Problem Transformed CSR 04 Ex4 Zan97 Z

Tool CaT

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

stdout:

MAYBE

Problem:
f(X) -> cons(X,n__f(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)
activate(n__f(X)) -> f(X)
activate(X) -> X

Proof:
Open

Tool IRC1

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

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^3))
Input	Transformed CSR 04 Ex4 Zan97 Z

stdout:

YES(?,O(n^3))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^3))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(X) -> cons(X, n__f(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)
     , activate(n__f(X)) -> f(X)
     , activate(X) -> X}

Proof Output:
  'wdg' proved the best result:

  Details:
  --------
    'wdg' succeeded with the following output:
     'wdg'
     -----
     Answer:           YES(?,O(n^3))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  f(X) -> cons(X, n__f(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)
          , activate(n__f(X)) -> f(X)
          , activate(X) -> X}

     Proof Output:
       Transformation Details:
       -----------------------
         We have computed the following set of weak (innermost) dependency pairs:

           {  1: f^#(X) -> c_0(g^#(X))
            , 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: activate^#(n__f(X)) -> c_6(f^#(X))
            , 8: activate^#(X) -> c_7()}

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

           ->{8}                                                       [    YES(?,O(1))     ]

           ->{7}                                                       [   YES(?,O(n^2))    ]
              |
              |->{1}                                                   [   YES(?,O(n^2))    ]
              |   |
              |   |->{2}                                               [   YES(?,O(n^2))    ]
              |   |
              |   `->{3}                                               [   YES(?,O(n^3))    ]
              |       |
              |       `->{2}                                           [   YES(?,O(n^2))    ]
              |
              `->{6}                                                   [   YES(?,O(n^2))    ]

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



       Sub-problems:
       -------------
         * Path {5}: YES(?,O(n^3))
           -----------------------

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {2}, Uargs(n__f) = {1}, Uargs(g) = {},
               Uargs(s) = {1}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
               Uargs(sel^#) = {2}, Uargs(c_4) = {1}, Uargs(activate^#) = {},
               Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1) = [1 1 3] x1 + [3]
                      [0 1 3]      [3]
                      [0 0 0]      [3]
              cons(x1, x2) = [1 1 1] x1 + [1 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [3]
                             [0 0 0]      [0 0 1]      [0]
              n__f(x1) = [1 0 0] x1 + [0]
                         [0 1 3]      [1]
                         [0 0 0]      [3]
              g(x1) = [0 0 2] x1 + [2]
                      [3 1 3]      [0]
                      [0 0 2]      [0]
              0() = [2]
                    [1]
                    [3]
              s(x1) = [1 0 0] x1 + [2]
                      [0 0 0]      [1]
                      [0 0 1]      [3]
              sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
              activate(x1) = [1 1 1] x1 + [3]
                             [0 2 0]      [3]
                             [0 0 2]      [1]
              f^#(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]
              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 + [1 0 0] x2 + [0]
                              [3 3 3]      [3 3 3]      [0]
                              [3 3 3]      [3 3 3]      [0]
              c_3() = [0]
                      [0]
                      [0]
              c_4(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              c_5() = [0]
                      [0]
                      [0]
              activate^#(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]
              c_7() = [0]
                      [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           'matrix-interpretation of dimension 3'
           --------------------------------------
           Answer:           YES(?,O(n^2))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules:
               {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))}
             Weak Rules:
               {  activate(n__f(X)) -> f(X)
                , activate(X) -> X
                , f(X) -> cons(X, n__f(g(X)))
                , f(X) -> n__f(X)
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))}

           Proof Output:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
               Uargs(s) = {}, Uargs(activate) = {}, Uargs(sel^#) = {},
               Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1) = [0 0 0] x1 + [4]
                      [0 0 0]      [0]
                      [0 0 0]      [0]
              cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
              n__f(x1) = [0 0 0] x1 + [2]
                         [0 0 0]      [0]
                         [0 0 0]      [0]
              g(x1) = [4 0 0] x1 + [1]
                      [4 0 1]      [0]
                      [0 0 3]      [0]
              0() = [2]
                    [0]
                    [4]
              s(x1) = [0 0 0] x1 + [1]
                      [0 0 0]      [0]
                      [0 0 1]      [5]
              activate(x1) = [1 0 0] x1 + [2]
                             [0 2 0]      [0]
                             [0 0 1]      [0]
              sel^#(x1, x2) = [0 0 2] x1 + [4 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [4]
                              [0 0 0]      [0 0 0]      [0]
              c_4(x1) = [1 0 0] x1 + [1]
                        [0 0 0]      [3]
                        [0 0 0]      [0]

