Problem Transformed CSR 04 Ex1 2 Luc02c FR

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex1 2 Luc02c FR

stdout:

MAYBE

Problem:
2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
from(X) -> cons(X,n__from(n__s(X)))
cons(X1,X2) -> n__cons(X1,X2)
from(X) -> n__from(X)
s(X) -> n__s(X)
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex1 2 Luc02c FR

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex1 2 Luc02c FR

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  2nd(cons(X, n__cons(Y, Z))) -> activate(Y)
     , from(X) -> cons(X, n__from(n__s(X)))
     , cons(X1, X2) -> n__cons(X1, X2)
     , from(X) -> n__from(X)
     , s(X) -> n__s(X)
     , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
     , activate(n__from(X)) -> from(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(X) -> X}

Proof Output:
  'wdg' proved the best result:

  Details:
  --------
    'wdg' succeeded with the following output:
     'wdg'
     -----
     Answer:           YES(?,O(n^1))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  2nd(cons(X, n__cons(Y, Z))) -> activate(Y)
          , from(X) -> cons(X, n__from(n__s(X)))
          , cons(X1, X2) -> n__cons(X1, X2)
          , from(X) -> n__from(X)
          , s(X) -> n__s(X)
          , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
          , activate(n__from(X)) -> from(activate(X))
          , activate(n__s(X)) -> s(activate(X))
          , activate(X) -> X}

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

           {  1: 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
            , 2: from^#(X) -> c_1(cons^#(X, n__from(n__s(X))))
            , 3: cons^#(X1, X2) -> c_2()
            , 4: from^#(X) -> c_3()
            , 5: s^#(X) -> c_4()
            , 6: activate^#(n__cons(X1, X2)) -> c_5(cons^#(activate(X1), X2))
            , 7: activate^#(n__from(X)) -> c_6(from^#(activate(X)))
            , 8: activate^#(n__s(X)) -> c_7(s^#(activate(X)))
            , 9: activate^#(X) -> c_8()}

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

           ->{1}                                                       [   YES(?,O(n^1))    ]
              |
              |->{6}                                                   [   YES(?,O(n^1))    ]
              |   |
              |   `->{3}                                               [   YES(?,O(n^1))    ]
              |
              |->{7}                                                   [   YES(?,O(n^1))    ]
              |   |
              |   |->{2}                                               [   YES(?,O(n^1))    ]
              |   |   |
              |   |   `->{3}                                           [   YES(?,O(n^1))    ]
              |   |
              |   `->{4}                                               [   YES(?,O(n^1))    ]
              |
              |->{8}                                                   [   YES(?,O(n^1))    ]
              |   |
              |   `->{5}                                               [   YES(?,O(n^1))    ]
              |
              `->{9}                                                   [   YES(?,O(n^1))    ]



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

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
             Weak Rules: {}

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

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

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons) = {1}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {},
               Uargs(c_1) = {}, Uargs(cons^#) = {1}, Uargs(s^#) = {},
               Uargs(c_5) = {1}, Uargs(c_6) = {}, Uargs(c_7) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [0] x2 + [2]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [1]
              activate(x1) = [3] x1 + [1]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              n__s(x1) = [1] x1 + [1]
              s(x1) = [1] x1 + [2]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [3] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              cons^#(x1, x2) = [1] x1 + [0] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [0] x1 + [0]
              c_4() = [0]
              c_5(x1) = [1] x1 + [0]
              c_6(x1) = [0] x1 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules:
               {activate^#(n__cons(X1, X2)) -> c_5(cons^#(activate(X1), X2))}
             Weak Rules:
               {  2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , cons(X1, X2) -> n__cons(X1, X2)
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(n__s) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {1},
               Uargs(activate^#) = {}, Uargs(cons^#) = {}, Uargs(c_5) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [1] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              n__s(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [7]
              c_0(x1) = [4] x1 + [3]
              activate^#(x1) = [0] x1 + [1]
              cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_5(x1) = [1] x1 + [0]

