Problem Strategy outermost added 08 Ex14 AEGL02 FR

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 AEGL02 FR

stdout:

MAYBE

Problem:
from(X) -> cons(X,n__from(n__s(X)))
length(n__nil()) -> 0()
length(n__cons(X,Y)) -> s(length1(activate(Y)))
length1(X) -> length(activate(X))
from(X) -> n__from(X)
s(X) -> n__s(X)
nil() -> n__nil()
cons(X1,X2) -> n__cons(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__nil()) -> nil()
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 AEGL02 FR

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Strategy outermost added 08 Ex14 AEGL02 FR

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, n__from(n__s(X)))
     , length(n__nil()) -> 0()
     , length(n__cons(X, Y)) -> s(length1(activate(Y)))
     , length1(X) -> length(activate(X))
     , from(X) -> n__from(X)
     , s(X) -> n__s(X)
     , nil() -> n__nil()
     , cons(X1, X2) -> n__cons(X1, X2)
     , activate(n__from(X)) -> from(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(n__nil()) -> nil()
     , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
     , activate(X) -> X}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 AEGL02 FR

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 AEGL02 FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, n__from(n__s(X)))
     , length(n__nil()) -> 0()
     , length(n__cons(X, Y)) -> s(length1(activate(Y)))
     , length1(X) -> length(activate(X))
     , from(X) -> n__from(X)
     , s(X) -> n__s(X)
     , nil() -> n__nil()
     , cons(X1, X2) -> n__cons(X1, X2)
     , activate(n__from(X)) -> from(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(n__nil()) -> nil()
     , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
     , activate(X) -> 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: from^#(X) -> c_0(cons^#(X, n__from(n__s(X))))
              , 2: length^#(n__nil()) -> c_1()
              , 3: length^#(n__cons(X, Y)) -> c_2(s^#(length1(activate(Y))))
              , 4: length1^#(X) -> c_3(length^#(activate(X)))
              , 5: from^#(X) -> c_4(X)
              , 6: s^#(X) -> c_5(X)
              , 7: nil^#() -> c_6()
              , 8: cons^#(X1, X2) -> c_7(X1, X2)
              , 9: activate^#(n__from(X)) -> c_8(from^#(activate(X)))
              , 10: activate^#(n__s(X)) -> c_9(s^#(activate(X)))
              , 11: activate^#(n__nil()) -> c_10(nil^#())
              , 12: activate^#(n__cons(X1, X2)) -> c_11(cons^#(activate(X1), X2))
              , 13: activate^#(X) -> c_12(X)}

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

             ->{13}                                                      [    YES(?,O(1))     ]

             ->{12}                                                      [     inherited      ]
                |
                `->{8}                                                   [         NA         ]

             ->{11}                                                      [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]

             ->{10}                                                      [     inherited      ]
                |
                `->{6}                                                   [         NA         ]

             ->{9}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{8}                                               [         NA         ]
                |
                `->{5}                                                   [       MAYBE        ]

             ->{4}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{6}                                               [         NA         ]



         Sub-problems:
         -------------
           * Path {4}: inherited
             -------------------

             This path is subsumed by the proof of path {4}->{3}->{6}.

           * Path {4}->{2}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {4}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {4}->{3}->{6}.

           * Path {4}->{3}->{6}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)
                , length1(X) -> length(activate(X))
                , length(n__nil()) -> 0()
                , length(n__cons(X, Y)) -> s(length1(activate(Y)))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {9}: inherited
             -------------------

             This path is subsumed by the proof of path {9}->{1}->{8}.

           * Path {9}->{1}: inherited
             ------------------------

             This path is subsumed by the proof of path {9}->{1}->{8}.

           * Path {9}->{1}->{8}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {9}->{5}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  activate^#(n__from(X)) -> c_8(from^#(activate(X)))
                  , from^#(X) -> c_4(X)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(n__nil()) -> nil()
                  , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)
                  , nil() -> n__nil()
                  , cons(X1, X2) -> n__cons(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

           * Path {10}: inherited
             --------------------

             This path is subsumed by the proof of path {10}->{6}.

           * Path {10}->{6}: NA
             ------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {11}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                n__from(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                n__s(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                n__nil() = [0]
                           [0]
                           [0]
                0() = [0]
                      [0]
                      [0]
                n__cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length1(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length1^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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]
                nil^#() = [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]
                activate^#(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]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(n__nil()) -> c_10(nil^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__nil() = [2]
                           [2]
                           [2]
                nil^#() = [2]
                          [2]
                          [0]
                activate^#(x1) = [0 2 2] x1 + [7]
                                 [2 2 0]      [7]
                                 [0 0 0]      [7]
                c_10(x1) = [2 2 0] x1 + [1]
                           [0 0 0]      [7]
                           [2 0 0]      [3]