         * Path {5}->{4}: YES(?,O(n^3))
           ----------------------------

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {2}, Uargs(n__f) = {1}, Uargs(g) = {},
               Uargs(s) = {1}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
               Uargs(sel^#) = {2}, Uargs(c_4) = {1}, Uargs(activate^#) = {},
               Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1) = [2 3 3] x1 + [3]
                      [0 1 1]      [3]
                      [0 0 1]      [3]
              cons(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                             [0 0 1]      [0 0 0]      [1]
                             [0 0 1]      [0 0 0]      [3]
              n__f(x1) = [1 1 0] x1 + [0]
                         [0 1 1]      [3]
                         [0 0 1]      [2]
              g(x1) = [1 0 3] x1 + [2]
                      [1 1 0]      [0]
                      [0 0 2]      [2]
              0() = [2]
                    [2]
                    [0]
              s(x1) = [1 0 0] x1 + [1]
                      [0 0 0]      [0]
                      [0 0 1]      [1]
              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) = [2 2 1] x1 + [3]
                             [0 1 2]      [3]
                             [0 0 1]      [1]
              f^#(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]
              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 + [3 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
              c_3() = [0]
                      [0]
                      [0]
              c_4(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              c_5() = [0]
                      [0]
                      [0]
              activate^#(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]
              c_7() = [0]
                      [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           'matrix-interpretation of dimension 3'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {sel^#(0(), cons(X, Y)) -> c_3()}
             Weak Rules:
               {  sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
                , activate(n__f(X)) -> f(X)
                , activate(X) -> X
                , f(X) -> cons(X, n__f(g(X)))
                , f(X) -> n__f(X)
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))}

           Proof Output:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
               Uargs(s) = {}, Uargs(activate) = {}, Uargs(sel^#) = {},
               Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1) = [0 3 0] x1 + [3]
                      [0 1 0]      [0]
                      [0 0 0]      [3]
              cons(x1, x2) = [0 3 0] x1 + [0 0 2] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
              n__f(x1) = [0 0 0] x1 + [2]
                         [0 1 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 4]      [0]
                      [0 0 0]      [0]
              activate(x1) = [4 4 0] x1 + [0]
                             [0 1 0]      [0]
                             [2 0 4]      [0]
              sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [1]
                              [0 0 0]      [4 0 0]      [0]
                              [0 1 0]      [0 0 0]      [0]
              c_3() = [0]
                      [0]
                      [0]
              c_4(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]

         * Path {7}: YES(?,O(n^2))
           -----------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
               Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(activate^#) = {},
               Uargs(c_6) = {}
             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) = [1 0 0] x1 + [0]
                         [0 0 0]      [0]
                         [0 0 1]      [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) = [3 0 0] x1 + [0]
                        [3 0 0]      [0]
                        [3 0 0]      [0]
              c_0(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]
              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]
              activate^#(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]
              c_7() = [0]
                      [0]
                      [0]

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

           'matrix-interpretation of dimension 3'
           --------------------------------------
           Answer:           YES(?,O(n^2))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}
             Weak Rules: {}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(activate^#) = {},
               Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__f(x1) = [1 2 2] x1 + [2]
                         [0 1 2]      [2]
                         [0 0 0]      [2]
              f^#(x1) = [0 2 0] x1 + [2]
                        [0 2 2]      [2]
                        [0 0 2]      [2]
              activate^#(x1) = [3 2 0] x1 + [5]
                               [2 0 2]      [6]
                               [2 3 2]      [0]
              c_6(x1) = [2 2 2] x1 + [1]
                        [2 0 0]      [6]
                        [0 0 0]      [7]

