x1 + [0]
                             [2 0 2]      [0]
                             [0 0 4]      [2]
                c_3(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 2]      [0 0 2]      [0]
                              [2 2 0]      [0 0 0]      [3]

           * Path {4}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {4}->{2}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {8}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
                               [0 0 0]      [0 1 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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [3 0 0] x1 + [3 3 3] x2 + [0]
                                [3 0 0]      [3 3 3]      [0]
                                [3 0 0]      [3 3 3]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, x2) = [0 0 0] x1 + [0 0 3] x2 + [0]
                                 [3 3 3]      [3 3 3]      [0]
                                 [3 3 3]      [3 3 3]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 1 1] x1 + [0 0 0] x2 + [0]
                               [0 0 3]      [0 0 0]      [0]
                               [0 0 1]      [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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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 2 1] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [1 1 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {8}->{6}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
                               [0 0 3]      [0 0 0]      [0]
                               [0 0 1]      [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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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 + [1 2 1] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [1 1 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

           * Path {9}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0 0] x1 + [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]
                dbls(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                nil() = [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]
                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]
                indx(x1, 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]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                dbls^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1, 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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                indx^#(x1, 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_6() = [0]
                        [0]
                        [0]
                c_7(x1, 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_8(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]

             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_8(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: dbl^#(0()) -> c_0()
              , 2: dbl^#(s(X)) -> c_1(dbl^#(X))
              , 3: dbls^#(nil()) -> c_2()
              , 4: dbls^#(cons(X, Y)) -> c_3(dbl^#(X), dbls^#(Y))
              , 5: sel^#(0(), cons(X, Y)) -> c_4(X)
              , 6: sel^#(s(X), cons(Y, Z)) -> c_5(sel^#(X, Z))
              , 7: indx^#(nil(), X) -> c_6()
              , 8: indx^#(cons(X, Y), Z) -> c_7(sel^#(X, Z), indx^#(Y, Z))
              , 9: from^#(X) -> c_8(X, from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                |->{5}                                                   [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |   |
                |   `->{5}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{4}                                                       [         NA         ]
                |
                |->{1}                                                   [         NA         ]
                |
                |->{2}                                                   [         NA         ]
                |   |
                |   `->{1}                                               [         NA         ]
                |
                `->{3}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 1] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [3 0] x1 + [0]
                            [3 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {4}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [3 3] x1 + [0]
                            [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {4}->{2}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {8}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 1] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
                                [3 0]      [3 3]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [1 3] x1 + [3 3] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 2] 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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {8}->{6}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]

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

           * Path {9}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                dbls(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                nil() = [0]
                        [0]
                cons(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]
                indx(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbls^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(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_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                indx^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_8(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [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_8(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: dbl^#(0()) -> c_0()
              , 2: dbl^#(s(X)) -> c_1(dbl^#(X))
              , 3: dbls^#(nil()) -> c_2()
              , 4: dbls^#(cons(X, Y)) -> c_3(dbl^#(X), dbls^#(Y))
              , 5: sel^#(0(), cons(X, Y)) -> c_4(X)
              , 6: sel^#(s(X), cons(Y, Z)) -> c_5(sel^#(X, Z))
              , 7: indx^#(nil(), X) -> c_6()
              , 8: indx^#(cons(X, Y), Z) -> c_7(sel^#(X, Z), indx^#(Y, Z))
              , 9: from^#(X) -> c_8(X, from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                |->{5}                                                   [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |   |
                |   `->{5}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{4}                                                       [         NA         ]
                |
                |->{1}                                                   [         NA         ]
                |
                |->{2}                                                   [         NA         ]
                |   |
                |   `->{1}                                               [         NA         ]
                |
                `->{3}                                                   [         NA         ]



         Sub-problems:
         -------------
           * Path {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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [1] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [3] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [3] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {4}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [1] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [3] x1 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [1] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {4}->{2}->{1}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {1}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {1, 2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [1] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {4}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {2}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {8}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {8}->{6}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(indx^#) = {}, Uargs(c_7) = {1, 2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {8}->{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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {2},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {9}: 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(dbl) = {}, Uargs(s) = {}, Uargs(dbls) = {}, Uargs(cons) = {},
                 Uargs(sel) = {}, Uargs(indx) = {}, Uargs(from) = {},
                 Uargs(dbl^#) = {}, Uargs(c_1) = {}, Uargs(dbls^#) = {},
                 Uargs(c_3) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(indx^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                dbl(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                dbls(x1) = [0] x1 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                indx(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                dbls^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                indx^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [3] x1 + [0]
                c_8(x1, x2) = [2] x1 + [1] x2 + [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_8(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

    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.