Problem Transformed CSR 04 Ex6 9 Luc02c Z

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex6 9 Luc02c Z

stdout:

MAYBE

Problem:
2nd(cons1(X,cons(Y,Z))) -> Y
2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
activate(n__from(X)) -> from(X)
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex6 9 Luc02c Z

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex6 9 Luc02c Z

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(cons1(X, cons(Y, Z))) -> Y
     , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
     , from(X) -> cons(X, n__from(s(X)))
     , from(X) -> n__from(X)
     , activate(n__from(X)) -> from(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(cons1(X, cons(Y, Z))) -> Y
          , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
          , from(X) -> cons(X, n__from(s(X)))
          , from(X) -> n__from(X)
          , activate(n__from(X)) -> from(X)
          , activate(X) -> X}

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

           {  1: 2nd^#(cons1(X, cons(Y, Z))) -> c_0()
            , 2: 2nd^#(cons(X, X1)) -> c_1(2nd^#(cons1(X, activate(X1))))
            , 3: from^#(X) -> c_2()
            , 4: from^#(X) -> c_3()
            , 5: activate^#(n__from(X)) -> c_4(from^#(X))
            , 6: activate^#(X) -> c_5()}

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

           ->{6}                                                       [    YES(?,O(1))     ]

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

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



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

           The usable rules for this path are:

             {  activate(n__from(X)) -> from(X)
              , activate(X) -> X
              , from(X) -> cons(X, n__from(s(X)))
              , from(X) -> n__from(X)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons1) = {2}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {1}, Uargs(c_1) = {1},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [1] x2 + [0]
              cons(x1, x2) = [0] x1 + [1] x2 + [0]
              activate(x1) = [1] x1 + [2]
              from(x1) = [1] x1 + [2]
              n__from(x1) = [1] x1 + [1]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [3] x1 + [0]
              c_0() = [0]
              c_1(x1) = [1] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_2() = [0]
              c_3() = [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [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:
               {2nd^#(cons(X, X1)) -> c_1(2nd^#(cons1(X, activate(X1))))}
             Weak Rules:
               {  activate(n__from(X)) -> from(X)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(s(X)))
                , from(X) -> n__from(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons1) = {}, Uargs(cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(2nd^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [4]
              activate(x1) = [4] x1 + [0]
              from(x1) = [3] x1 + [5]
              n__from(x1) = [1] x1 + [2]
              s(x1) = [1] x1 + [7]
              2nd^#(x1) = [2] x1 + [0]
              c_1(x1) = [2] x1 + [7]

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

           The usable rules for this path are:

             {  activate(n__from(X)) -> from(X)
              , activate(X) -> X
              , from(X) -> cons(X, n__from(s(X)))
              , from(X) -> n__from(X)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons1) = {2}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {1}, Uargs(c_1) = {1},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [1] x2 + [0]
              cons(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [3] x1 + [3]
              from(x1) = [3] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [3] x1 + [0]
              c_0() = [0]
              c_1(x1) = [1] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_2() = [0]
              c_3() = [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [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: {2nd^#(cons1(X, cons(Y, Z))) -> c_0()}
             Weak Rules:
               {  2nd^#(cons(X, X1)) -> c_1(2nd^#(cons1(X, activate(X1))))
                , activate(n__from(X)) -> from(X)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(s(X)))
                , from(X) -> n__from(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons1) = {}, Uargs(cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(2nd^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons1(x1, x2) = [0] x1 + [0] x2 + [2]
              cons(x1, x2) = [1] x1 + [0] x2 + [4]
              activate(x1) = [3] x1 + [2]
              from(x1) = [3] x1 + [7]
              n__from(x1) = [1] x1 + [2]
              s(x1) = [1] x1 + [7]
              2nd^#(x1) = [2] x1 + [0]
              c_0() = [1]
              c_1(x1) = [1] x1 + [4]

         * Path {5}: 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(cons1) = {}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_1) = {},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0() = [0]
              c_1(x1) = [0] x1 + [0]
              from^#(x1) = [1] x1 + [0]
              c_2() = [0]
              c_3() = [0]
              activate^#(x1) = [3] x1 + [0]
              c_4(x1) = [3] x1 + [0]
              c_5() = [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^#(n__from(X)) -> c_4(from^#(X))}
             Weak Rules: {}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__from) = {}, Uargs(from^#) = {}, Uargs(activate^#) = {},
               Uargs(c_4) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__from(x1) = [1] x1 + [0]
              from^#(x1) = [7] x1 + [0]
              activate^#(x1) = [0] x1 + [7]
              c_4(x1) = [0] x1 + [5]