         * Path {7}->{1}: YES(?,O(n^2))
           ----------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
               Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(activate^#) = {},
               Uargs(c_6) = {1}
             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]
              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) = [3 3 3] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
              c_0(x1) = [0 0 3] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
              g^#(x1) = [3 3 3] x1 + [0]
                        [3 3 3]      [0]
                        [1 1 1]      [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]
              activate^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
              c_6(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              c_7() = [0]
                      [0]
                      [0]

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

           'matrix-interpretation of dimension 3'
           --------------------------------------
           Answer:           YES(?,O(n^2))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {f^#(X) -> c_0(g^#(X))}
             Weak Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
               Uargs(g^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__f(x1) = [1 3 2] x1 + [0]
                         [0 1 0]      [2]
                         [0 0 0]      [2]
              f^#(x1) = [0 2 0] x1 + [2]
                        [0 2 0]      [2]
                        [0 0 2]      [0]
              c_0(x1) = [0 0 0] x1 + [1]
                        [0 0 0]      [2]
                        [0 0 0]      [0]
              g^#(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
              activate^#(x1) = [2 5 2] x1 + [1]
                               [4 2 2]      [7]
                               [4 2 2]      [6]
              c_6(x1) = [2 2 0] x1 + [3]
                        [2 2 0]      [3]
                        [2 0 0]      [7]

         * Path {7}->{1}->{2}: YES(?,O(n^2))
           ---------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
               Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {1}, Uargs(g^#) = {},
               Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {},
               Uargs(activate^#) = {}, Uargs(c_6) = {1}
             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]
              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) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [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]
              activate^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
              c_6(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              c_7() = [0]
                      [0]
                      [0]

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

           'matrix-interpretation of dimension 3'
           --------------------------------------
           Answer:           YES(?,O(n^2))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {g^#(0()) -> c_1()}
             Weak Rules:
               {  f^#(X) -> c_0(g^#(X))
                , activate^#(n__f(X)) -> c_6(f^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
               Uargs(g^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__f(x1) = [1 6 1] x1 + [2]
                         [0 0 3]      [2]
                         [0 0 1]      [2]
              0() = [2]
                    [2]
                    [2]
              f^#(x1) = [0 2 0] x1 + [2]
                        [0 2 2]      [2]
                        [0 2 0]      [2]
              c_0(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [1]
                        [0 0 0]      [1]
              g^#(x1) = [0 0 0] x1 + [2]
                        [2 0 2]      [0]
                        [0 2 2]      [0]
              c_1() = [1]
                      [0]
                      [0]
              activate^#(x1) = [2 2 0] x1 + [7]
                               [2 2 0]      [7]
                               [2 2 2]      [2]
              c_6(x1) = [2 2 2] x1 + [3]
                        [2 0 0]      [7]
                        [0 0 0]      [3]

         * Path {7}->{1}->{3}: YES(?,O(n^3))
           ---------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
               Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {1}, Uargs(g^#) = {},
               Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {},
               Uargs(activate^#) = {}, Uargs(c_6) = {1}
             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]
              g(x1) = [0 0 0] x1 + [0]
                      [0 0 0]      [0]
                      [0 0 0]      [0]
              0() = [0]
                    [0]
                    [0]
              s(x1) = [1 3 0] x1 + [0]
                      [0 1 0]      [0]
                      [0 0 0]      [0]
              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) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              g^#(x1) = [0 0 0] x1 + [0]
                        [3 3 3]      [0]
                        [3 3 3]      [0]
              c_1() = [0]
                      [0]
                      [0]
              c_2(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
              c_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]
              activate^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
              c_6(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              c_7() = [0]
                      [0]
                      [0]