         * Path {1}->{6}->{3}: YES(?,O(n^1))
           ---------------------------------

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons) = {1}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {},
               Uargs(c_1) = {}, Uargs(cons^#) = {1}, Uargs(s^#) = {},
               Uargs(c_5) = {1}, Uargs(c_6) = {}, Uargs(c_7) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [0] x2 + [2]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [1]
              activate(x1) = [3] x1 + [1]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              n__s(x1) = [1] x1 + [1]
              s(x1) = [1] x1 + [2]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              cons^#(x1, x2) = [3] x1 + [0] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [0] x1 + [0]
              c_4() = [0]
              c_5(x1) = [1] x1 + [0]
              c_6(x1) = [0] x1 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {cons^#(X1, X2) -> c_2()}
             Weak Rules:
               {  activate^#(n__cons(X1, X2)) -> c_5(cons^#(activate(X1), X2))
                , 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , cons(X1, X2) -> n__cons(X1, X2)
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(n__s) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {1},
               Uargs(activate^#) = {}, Uargs(cons^#) = {}, Uargs(c_5) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [0]
              n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [1] x1 + [0]
              from(x1) = [1] x1 + [0]
              n__from(x1) = [1] x1 + [0]
              n__s(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [6] x1 + [6]
              c_0(x1) = [1] x1 + [4]
              activate^#(x1) = [4] x1 + [2]
              cons^#(x1, x2) = [2] x1 + [0] x2 + [1]
              c_2() = [0]
              c_5(x1) = [2] x1 + [0]

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

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons) = {1}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {1},
               Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(s^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {1}, Uargs(c_7) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [0] x2 + [2]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [1]
              activate(x1) = [3] x1 + [1]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              n__s(x1) = [1] x1 + [1]
              s(x1) = [1] x1 + [2]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [3] x1 + [0]
              from^#(x1) = [1] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [0] x1 + [0]
              c_4() = [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1) = [1] x1 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {activate^#(n__from(X)) -> c_6(from^#(activate(X)))}
             Weak Rules:
               {  2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , cons(X1, X2) -> n__cons(X1, X2)
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(n__s) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {1},
               Uargs(activate^#) = {}, Uargs(from^#) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [1] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              n__s(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [7]
              c_0(x1) = [1] x1 + [3]
              activate^#(x1) = [0] x1 + [1]
              from^#(x1) = [0] x1 + [0]
              c_6(x1) = [2] x1 + [0]

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

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons) = {1}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {1},
               Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(s^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {1}, Uargs(c_7) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [0] x2 + [2]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [1]
              activate(x1) = [3] x1 + [2]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              n__s(x1) = [1] x1 + [2]
              s(x1) = [1] x1 + [3]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              from^#(x1) = [3] x1 + [0]
              c_1(x1) = [3] x1 + [0]
              cons^#(x1, x2) = [0] x1 + [1] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [0] x1 + [0]
              c_4() = [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1) = [1] x1 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {from^#(X) -> c_1(cons^#(X, n__from(n__s(X))))}
             Weak Rules:
               {  activate^#(n__from(X)) -> c_6(from^#(activate(X)))
                , 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , cons(X1, X2) -> n__cons(X1, X2)
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(n__s) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {1},
               Uargs(activate^#) = {}, Uargs(from^#) = {}, Uargs(c_1) = {},
               Uargs(cons^#) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [0]
              n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [1] x1 + [0]
              from(x1) = [1] x1 + [1]
              n__from(x1) = [1] x1 + [1]
              n__s(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [1] x1 + [6]
              c_0(x1) = [1] x1 + [6]
              activate^#(x1) = [1] x1 + [0]
              from^#(x1) = [0] x1 + [1]
              c_1(x1) = [0] x1 + [0]
              cons^#(x1, x2) = [0] x1 + [0] x2 + [7]
              c_6(x1) = [1] x1 + [0]

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

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons) = {1}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {1},
               Uargs(c_1) = {1}, Uargs(cons^#) = {}, Uargs(s^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {1}, Uargs(c_7) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [0] x2 + [2]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [1]
              activate(x1) = [3] x1 + [1]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              n__s(x1) = [1] x1 + [2]
              s(x1) = [1] x1 + [3]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              from^#(x1) = [3] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [0] x1 + [0]
              c_4() = [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1) = [1] x1 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {cons^#(X1, X2) -> c_2()}
             Weak Rules:
               {  from^#(X) -> c_1(cons^#(X, n__from(n__s(X))))
                , activate^#(n__from(X)) -> c_6(from^#(activate(X)))
                , 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , cons(X1, X2) -> n__cons(X1, X2)
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}