           * Path {11}->{7}: YES(?,O(1))
             ---------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                n__from(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                n__s(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                n__nil() = [0]
                           [0]
                           [0]
                0() = [0]
                      [0]
                      [0]
                n__cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length1(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length1^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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]
                nil^#() = [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]
                activate^#(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]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {nil^#() -> c_6()}
               Weak Rules: {activate^#(n__nil()) -> c_10(nil^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__nil() = [2]
                           [6]
                           [2]
                nil^#() = [2]
                          [2]
                          [2]
                c_6() = [1]
                        [0]
                        [0]
                activate^#(x1) = [2 0 2] x1 + [7]
                                 [0 2 0]      [3]
                                 [2 0 0]      [7]
                c_10(x1) = [2 2 0] x1 + [2]
                           [2 2 2]      [3]
                           [2 2 0]      [3]

           * Path {12}: inherited
             --------------------

             This path is subsumed by the proof of path {12}->{8}.

           * Path {12}->{8}: NA
             ------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {13}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                n__from(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                n__s(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                n__nil() = [0]
                           [0]
                           [0]
                0() = [0]
                      [0]
                      [0]
                n__cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length1(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length1^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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]
                nil^#() = [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]
                activate^#(x1) = [3 3 3] 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]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [1 1 1] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_12(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7 7] x1 + [7]
                                 [7 7 7]      [7]
                                 [7 7 7]      [7]
                c_12(x1) = [3 3 3] x1 + [0]
                           [3 1 3]      [1]
                           [1 1 1]      [1]

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

             {  1: from^#(X) -> c_0(cons^#(X, n__from(n__s(X))))
              , 2: length^#(n__nil()) -> c_1()
              , 3: length^#(n__cons(X, Y)) -> c_2(s^#(length1(activate(Y))))
              , 4: length1^#(X) -> c_3(length^#(activate(X)))
              , 5: from^#(X) -> c_4(X)
              , 6: s^#(X) -> c_5(X)
              , 7: nil^#() -> c_6()
              , 8: cons^#(X1, X2) -> c_7(X1, X2)
              , 9: activate^#(n__from(X)) -> c_8(from^#(activate(X)))
              , 10: activate^#(n__s(X)) -> c_9(s^#(activate(X)))
              , 11: activate^#(n__nil()) -> c_10(nil^#())
              , 12: activate^#(n__cons(X1, X2)) -> c_11(cons^#(activate(X1), X2))
              , 13: activate^#(X) -> c_12(X)}

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

             ->{13}                                                      [    YES(?,O(1))     ]

             ->{12}                                                      [     inherited      ]
                |
                `->{8}                                                   [         NA         ]

             ->{11}                                                      [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]

             ->{10}                                                      [     inherited      ]
                |
                `->{6}                                                   [         NA         ]

             ->{9}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{8}                                               [         NA         ]
                |
                `->{5}                                                   [       MAYBE        ]

             ->{4}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{6}                                               [         NA         ]



         Sub-problems:
         -------------
           * Path {4}: inherited
             -------------------

             This path is subsumed by the proof of path {4}->{3}->{6}.

           * Path {4}->{2}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {4}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {4}->{3}->{6}.

           * Path {4}->{3}->{6}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)
                , length1(X) -> length(activate(X))
                , length(n__nil()) -> 0()
                , length(n__cons(X, Y)) -> s(length1(activate(Y)))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {9}: inherited
             -------------------

             This path is subsumed by the proof of path {9}->{1}->{8}.

           * Path {9}->{1}: inherited
             ------------------------

             This path is subsumed by the proof of path {9}->{1}->{8}.

           * Path {9}->{1}->{8}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {9}->{5}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  activate^#(n__from(X)) -> c_8(from^#(activate(X)))
                  , from^#(X) -> c_4(X)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(n__nil()) -> nil()
                  , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)
                  , nil() -> n__nil()
                  , cons(X1, X2) -> n__cons(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

           * Path {10}: inherited
             --------------------

             This path is subsumed by the proof of path {10}->{6}.

           * Path {10}->{6}: NA
             ------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {11}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__from(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                n__nil() = [0]
                           [0]
                0() = [0]
                      [0]
                n__cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length1(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length1^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                nil^#() = [0]
                          [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(n__nil()) -> c_10(nil^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__nil() = [2]
                           [2]
                nil^#() = [0]
                          [2]
                activate^#(x1) = [2 2] x1 + [7]
                                 [2 0]      [7]
                c_10(x1) = [0 2] x1 + [3]
                           [0 2]      [3]

           * Path {11}->{7}: YES(?,O(1))
             ---------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__from(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                n__nil() = [0]
                           [0]
                0() = [0]
                      [0]
                n__cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length1(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length1^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                nil^#() = [0]
                          [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {nil^#() -> c_6()}
               Weak Rules: {activate^#(n__nil()) -> c_10(nil^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__nil() = [2]
                           [2]
                nil^#() = [2]
                          [2]
                c_6() = [1]
                        [0]
                activate^#(x1) = [0 0] x1 + [7]
                                 [1 2]      [7]
                c_10(x1) = [2 0] x1 + [2]
                           [2 2]      [3]

           * Path {12}: inherited
             --------------------

             This path is subsumed by the proof of path {12}->{8}.