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

           'matrix-interpretation of dimension 3'
           --------------------------------------
           Answer:           YES(?,O(n^3))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
             Weak Rules:
               {  f^#(X) -> c_0(g^#(X))
                , activate^#(n__f(X)) -> c_6(f^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__f) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
               Uargs(g^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__f(x1) = [1 0 0] x1 + [0]
                         [0 1 3]      [2]
                         [0 0 1]      [2]
              s(x1) = [1 2 0] x1 + [0]
                      [0 1 2]      [2]
                      [0 0 0]      [0]
              f^#(x1) = [0 1 2] x1 + [2]
                        [0 0 2]      [2]
                        [4 0 0]      [0]
              c_0(x1) = [1 0 0] x1 + [1]
                        [0 0 0]      [2]
                        [0 0 0]      [0]
              g^#(x1) = [0 1 2] x1 + [0]
                        [2 2 0]      [2]
                        [6 2 0]      [4]
              c_2(x1) = [1 0 0] x1 + [1]
                        [2 0 0]      [3]
                        [0 2 0]      [3]
              activate^#(x1) = [0 4 0] x1 + [6]
                               [0 0 0]      [7]
                               [0 2 2]      [7]
              c_6(x1) = [1 2 0] x1 + [1]
                        [0 0 0]      [3]
                        [2 2 0]      [3]

         * Path {7}->{1}->{3}->{2}: YES(?,O(n^2))
           --------------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
               Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {1}, Uargs(g^#) = {},
               Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {},
               Uargs(activate^#) = {}, Uargs(c_6) = {1}
             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]
              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) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              g^#(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
              c_1() = [0]
                      [0]
                      [0]
              c_2(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
              c_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]
              activate^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
              c_6(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              c_7() = [0]
                      [0]
                      [0]

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

           'matrix-interpretation of dimension 3'
           --------------------------------------
           Answer:           YES(?,O(n^2))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {g^#(0()) -> c_1()}
             Weak Rules:
               {  g^#(s(X)) -> c_2(g^#(X))
                , f^#(X) -> c_0(g^#(X))
                , activate^#(n__f(X)) -> c_6(f^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__f) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
               Uargs(g^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__f(x1) = [1 1 0] x1 + [0]
                         [0 1 0]      [4]
                         [0 0 0]      [2]
              0() = [2]
                    [2]
                    [2]
              s(x1) = [1 1 0] x1 + [2]
                      [0 0 2]      [0]
                      [0 0 0]      [2]
              f^#(x1) = [7 4 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [2]
              c_0(x1) = [1 0 2] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [1]
              g^#(x1) = [3 2 0] x1 + [0]
                        [0 2 2]      [4]
                        [2 1 0]      [0]
              c_1() = [1]
                      [0]
                      [0]
              c_2(x1) = [1 0 0] x1 + [3]
                        [0 0 0]      [2]
                        [0 0 0]      [3]
              activate^#(x1) = [7 1 2] x1 + [6]
                               [2 0 0]      [7]
                               [0 0 2]      [7]
              c_6(x1) = [1 4 0] x1 + [7]
                        [0 0 2]      [3]
                        [0 0 0]      [7]

         * Path {7}->{6}: YES(?,O(n^2))
           ----------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
               Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(activate^#) = {},
               Uargs(c_6) = {1}
             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]
              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) = [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]
              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]
              activate^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
              c_6(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
              c_7() = [0]
                      [0]
                      [0]

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

           'matrix-interpretation of dimension 3'
           --------------------------------------
           Answer:           YES(?,O(n^2))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {f^#(X) -> c_5()}
             Weak Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(activate^#) = {},
               Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__f(x1) = [1 2 2] x1 + [2]
                         [0 1 2]      [0]
                         [0 0 0]      [0]
              f^#(x1) = [0 0 2] x1 + [2]
                        [0 0 0]      [0]
                        [0 0 0]      [2]
              c_5() = [1]
                      [0]
                      [0]
              activate^#(x1) = [2 4 0] x1 + [7]
                               [6 0 0]      [2]
                               [2 2 0]      [6]
              c_6(x1) = [1 0 2] x1 + [1]
                        [0 0 2]      [7]
                        [2 0 2]      [2]

         * Path {8}: 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(g) = {},
               Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
               Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(activate^#) = {},
               Uargs(c_6) = {}
             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]
              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) = [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]
              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]
              activate^#(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]
              c_7() = [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_7()}
             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_7() = [0]
                      [3]
                      [3]

Tool RC1

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

stdout:

MAYBE

Tool RC2

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

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(X) -> cons(X, n__f(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)
     , activate(n__f(X)) -> f(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, g^#(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: activate^#(n__f(X)) -> c_6(f^#(X))
              , 8: activate^#(X) -> c_7(X)}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |   |
                |   |->{2}                                               [   YES(?,O(n^2))    ]
                |   |
                |   `->{3}                                               [   YES(?,O(n^2))    ]
                |       |
                |       `->{2}                                           [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]

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



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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(X)
                , activate(X) -> X
                , f(X) -> cons(X, n__f(g(X)))
                , f(X) -> n__f(X)
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(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:
                 {  sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
                  , activate(n__f(X)) -> f(X)
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(g(X)))
                  , f(X) -> n__f(X)
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

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

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {1}, Uargs(cons) = {1, 2}, Uargs(n__f) = {1},
                 Uargs(g) = {1}, Uargs(s) = {1}, Uargs(sel) = {},
                 Uargs(activate) = {1}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(sel^#) = {2},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {},
                 Uargs(activate^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [2 2 0] x1 + [3]
                        [0 1 0]      [3]
                        [0 0 0]      [3]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [3]
                               [0 0 0]      [0 0 1]      [1]
                n__f(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 0]      [1]
                g(x1) = [1 2 0] x1 + [2]
                        [3 3 0]      [0]
                        [0 0 0]      [0]
                0() = [1]
                      [1]
                      [0]
                s(x1) = [1 0 2] x1 + [3]
                        [0 1 3]      [3]
                        [0 0 0]      [0]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                activate(x1) = [2 2 1] x1 + [3]
                               [0 2 0]      [3]
                               [0 0 1]      [3]
                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 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5(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]
                c_6(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]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {7}: YES(?,O(n^2))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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) = [1 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 1]      [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) = [3 0 0] x1 + [0]
                          [3 0 0]      [0]
                          [3 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]
                activate^#(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]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 2 2] x1 + [2]
                           [0 1 2]      [2]
                           [0 0 0]      [2]
                f^#(x1) = [0 2 0] x1 + [2]
                          [0 2 2]      [2]
                          [0 0 2]      [2]
                activate^#(x1) = [3 2 0] x1 + [5]
                                 [2 0 2]      [6]
                                 [2 3 2]      [0]
                c_6(x1) = [2 2 2] x1 + [1]
                          [2 0 0]      [6]
                          [0 0 0]      [7]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               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]
                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) = [3 3 3] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1, x2) = [0 0 1] x1 + [1 1 1] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                g^#(x1) = [0 1 1] x1 + [0]
                          [3 1 0]      [0]
                          [0 1 1]      [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]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(X) -> c_0(X, g^#(X))}
               Weak Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 2 2] x1 + [2]
                           [0 0 2]      [0]
                           [0 0 0]      [1]
                f^#(x1) = [0 0 0] x1 + [1]
                          [0 0 2]      [0]
                          [0 2 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 2]      [0 0 0]      [0]
                              [0 2 0]      [0 0 0]      [0]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                activate^#(x1) = [2 2 4] x1 + [6]
                                 [0 0 1]      [7]
                                 [4 2 0]      [7]
                c_6(x1) = [1 2 2] x1 + [1]
                          [0 0 0]      [7]
                          [0 0 0]      [7]

           * Path {7}->{1}->{2}: YES(?,O(n^2))
             ---------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                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 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [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]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules:
                 {  f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 1 0] x1 + [0]
                           [0 1 2]      [0]
                           [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                f^#(x1) = [2 2 2] x1 + [0]
                          [0 2 0]      [0]
                          [0 2 0]      [0]
                c_0(x1, x2) = [0 0 2] x1 + [1 0 0] x2 + [0]
                              [0 2 0]      [0 0 0]      [0]
                              [0 1 0]      [0 0 0]      [0]
                g^#(x1) = [2 2 0] x1 + [0]
                          [2 2 2]      [0]
                          [2 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]
                activate^#(x1) = [7 6 0] x1 + [7]
                                 [0 0 0]      [7]
                                 [4 4 0]      [7]
                c_6(x1) = [2 2 2] x1 + [3]
                          [0 0 0]      [3]
                          [0 0 0]      [2]

