Problem Transformed CSR 04 Ex4 Zan97 FR

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 Zan97 FR

stdout:

MAYBE

Problem:
f(X) -> cons(X,n__f(n__g(X)))
g(0()) -> s(0())
g(s(X)) -> s(s(g(X)))
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
f(X) -> n__f(X)
g(X) -> n__g(X)
activate(n__f(X)) -> f(activate(X))
activate(n__g(X)) -> g(activate(X))
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 Zan97 FR

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 Zan97 FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(X) -> cons(X, n__f(n__g(X)))
     , g(0()) -> s(0())
     , g(s(X)) -> s(s(g(X)))
     , sel(0(), cons(X, Y)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
     , f(X) -> n__f(X)
     , g(X) -> n__g(X)
     , activate(n__f(X)) -> f(activate(X))
     , activate(n__g(X)) -> g(activate(X))
     , 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: f^#(X) -> c_0()
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3()
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
              , 6: f^#(X) -> c_5()
              , 7: g^#(X) -> c_6()
              , 8: activate^#(n__f(X)) -> c_7(f^#(activate(X)))
              , 9: activate^#(n__g(X)) -> c_8(g^#(activate(X)))
              , 10: activate^#(X) -> c_9()}

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

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

             ->{9}                                                       [     inherited      ]
                |
                |->{2}                                                   [       MAYBE        ]
                |
                |->{3}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [         NA         ]
                |   |
                |   `->{7}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{6}                                                   [         NA         ]

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



         Sub-problems:
         -------------
           * Path {5}: inherited
             -------------------

             This path is subsumed by the proof of path {5}->{4}.

           * Path {5}->{4}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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 {8}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{2}.

           * Path {9}->{2}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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:    innermost runtime-complexity with respect to
               Rules:
                 {  activate^#(n__g(X)) -> c_8(g^#(activate(X)))
                  , g^#(0()) -> c_1()
                  , activate(n__f(X)) -> f(activate(X))
                  , activate(n__g(X)) -> g(activate(X))
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(n__g(X)))
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))
                  , f(X) -> n__f(X)
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

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

             This path is subsumed by the proof of path {9}->{3}->{2}.

           * Path {9}->{3}->{2}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{7}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{7}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {},
                 Uargs(n__g) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(activate^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(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__f(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                g(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]
                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]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                g^#(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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]

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

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

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

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

             {  1: f^#(X) -> c_0()
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3()
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
              , 6: f^#(X) -> c_5()
              , 7: g^#(X) -> c_6()
              , 8: activate^#(n__f(X)) -> c_7(f^#(activate(X)))
              , 9: activate^#(n__g(X)) -> c_8(g^#(activate(X)))
              , 10: activate^#(X) -> c_9()}

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

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

             ->{9}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                |->{3}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [         NA         ]
                |   |
                |   `->{7}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{6}                                                   [         NA         ]

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



         Sub-problems:
         -------------
           * Path {5}: inherited
             -------------------

             This path is subsumed by the proof of path {5}->{4}.

           * Path {5}->{4}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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 {8}: inherited
             -------------------

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

           * Path {8}->{1}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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:    innermost runtime-complexity with respect to
               Rules:
                 {  activate^#(n__f(X)) -> c_7(f^#(activate(X)))
                  , f^#(X) -> c_0()
                  , activate(n__f(X)) -> f(activate(X))
                  , activate(n__g(X)) -> g(activate(X))
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(n__g(X)))
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))
                  , f(X) -> n__f(X)
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{2}.

           * Path {9}->{2}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {9}->{3}->{2}.

           * Path {9}->{3}->{2}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{7}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{7}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {},
                 Uargs(n__g) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(activate^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__f(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0() = [0]
                        [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]

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

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

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

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

             {  1: f^#(X) -> c_0()
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3()
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
              , 6: f^#(X) -> c_5()
              , 7: g^#(X) -> c_6()
              , 8: activate^#(n__f(X)) -> c_7(f^#(activate(X)))
              , 9: activate^#(n__g(X)) -> c_8(g^#(activate(X)))
              , 10: activate^#(X) -> c_9()}

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

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

             ->{9}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                |->{3}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [         NA         ]
                |   |
                |   `->{7}                                               [         NA         ]
                |
                `->{7}                                                   [       MAYBE        ]

             ->{8}                                                       [     inherited      ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{6}                                                   [         NA         ]

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



         Sub-problems:
         -------------
           * Path {5}: inherited
             -------------------

             This path is subsumed by the proof of path {5}->{4}.

           * Path {5}->{4}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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 {8}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{2}.

           * Path {9}->{2}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {9}->{3}->{2}.