         * Path {5}->{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(2nd) = {}, Uargs(cons1) = {}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_1) = {},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0() = [0]
              c_1(x1) = [0] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_2() = [0]
              c_3() = [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [1] x1 + [0]
              c_5() = [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: {from^#(X) -> c_2()}
             Weak Rules: {activate^#(n__from(X)) -> c_4(from^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__from) = {}, Uargs(from^#) = {}, Uargs(activate^#) = {},
               Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__from(x1) = [1] x1 + [2]
              from^#(x1) = [0] x1 + [2]
              c_2() = [1]
              activate^#(x1) = [2] x1 + [6]
              c_4(x1) = [4] x1 + [2]

         * Path {5}->{4}: 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(cons1) = {}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_1) = {},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0() = [0]
              c_1(x1) = [0] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_2() = [0]
              c_3() = [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [1] x1 + [0]
              c_5() = [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: {from^#(X) -> c_3()}
             Weak Rules: {activate^#(n__from(X)) -> c_4(from^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__from) = {}, Uargs(from^#) = {}, Uargs(activate^#) = {},
               Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__from(x1) = [1] x1 + [2]
              from^#(x1) = [0] x1 + [2]
              c_3() = [1]
              activate^#(x1) = [2] x1 + [6]
              c_4(x1) = [4] x1 + [2]

         * Path {6}: 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(cons1) = {}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_1) = {},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0() = [0]
              c_1(x1) = [0] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_2() = [0]
              c_3() = [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [0]

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex6 9 Luc02c Z

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex6 9 Luc02c Z

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    runtime-complexity with respect to
  Rules:
    {  2nd(cons1(X, cons(Y, Z))) -> Y
     , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
     , from(X) -> cons(X, n__from(s(X)))
     , from(X) -> n__from(X)
     , activate(n__from(X)) -> from(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:    runtime-complexity with respect to
       Rules:
         {  2nd(cons1(X, cons(Y, Z))) -> Y
          , 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
          , from(X) -> cons(X, n__from(s(X)))
          , from(X) -> n__from(X)
          , activate(n__from(X)) -> from(X)
          , activate(X) -> X}

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

           {  1: 2nd^#(cons1(X, cons(Y, Z))) -> c_0(Y)
            , 2: 2nd^#(cons(X, X1)) -> c_1(2nd^#(cons1(X, activate(X1))))
            , 3: from^#(X) -> c_2(X, X)
            , 4: from^#(X) -> c_3(X)
            , 5: activate^#(n__from(X)) -> c_4(from^#(X))
            , 6: activate^#(X) -> c_5(X)}

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

           ->{6}                                                       [    YES(?,O(1))     ]

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

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



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

           The usable rules for this path are:

             {  activate(n__from(X)) -> from(X)
              , activate(X) -> X
              , from(X) -> cons(X, n__from(s(X)))
              , from(X) -> n__from(X)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons1) = {2}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {1}, Uargs(c_0) = {},
               Uargs(c_1) = {1}, Uargs(from^#) = {}, Uargs(c_2) = {},
               Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
               Uargs(c_5) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [1] x2 + [0]
              cons(x1, x2) = [0] x1 + [1] x2 + [0]
              activate(x1) = [1] x1 + [2]
              from(x1) = [1] x1 + [2]
              n__from(x1) = [1] x1 + [1]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [3] x1 + [0]
              c_0(x1) = [0] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_2(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [0] x1 + [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:    DP runtime-complexity with respect to
             Strict Rules:
               {2nd^#(cons(X, X1)) -> c_1(2nd^#(cons1(X, activate(X1))))}
             Weak Rules:
               {  activate(n__from(X)) -> from(X)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(s(X)))
                , from(X) -> n__from(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons1) = {}, Uargs(cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(2nd^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [4]
              activate(x1) = [4] x1 + [0]
              from(x1) = [3] x1 + [5]
              n__from(x1) = [1] x1 + [2]
              s(x1) = [1] x1 + [7]
              2nd^#(x1) = [2] x1 + [0]
              c_1(x1) = [2] x1 + [7]

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

           The usable rules for this path are:

             {  activate(n__from(X)) -> from(X)
              , activate(X) -> X
              , from(X) -> cons(X, n__from(s(X)))
              , from(X) -> n__from(X)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(2nd) = {}, Uargs(cons1) = {2}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {1}, Uargs(c_0) = {},
               Uargs(c_1) = {1}, Uargs(from^#) = {}, Uargs(c_2) = {},
               Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
               Uargs(c_5) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [1] x1 + [1] x2 + [0]
              cons(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [3] x1 + [3]
              from(x1) = [1] x1 + [3]
              n__from(x1) = [1] x1 + [2]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [3] x1 + [0]
              c_0(x1) = [1] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_2(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [0] x1 + [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:    DP runtime-complexity with respect to
             Strict Rules: {2nd^#(cons1(X, cons(Y, Z))) -> c_0(Y)}
             Weak Rules:
               {  2nd^#(cons(X, X1)) -> c_1(2nd^#(cons1(X, activate(X1))))
                , activate(n__from(X)) -> from(X)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(s(X)))
                , from(X) -> n__from(X)}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons1) = {}, Uargs(cons) = {}, Uargs(activate) = {},
               Uargs(from) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(2nd^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons1(x1, x2) = [0] x1 + [0] x2 + [2]
              cons(x1, x2) = [1] x1 + [0] x2 + [4]
              activate(x1) = [1] x1 + [2]
              from(x1) = [1] x1 + [5]
              n__from(x1) = [1] x1 + [4]
              s(x1) = [1] x1 + [7]
              2nd^#(x1) = [2] x1 + [4]
              c_0(x1) = [0] x1 + [1]
              c_1(x1) = [1] x1 + [2]

         * Path {5}: 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(cons1) = {}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
               Uargs(from^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
               Uargs(activate^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              from^#(x1) = [1] x1 + [0]
              c_2(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              activate^#(x1) = [3] x1 + [0]
              c_4(x1) = [3] x1 + [0]
              c_5(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__from(X)) -> c_4(from^#(X))}
             Weak Rules: {}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__from) = {}, Uargs(from^#) = {}, Uargs(activate^#) = {},
               Uargs(c_4) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__from(x1) = [1] x1 + [0]
              from^#(x1) = [7] x1 + [0]
              activate^#(x1) = [0] x1 + [7]
              c_4(x1) = [0] x1 + [5]

         * Path {5}->{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(2nd) = {}, Uargs(cons1) = {}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
               Uargs(from^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
               Uargs(activate^#) = {}, Uargs(c_4) = {1}, Uargs(c_5) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              from^#(x1) = [3] x1 + [0]
              c_2(x1, x2) = [2] x1 + [1] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [1] x1 + [0]
              c_5(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: {from^#(X) -> c_2(X, X)}
             Weak Rules: {activate^#(n__from(X)) -> c_4(from^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__from) = {}, Uargs(from^#) = {}, Uargs(c_2) = {},
               Uargs(activate^#) = {}, Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__from(x1) = [1] x1 + [0]
              from^#(x1) = [0] x1 + [2]
              c_2(x1, x2) = [0] x1 + [0] x2 + [1]
              activate^#(x1) = [0] x1 + [7]
              c_4(x1) = [2] x1 + [3]

         * Path {5}->{4}: 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(cons1) = {}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
               Uargs(from^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
               Uargs(activate^#) = {}, Uargs(c_4) = {1}, Uargs(c_5) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              from^#(x1) = [3] x1 + [0]
              c_2(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              c_4(x1) = [1] x1 + [0]
              c_5(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: {from^#(X) -> c_3(X)}
             Weak Rules: {activate^#(n__from(X)) -> c_4(from^#(X))}

           Proof Output:
             The following argument positions are usable:
               Uargs(n__from) = {}, Uargs(from^#) = {}, Uargs(c_3) = {},
               Uargs(activate^#) = {}, Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              n__from(x1) = [1] x1 + [2]
              from^#(x1) = [1] x1 + [2]
              c_3(x1) = [1] x1 + [1]
              activate^#(x1) = [2] x1 + [7]
              c_4(x1) = [2] x1 + [7]

         * Path {6}: 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(cons1) = {}, Uargs(cons) = {},
               Uargs(activate) = {}, Uargs(from) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(2nd^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
               Uargs(from^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
               Uargs(activate^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              2nd(x1) = [0] x1 + [0]
              cons1(x1, x2) = [0] x1 + [0] x2 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              from(x1) = [0] x1 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              2nd^#(x1) = [0] x1 + [0]
              c_0(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              from^#(x1) = [0] x1 + [0]
              c_2(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              activate^#(x1) = [3] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(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_5(X)}
             Weak Rules: {}

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