           * Path {7}->{1}->{3}: YES(?,O(n^2))
             ---------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                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 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                g^#(x1) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_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]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
               Weak Rules:
                 {  f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 0]      [0]
                s(x1) = [1 1 7] x1 + [7]
                        [0 0 1]      [1]
                        [0 0 0]      [0]
                f^#(x1) = [1 1 0] x1 + [0]
                          [0 0 0]      [4]
                          [0 0 2]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [3]
                              [0 0 1]      [0 0 0]      [0]
                g^#(x1) = [1 1 0] x1 + [0]
                          [0 0 0]      [2]
                          [0 4 0]      [4]
                c_2(x1) = [1 2 0] x1 + [3]
                          [0 0 0]      [2]
                          [0 0 0]      [6]
                activate^#(x1) = [4 4 0] x1 + [6]
                                 [0 1 0]      [7]
                                 [4 4 0]      [6]
                c_6(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [3]
                          [0 0 0]      [3]

           * Path {7}->{1}->{3}->{2}: YES(?,O(n^1))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                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 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_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]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules:
                 {  g^#(s(X)) -> c_2(g^#(X))
                  , f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 2 2] x1 + [2]
                           [0 0 2]      [0]
                           [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 2 2] x1 + [0]
                        [0 0 4]      [2]
                        [0 0 0]      [2]
                f^#(x1) = [2 2 4] x1 + [2]
                          [0 0 0]      [2]
                          [0 2 2]      [2]
                c_0(x1, x2) = [0 2 2] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 0 0]      [1]
                              [0 2 2]      [0 0 0]      [1]
                g^#(x1) = [2 0 2] x1 + [0]
                          [2 2 2]      [0]
                          [2 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
                activate^#(x1) = [2 4 2] x1 + [7]
                                 [6 0 0]      [2]
                                 [2 0 2]      [6]
                c_6(x1) = [1 0 0] x1 + [5]
                          [2 0 2]      [3]
                          [0 2 0]      [7]

           * Path {7}->{6}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               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]
                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) = [3 3 3] 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) = [1 1 1] 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]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(X) -> c_5(X)}
               Weak Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_5) = {},
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 2 1] x1 + [0]
                           [0 0 0]      [2]
                           [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [2]
                          [0 0 0]      [2]
                          [0 0 0]      [2]
                c_5(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                activate^#(x1) = [6 0 0] x1 + [7]
                                 [0 0 0]      [7]
                                 [4 2 0]      [7]
                c_6(x1) = [1 0 0] x1 + [1]
                          [0 2 0]      [3]
                          [2 0 2]      [3]

           * Path {8}: 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(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(activate^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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]
                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]
                activate^#(x1) = [3 3 3] 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]
                c_7(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_7(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               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_7(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, g^#(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: activate^#(n__f(X)) -> c_6(f^#(X))
              , 8: activate^#(X) -> c_7(X)}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                |->{1}                                                   [   YES(?,O(n^2))    ]
                |   |
                |   |->{2}                                               [   YES(?,O(n^2))    ]
                |   |
                |   `->{3}                                               [   YES(?,O(n^2))    ]
                |       |
                |       `->{2}                                           [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]

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



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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(X)
                , activate(X) -> X
                , f(X) -> cons(X, n__f(g(X)))
                , f(X) -> n__f(X)
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(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:
                 {  sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
                  , activate(n__f(X)) -> f(X)
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(g(X)))
                  , f(X) -> n__f(X)
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

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

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {1}, Uargs(cons) = {1, 2}, Uargs(n__f) = {1},
                 Uargs(g) = {1}, Uargs(s) = {1}, Uargs(sel) = {},
                 Uargs(activate) = {1}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_2) = {}, Uargs(sel^#) = {2},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {},
                 Uargs(activate^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [3 1] x1 + [3]
                        [3 3]      [1]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [1]
                n__f(x1) = [1 0] x1 + [2]
                           [0 1]      [0]
                g(x1) = [2 1] x1 + [0]
                        [0 2]      [0]
                0() = [0]
                      [2]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                activate(x1) = [3 1] x1 + [1]
                               [3 3]      [3]
                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 + [3 2] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {7}: YES(?,O(n^2))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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) = [1 1] x1 + [0]
                           [0 1]      [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) = [3 0] x1 + [0]
                          [3 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]
                activate^#(x1) = [1 3] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