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

         * Path {1}->{7}->{4}: YES(?,O(n^1))
           ---------------------------------

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons) = {1}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {1},
               Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(s^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {1}, Uargs(c_7) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [0] x2 + [2]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [1]
              activate(x1) = [3] x1 + [1]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              n__s(x1) = [1] x1 + [1]
              s(x1) = [1] x1 + [2]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              from^#(x1) = [3] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [0] x1 + [0]
              c_4() = [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1) = [1] x1 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {from^#(X) -> c_3()}
             Weak Rules:
               {  activate^#(n__from(X)) -> c_6(from^#(activate(X)))
                , 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , cons(X1, X2) -> n__cons(X1, X2)
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(n__s) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {1},
               Uargs(activate^#) = {}, Uargs(from^#) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [0]
              n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [1] x1 + [0]
              from(x1) = [1] x1 + [0]
              n__from(x1) = [1] x1 + [0]
              n__s(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [6] x1 + [7]
              c_0(x1) = [1] x1 + [3]
              activate^#(x1) = [4] x1 + [4]
              from^#(x1) = [2] x1 + [2]
              c_3() = [1]
              c_6(x1) = [2] x1 + [0]

         * Path {1}->{8}: YES(?,O(n^1))
           ----------------------------

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons) = {1}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {},
               Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(s^#) = {1},
               Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [0] x2 + [2]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [1]
              activate(x1) = [3] x1 + [1]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              n__s(x1) = [1] x1 + [1]
              s(x1) = [1] x1 + [2]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [3] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [1] x1 + [0]
              c_4() = [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1) = [0] x1 + [0]
              c_7(x1) = [1] x1 + [0]
              c_8() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {activate^#(n__s(X)) -> c_7(s^#(activate(X)))}
             Weak Rules:
               {  2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , cons(X1, X2) -> n__cons(X1, X2)
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(n__s) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {1},
               Uargs(activate^#) = {}, Uargs(s^#) = {}, Uargs(c_7) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [0] x2 + [4]
              n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [4] x1 + [5]
              from(x1) = [0] x1 + [4]
              n__from(x1) = [0] x1 + [0]
              n__s(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [3]
              2nd^#(x1) = [0] x1 + [7]
              c_0(x1) = [1] x1 + [3]
              activate^#(x1) = [0] x1 + [2]
              s^#(x1) = [0] x1 + [0]
              c_7(x1) = [4] x1 + [1]

         * Path {1}->{8}->{5}: YES(?,O(n^1))
           ---------------------------------

           The usable rules for this path are:

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

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons) = {1}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {},
               Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(s^#) = {1},
               Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [0] x2 + [2]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [1]
              activate(x1) = [3] x1 + [1]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              n__s(x1) = [1] x1 + [1]
              s(x1) = [1] x1 + [2]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [3] x1 + [0]
              c_4() = [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1) = [0] x1 + [0]
              c_7(x1) = [1] x1 + [0]
              c_8() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {s^#(X) -> c_4()}
             Weak Rules:
               {  activate^#(n__s(X)) -> c_7(s^#(activate(X)))
                , 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , cons(X1, X2) -> n__cons(X1, X2)
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(n__s) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {1},
               Uargs(activate^#) = {}, Uargs(s^#) = {}, Uargs(c_7) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [4] x1 + [4]
              from(x1) = [0] x1 + [1]
              n__from(x1) = [0] x1 + [1]
              n__s(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [7]
              c_0(x1) = [1] x1 + [3]
              activate^#(x1) = [0] x1 + [2]
              s^#(x1) = [0] x1 + [2]
              c_4() = [1]
              c_7(x1) = [1] x1 + [0]

         * Path {1}->{9}: 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(2nd) = {}, Uargs(cons) = {}, Uargs(n__cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {},
               Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(s^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              n__s(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_2() = [0]
              c_3() = [0]
              s^#(x1) = [0] x1 + [0]
              c_4() = [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1) = [0] x1 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8() = [0]

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {activate^#(X) -> c_8()}
             Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(2nd^#) = {},
               Uargs(c_0) = {1}, Uargs(activate^#) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [2] x2 + [0]
              n__cons(x1, x2) = [1] x1 + [0] x2 + [0]
              2nd^#(x1) = [2] x1 + [6]
              c_0(x1) = [2] x1 + [2]
              activate^#(x1) = [0] x1 + [2]
              c_8() = [1]

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex1 2 Luc02c FR

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex1 2 Luc02c FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  2nd(cons(X, n__cons(Y, Z))) -> activate(Y)
     , from(X) -> cons(X, n__from(n__s(X)))
     , cons(X1, X2) -> n__cons(X1, X2)
     , from(X) -> n__from(X)
     , s(X) -> n__s(X)
     , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
     , activate(n__from(X)) -> from(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(X) -> X}

Proof Output:
  None of the processors succeeded.