           * Path {9}->{3}->{2}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{7}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{7}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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:    innermost runtime-complexity with respect to
               Rules:
                 {  activate^#(n__g(X)) -> c_8(g^#(activate(X)))
                  , g^#(X) -> c_6()
                  , activate(n__f(X)) -> f(activate(X))
                  , activate(n__g(X)) -> g(activate(X))
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(n__g(X)))
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))
                  , f(X) -> n__f(X)
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {},
                 Uargs(n__g) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
                 Uargs(activate^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                n__f(x1) = [0] x1 + [0]
                n__g(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1) = [0] x1 + [0]
                c_0() = [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 Zan97 FR

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 Zan97 FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(X) -> cons(X, n__f(n__g(X)))
     , g(0()) -> s(0())
     , g(s(X)) -> s(s(g(X)))
     , sel(0(), cons(X, Y)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
     , f(X) -> n__f(X)
     , g(X) -> n__g(X)
     , activate(n__f(X)) -> f(activate(X))
     , activate(n__g(X)) -> g(activate(X))
     , 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: f^#(X) -> c_0(X, X)
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3(X)
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
              , 6: f^#(X) -> c_5(X)
              , 7: g^#(X) -> c_6(X)
              , 8: activate^#(n__f(X)) -> c_7(f^#(activate(X)))
              , 9: activate^#(n__g(X)) -> c_8(g^#(activate(X)))
              , 10: activate^#(X) -> c_9(X)}

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

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

             ->{9}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                |->{3}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [         NA         ]
                |   |
                |   `->{7}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{6}                                                   [         NA         ]

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



         Sub-problems:
         -------------
           * Path {5}: inherited
             -------------------

             This path is subsumed by the proof of path {5}->{4}.

           * Path {5}->{4}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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 {8}: inherited
             -------------------

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

           * Path {8}->{1}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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__f(X)) -> c_7(f^#(activate(X)))
                  , f^#(X) -> c_0(X, X)
                  , activate(n__f(X)) -> f(activate(X))
                  , activate(n__g(X)) -> g(activate(X))
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(n__g(X)))
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))
                  , f(X) -> n__f(X)
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{2}.

           * Path {9}->{2}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {9}->{3}->{2}.

           * Path {9}->{3}->{2}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{7}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{7}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {},
                 Uargs(n__g) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(sel^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(activate^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(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__f(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                g(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]
                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]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                g^#(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]
                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_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]
                c_6(x1) = [0 0 0] x1 + [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_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(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_9(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_9) = {}
               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_9(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: f^#(X) -> c_0(X, X)
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3(X)
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
              , 6: f^#(X) -> c_5(X)
              , 7: g^#(X) -> c_6(X)
              , 8: activate^#(n__f(X)) -> c_7(f^#(activate(X)))
              , 9: activate^#(n__g(X)) -> c_8(g^#(activate(X)))
              , 10: activate^#(X) -> c_9(X)}

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

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

             ->{9}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                |->{3}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [         NA         ]
                |   |
                |   `->{7}                                               [         NA         ]
                |
                `->{7}                                                   [       MAYBE        ]

             ->{8}                                                       [     inherited      ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{6}                                                   [         NA         ]

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



         Sub-problems:
         -------------
           * Path {5}: inherited
             -------------------

             This path is subsumed by the proof of path {5}->{4}.

           * Path {5}->{4}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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 {8}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{2}.

           * Path {9}->{2}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {9}->{3}->{2}.

           * Path {9}->{3}->{2}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{7}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{7}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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__g(X)) -> c_8(g^#(activate(X)))
                  , g^#(X) -> c_6(X)
                  , activate(n__f(X)) -> f(activate(X))
                  , activate(n__g(X)) -> g(activate(X))
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(n__g(X)))
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))
                  , f(X) -> n__f(X)
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {},
                 Uargs(n__g) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(sel^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(activate^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__f(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(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_9(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7] x1 + [7]
                                 [7 7]      [7]
                c_9(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: f^#(X) -> c_0(X, X)
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3(X)
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
              , 6: f^#(X) -> c_5(X)
              , 7: g^#(X) -> c_6(X)
              , 8: activate^#(n__f(X)) -> c_7(f^#(activate(X)))
              , 9: activate^#(n__g(X)) -> c_8(g^#(activate(X)))
              , 10: activate^#(X) -> c_9(X)}

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

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

             ->{9}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                |->{3}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [         NA         ]
                |   |
                |   `->{7}                                               [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{6}                                                   [         NA         ]

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



         Sub-problems:
         -------------
           * Path {5}: inherited
             -------------------

             This path is subsumed by the proof of path {5}->{4}.

           * Path {5}->{4}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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 {8}: inherited
             -------------------

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

           * Path {8}->{1}: MAYBE
             --------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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__f(X)) -> c_7(f^#(activate(X)))
                  , f^#(X) -> c_0(X, X)
                  , activate(n__f(X)) -> f(activate(X))
                  , activate(n__g(X)) -> g(activate(X))
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(n__g(X)))
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))
                  , f(X) -> n__f(X)
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{2}.

           * Path {9}->{2}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {9}->{3}->{2}.

           * Path {9}->{3}->{2}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{3}->{7}: NA
             ----------------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

             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}->{7}: NA
             -----------------

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(activate(X))
                , activate(n__g(X)) -> g(activate(X))
                , activate(X) -> X
                , f(X) -> cons(X, n__f(n__g(X)))
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))
                , f(X) -> n__f(X)
                , g(X) -> n__g(X)}

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {},
                 Uargs(n__g) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(sel^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(activate^#) = {}, Uargs(c_7) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                n__f(x1) = [0] x1 + [0]
                n__g(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                activate^#(x1) = [3] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(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_9(X)}
               Weak Rules: {}

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