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

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

           * Path {7}->{1}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               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]
                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) = [3 3] x1 + [0]
                          [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                g^#(x1) = [1 1] x1 + [0]
                          [3 3]      [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]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(X) -> c_0(X, g^#(X))}
               Weak Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 2] x1 + [4]
                           [0 1]      [0]
                f^#(x1) = [0 2] x1 + [2]
                          [0 0]      [2]
                c_0(x1, x2) = [0 2] x1 + [0 0] x2 + [1]
                              [0 0]      [0 0]      [2]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [2 1] x1 + [7]
                                 [2 5]      [7]
                c_6(x1) = [1 0] x1 + [5]
                          [2 2]      [3]

           * Path {7}->{1}->{2}: YES(?,O(n^2))
             ---------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                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 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [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]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules:
                 {  f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 2] x1 + [2]
                           [0 1]      [0]
                0() = [2]
                      [2]
                f^#(x1) = [2 2] x1 + [2]
                          [0 2]      [2]
                c_0(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                              [0 1]      [0 0]      [1]
                g^#(x1) = [2 2] x1 + [0]
                          [2 2]      [0]
                c_1() = [1]
                        [0]
                activate^#(x1) = [4 1] x1 + [7]
                                 [6 0]      [2]
                c_6(x1) = [2 2] x1 + [3]
                          [2 0]      [7]

           * Path {7}->{1}->{3}: YES(?,O(n^2))
             ---------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] 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 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                g^#(x1) = [3 3] x1 + [0]
                          [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [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]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
               Weak Rules:
                 {  f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 2] x1 + [0]
                           [0 1]      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 1]      [2]
                f^#(x1) = [0 2] x1 + [0]
                          [4 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [3 0]      [0 0]      [0]
                g^#(x1) = [0 2] x1 + [0]
                          [4 4]      [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
                activate^#(x1) = [4 4] x1 + [7]
                                 [4 0]      [7]
                c_6(x1) = [4 0] x1 + [2]
                          [0 0]      [3]

           * Path {7}->{1}->{3}->{2}: YES(?,O(n^2))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                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 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [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]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules:
                 {  g^#(s(X)) -> c_2(g^#(X))
                  , f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 1] x1 + [0]
                           [0 1]      [2]
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 0]      [2]
                f^#(x1) = [2 2] x1 + [2]
                          [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
                              [0 0]      [0 0]      [0]
                g^#(x1) = [2 2] x1 + [0]
                          [4 2]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [7]
                          [2 0]      [7]
                activate^#(x1) = [6 7] x1 + [1]
                                 [6 2]      [7]
                c_6(x1) = [2 0] x1 + [7]
                          [2 0]      [3]

           * Path {7}->{6}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               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]
                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) = [3 3] 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) = [1 1] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(X) -> c_5(X)}
               Weak Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_5) = {},
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1 2] x1 + [2]
                           [0 1]      [1]
                f^#(x1) = [0 2] x1 + [2]
                          [0 4]      [0]
                c_5(x1) = [0 0] x1 + [1]
                          [0 0]      [0]
                activate^#(x1) = [2 5] x1 + [5]
                                 [6 0]      [2]
                c_6(x1) = [2 0] x1 + [6]
                          [2 1]      [3]

           * Path {8}: 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(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(activate^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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]
                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]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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_7(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7] x1 + [7]
                                 [7 7]      [7]
                c_7(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, g^#(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: activate^#(n__f(X)) -> c_6(f^#(X))
              , 8: activate^#(X) -> c_7(X)}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |   |
                |   |->{2}                                               [   YES(?,O(n^1))    ]
                |   |
                |   `->{3}                                               [   YES(?,O(n^1))    ]
                |       |
                |       `->{2}                                           [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]

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



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

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

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

             The usable rules for this path are:

               {  activate(n__f(X)) -> f(X)
                , activate(X) -> X
                , f(X) -> cons(X, n__f(g(X)))
                , f(X) -> n__f(X)
                , g(0()) -> s(0())
                , g(s(X)) -> s(s(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:
                 {  sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
                  , sel^#(0(), cons(X, Y)) -> c_3(X)
                  , activate(n__f(X)) -> f(X)
                  , activate(X) -> X
                  , f(X) -> cons(X, n__f(g(X)))
                  , f(X) -> n__f(X)
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))}

             Proof Output:
               The input cannot be shown compatible

           * Path {7}: YES(?,O(n^1))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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) = [1] 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) = [1] 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]
                activate^#(x1) = [3] x1 + [0]
                c_6(x1) = [3] x1 + [0]
                c_7(x1) = [0] x1 + [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1] x1 + [0]
                f^#(x1) = [7] x1 + [0]
                activate^#(x1) = [0] x1 + [7]
                c_6(x1) = [0] x1 + [5]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               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]
                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) = [3] x1 + [0]
                c_0(x1, x2) = [0] x1 + [3] x2 + [0]
                g^#(x1) = [1] 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]
                activate^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(X) -> c_0(X, g^#(X))}
               Weak Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1] x1 + [2]
                f^#(x1) = [0] x1 + [2]
                c_0(x1, x2) = [0] x1 + [7] x2 + [1]
                g^#(x1) = [0] x1 + [0]
                activate^#(x1) = [2] x1 + [7]
                c_6(x1) = [4] x1 + [3]

           * Path {7}->{1}->{2}: YES(?,O(n^1))
             ---------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                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 + [1] 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]
                activate^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules:
                 {  f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1] x1 + [2]
                0() = [2]
                f^#(x1) = [2] x1 + [2]
                c_0(x1, x2) = [0] x1 + [1] x2 + [2]
                g^#(x1) = [2] x1 + [0]
                c_1() = [1]
                activate^#(x1) = [6] x1 + [3]
                c_6(x1) = [2] x1 + [7]

           * Path {7}->{1}->{3}: YES(?,O(n^1))
             ---------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] 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 + [1] x2 + [0]
                g^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] 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]
                activate^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
               Weak Rules:
                 {  f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1] x1 + [2]
                s(x1) = [1] x1 + [2]
                f^#(x1) = [2] x1 + [2]
                c_0(x1, x2) = [0] x1 + [1] x2 + [1]
                g^#(x1) = [2] x1 + [0]
                c_2(x1) = [1] x1 + [3]
                activate^#(x1) = [6] x1 + [2]
                c_6(x1) = [2] x1 + [6]

           * Path {7}->{1}->{3}->{2}: YES(?,O(n^1))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, Uargs(g) = {},
                 Uargs(s) = {}, Uargs(sel) = {}, Uargs(activate) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {2}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_7) = {}
               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]
                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 + [1] x2 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] 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]
                activate^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules:
                 {  g^#(s(X)) -> c_2(g^#(X))
                  , f^#(X) -> c_0(X, g^#(X))
                  , activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {2},
                 Uargs(g^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1] x1 + [0]
                0() = [2]
                s(x1) = [1] x1 + [0]
                f^#(x1) = [3] x1 + [2]
                c_0(x1, x2) = [0] x1 + [1] x2 + [0]
                g^#(x1) = [3] x1 + [2]
                c_1() = [1]
                c_2(x1) = [1] x1 + [0]
                activate^#(x1) = [6] x1 + [7]
                c_6(x1) = [2] x1 + [3]

           * Path {7}->{6}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cons) = {}, Uargs(n__f) = {}, 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(activate^#) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               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]
                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) = [3] 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) = [1] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(X) -> c_5(X)}
               Weak Rules: {activate^#(n__f(X)) -> c_6(f^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__f) = {}, Uargs(f^#) = {}, Uargs(c_5) = {},
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__f(x1) = [1] x1 + [2]
                f^#(x1) = [1] x1 + [2]
                c_5(x1) = [1] x1 + [1]
                activate^#(x1) = [2] x1 + [7]
                c_6(x1) = [2] x1 + [7]

           * Path {8}: 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(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(activate^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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]
                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]
                activate^#(x1) = [3] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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_7(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7] x1 + [7]
                c_7(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.