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

             {  1: 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
              , 2: from^#(X) -> c_1(cons^#(X, n__from(n__s(X))))
              , 3: cons^#(X1, X2) -> c_2(X1, X2)
              , 4: from^#(X) -> c_3(X)
              , 5: s^#(X) -> c_4(X)
              , 6: activate^#(n__cons(X1, X2)) -> c_5(cons^#(activate(X1), X2))
              , 7: activate^#(n__from(X)) -> c_6(from^#(activate(X)))
              , 8: activate^#(n__s(X)) -> c_7(s^#(activate(X)))
              , 9: activate^#(X) -> c_8(X)}

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

             ->{1}                                                       [     inherited      ]
                |
                |->{6}                                                   [     inherited      ]
                |   |
                |   `->{3}                                               [         NA         ]
                |
                |->{7}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [     inherited      ]
                |   |   |
                |   |   `->{3}                                           [         NA         ]
                |   |
                |   `->{4}                                               [         NA         ]
                |
                |->{8}                                                   [     inherited      ]
                |   |
                |   `->{5}                                               [       MAYBE        ]
                |
                `->{9}                                                   [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

             This path is subsumed by the proof of path {1}->{8}->{5}.

           * Path {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{3}.

           * Path {1}->{6}->{3}: NA
             ----------------------

             The usable rules for this path are:

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

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

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

           * Path {1}->{7}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{7}->{2}->{3}.

           * Path {1}->{7}->{2}: inherited
             -----------------------------

             This path is subsumed by the proof of path {1}->{7}->{2}->{3}.

           * Path {1}->{7}->{2}->{3}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

           * Path {1}->{7}->{4}: NA
             ----------------------

             The usable rules for this path are:

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

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

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

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

             This path is subsumed by the proof of path {1}->{8}->{5}.

           * Path {1}->{8}->{5}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  activate^#(n__s(X)) -> c_7(s^#(activate(X)))
                  , 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                  , s^#(X) -> c_4(X)
                  , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , cons(X1, X2) -> n__cons(X1, X2)
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)}

             Proof Output:
               The input cannot be shown compatible

           * Path {1}->{9}: 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(2nd) = {}, Uargs(cons) = {}, Uargs(n__cons) = {},
                 Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(s) = {}, Uargs(2nd^#) = {},
                 Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(s^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                2nd(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__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]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                n__from(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                n__s(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                2nd^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                activate^#(x1) = [3 3 3] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_1(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]
                c_2(x1, 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]
                s^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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_8(X)}
               Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(2nd^#) = {},
                 Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [2 0 0] x2 + [0]
                               [0 0 0]      [0 1 0]      [2]
                               [0 0 0]      [0 0 0]      [0]
                n__cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [2]
                                  [0 0 0]      [0 0 0]      [0]
                2nd^#(x1) = [2 0 0] x1 + [7]
                            [0 2 0]      [7]
                            [0 2 0]      [7]
                c_0(x1) = [1 0 0] x1 + [3]
                          [0 0 2]      [7]
                          [0 0 0]      [7]
                activate^#(x1) = [0 0 0] x1 + [4]
                                 [0 0 0]      [0]
                                 [0 0 0]      [2]
                c_8(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             {  1: 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
              , 2: from^#(X) -> c_1(cons^#(X, n__from(n__s(X))))
              , 3: cons^#(X1, X2) -> c_2(X1, X2)
              , 4: from^#(X) -> c_3(X)
              , 5: s^#(X) -> c_4(X)
              , 6: activate^#(n__cons(X1, X2)) -> c_5(cons^#(activate(X1), X2))
              , 7: activate^#(n__from(X)) -> c_6(from^#(activate(X)))
              , 8: activate^#(n__s(X)) -> c_7(s^#(activate(X)))
              , 9: activate^#(X) -> c_8(X)}

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

             ->{1}                                                       [     inherited      ]
                |
                |->{6}                                                   [     inherited      ]
                |   |
                |   `->{3}                                               [         NA         ]
                |
                |->{7}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [     inherited      ]
                |   |   |
                |   |   `->{3}                                           [         NA         ]
                |   |
                |   `->{4}                                               [       MAYBE        ]
                |
                |->{8}                                                   [     inherited      ]
                |   |
                |   `->{5}                                               [         NA         ]
                |
                `->{9}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

             This path is subsumed by the proof of path {1}->{8}->{5}.

           * Path {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{3}.

           * Path {1}->{6}->{3}: NA
             ----------------------

             The usable rules for this path are:

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

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

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

           * Path {1}->{7}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{7}->{2}->{3}.