           * Path {12}->{8}: NA
             ------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {13}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__from(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                n__nil() = [0]
                           [0]
                0() = [0]
                      [0]
                n__cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length1(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length1^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                nil^#() = [0]
                          [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [1 1] x1 + [0]
                           [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_12(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7] x1 + [7]
                                 [7 7]      [7]
                c_12(x1) = [1 3] x1 + [0]
                           [3 1]      [3]

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

             {  1: from^#(X) -> c_0(cons^#(X, n__from(n__s(X))))
              , 2: length^#(n__nil()) -> c_1()
              , 3: length^#(n__cons(X, Y)) -> c_2(s^#(length1(activate(Y))))
              , 4: length1^#(X) -> c_3(length^#(activate(X)))
              , 5: from^#(X) -> c_4(X)
              , 6: s^#(X) -> c_5(X)
              , 7: nil^#() -> c_6()
              , 8: cons^#(X1, X2) -> c_7(X1, X2)
              , 9: activate^#(n__from(X)) -> c_8(from^#(activate(X)))
              , 10: activate^#(n__s(X)) -> c_9(s^#(activate(X)))
              , 11: activate^#(n__nil()) -> c_10(nil^#())
              , 12: activate^#(n__cons(X1, X2)) -> c_11(cons^#(activate(X1), X2))
              , 13: activate^#(X) -> c_12(X)}

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

             ->{13}                                                      [    YES(?,O(1))     ]

             ->{12}                                                      [     inherited      ]
                |
                `->{8}                                                   [         NA         ]

             ->{11}                                                      [    YES(?,O(1))     ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]

             ->{10}                                                      [     inherited      ]
                |
                `->{6}                                                   [         NA         ]

             ->{9}                                                       [     inherited      ]
                |
                |->{1}                                                   [     inherited      ]
                |   |
                |   `->{8}                                               [         NA         ]
                |
                `->{5}                                                   [       MAYBE        ]

             ->{4}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{6}                                               [         NA         ]



         Sub-problems:
         -------------
           * Path {4}: inherited
             -------------------

             This path is subsumed by the proof of path {4}->{3}->{6}.

           * Path {4}->{2}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {4}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {4}->{3}->{6}.

           * Path {4}->{3}->{6}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)
                , length1(X) -> length(activate(X))
                , length(n__nil()) -> 0()
                , length(n__cons(X, Y)) -> s(length1(activate(Y)))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {9}: inherited
             -------------------

             This path is subsumed by the proof of path {9}->{1}->{8}.

           * Path {9}->{1}: inherited
             ------------------------

             This path is subsumed by the proof of path {9}->{1}->{8}.

           * Path {9}->{1}->{8}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {9}->{5}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  activate^#(n__from(X)) -> c_8(from^#(activate(X)))
                  , from^#(X) -> c_4(X)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(n__nil()) -> nil()
                  , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)
                  , nil() -> n__nil()
                  , cons(X1, X2) -> n__cons(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

           * Path {10}: inherited
             --------------------

             This path is subsumed by the proof of path {10}->{6}.

           * Path {10}->{6}: NA
             ------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {11}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                n__from(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                n__nil() = [0]
                0() = [0]
                n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                length1(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                length1^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                nil^#() = [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(n__nil()) -> c_10(nil^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__nil() = [2]
                nil^#() = [2]
                activate^#(x1) = [2] x1 + [5]
                c_10(x1) = [2] x1 + [3]

           * Path {11}->{7}: YES(?,O(1))
             ---------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {1},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                n__from(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                n__nil() = [0]
                0() = [0]
                n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                length1(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                length1^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                nil^#() = [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [1] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {nil^#() -> c_6()}
               Weak Rules: {activate^#(n__nil()) -> c_10(nil^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__nil() = [0]
                nil^#() = [2]
                c_6() = [1]
                activate^#(x1) = [0] x1 + [6]
                c_10(x1) = [2] x1 + [2]

           * Path {12}: inherited
             --------------------

             This path is subsumed by the proof of path {12}->{8}.

           * Path {12}->{8}: NA
             ------------------

             The usable rules for this path are:

               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(n__nil()) -> nil()
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)
                , nil() -> n__nil()
                , cons(X1, X2) -> n__cons(X1, X2)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {13}: YES(?,O(1))
             ----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(length) = {}, Uargs(n__cons) = {},
                 Uargs(s) = {}, Uargs(length1) = {}, Uargs(activate) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(cons^#) = {},
                 Uargs(length^#) = {}, Uargs(c_2) = {}, Uargs(s^#) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(activate^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                n__from(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                n__nil() = [0]
                0() = [0]
                n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                length1(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                length1^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                nil^#() = [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                activate^#(x1) = [3] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_12(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7] x1 + [7]
                c_12(x1) = [1] x1 + [0]

    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.