           * Path {1}->{7}->{2}: inherited
             -----------------------------

             This path is subsumed by the proof of path {1}->{7}->{2}->{3}.

           * Path {1}->{7}->{2}->{3}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

           * Path {1}->{7}->{4}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  activate^#(n__from(X)) -> c_6(from^#(activate(X)))
                  , 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                  , from^#(X) -> c_3(X)
                  , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , cons(X1, X2) -> n__cons(X1, X2)
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)}

             Proof Output:
               The input cannot be shown compatible

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

             This path is subsumed by the proof of path {1}->{8}->{5}.

           * Path {1}->{8}->{5}: NA
             ----------------------

             The usable rules for this path are:

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

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

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

           * Path {1}->{9}: 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(2nd) = {}, Uargs(cons) = {}, Uargs(n__cons) = {},
                 Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(s) = {}, Uargs(2nd^#) = {},
                 Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(s^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                2nd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__from(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                2nd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_8(X)}
               Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(2nd^#) = {},
                 Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [2 0] x2 + [4]
                               [0 0]      [3 0]      [2]
                n__cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                2nd^#(x1) = [2 2] x1 + [3]
                            [1 2]      [6]
                c_0(x1) = [4 0] x1 + [3]
                          [2 2]      [3]
                activate^#(x1) = [0 0] x1 + [2]
                                 [0 0]      [2]
                c_8(x1) = [0 0] x1 + [1]
                          [0 0]      [0]

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

             {  1: 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
              , 2: from^#(X) -> c_1(cons^#(X, n__from(n__s(X))))
              , 3: cons^#(X1, X2) -> c_2(X1, X2)
              , 4: from^#(X) -> c_3(X)
              , 5: s^#(X) -> c_4(X)
              , 6: activate^#(n__cons(X1, X2)) -> c_5(cons^#(activate(X1), X2))
              , 7: activate^#(n__from(X)) -> c_6(from^#(activate(X)))
              , 8: activate^#(n__s(X)) -> c_7(s^#(activate(X)))
              , 9: activate^#(X) -> c_8(X)}

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

             ->{1}                                                       [     inherited      ]
                |
                |->{6}                                                   [     inherited      ]
                |   |
                |   `->{3}                                               [         NA         ]
                |
                |->{7}                                                   [     inherited      ]
                |   |
                |   |->{2}                                               [     inherited      ]
                |   |   |
                |   |   `->{3}                                           [         NA         ]
                |   |
                |   `->{4}                                               [       MAYBE        ]
                |
                |->{8}                                                   [     inherited      ]
                |   |
                |   `->{5}                                               [         NA         ]
                |
                `->{9}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

             This path is subsumed by the proof of path {1}->{8}->{5}.

           * Path {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{3}.

           * Path {1}->{6}->{3}: NA
             ----------------------

             The usable rules for this path are:

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

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

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

           * Path {1}->{7}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{7}->{2}->{3}.

           * Path {1}->{7}->{2}: inherited
             -----------------------------

             This path is subsumed by the proof of path {1}->{7}->{2}->{3}.

           * Path {1}->{7}->{2}->{3}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

           * Path {1}->{7}->{4}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  activate^#(n__from(X)) -> c_6(from^#(activate(X)))
                  , 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
                  , from^#(X) -> c_3(X)
                  , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , cons(X1, X2) -> n__cons(X1, X2)
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)}

             Proof Output:
               The input cannot be shown compatible

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

             This path is subsumed by the proof of path {1}->{8}->{5}.

           * Path {1}->{8}->{5}: NA
             ----------------------

             The usable rules for this path are:

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

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

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

           * Path {1}->{9}: 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(2nd) = {}, Uargs(cons) = {}, Uargs(n__cons) = {},
                 Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(s) = {}, Uargs(2nd^#) = {},
                 Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(cons^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(s^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                2nd(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                n__from(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                2nd^#(x1) = [0] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                activate^#(x1) = [3] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(x1) = [1] 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^#(X) -> c_8(X)}
               Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__cons) = {}, Uargs(2nd^#) = {},
                 Uargs(c_0) = {1}, Uargs(activate^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [2] x2 + [0]
                n__cons(x1, x2) = [1] x1 + [0] x2 + [0]
                2nd^#(x1) = [2] x1 + [7]
                c_0(x1) = [2] x1 + [3]
                activate^#(x1) = [0] x1 + [2]
                c_8(x1) = [0] x1 + [1]

    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.