Problem Transformed CSR 04 Ex1 Luc02b Z

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex1 Luc02b Z

stdout:

MAYBE

Problem:
from(X) -> cons(X,n__from(s(X)))
first(0(),Z) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
sel(0(),cons(X,Z)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
from(X) -> n__from(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__from(X)) -> from(X)
activate(n__first(X1,X2)) -> first(X1,X2)
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex1 Luc02b Z

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex1 Luc02b 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:
    {  from(X) -> cons(X, n__from(s(X)))
     , first(0(), Z) -> nil()
     , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
     , sel(0(), cons(X, Z)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
     , from(X) -> n__from(X)
     , first(X1, X2) -> n__first(X1, X2)
     , activate(n__from(X)) -> from(X)
     , activate(n__first(X1, X2)) -> first(X1, X2)
     , 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:
         {  from(X) -> cons(X, n__from(s(X)))
          , first(0(), Z) -> nil()
          , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
          , sel(0(), cons(X, Z)) -> X
          , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
          , from(X) -> n__from(X)
          , first(X1, X2) -> n__first(X1, X2)
          , activate(n__from(X)) -> from(X)
          , activate(n__first(X1, X2)) -> first(X1, X2)
          , activate(X) -> X}

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

           {  1: from^#(X) -> c_0()
            , 2: first^#(0(), Z) -> c_1()
            , 3: first^#(s(X), cons(Y, Z)) -> c_2(activate^#(Z))
            , 4: sel^#(0(), cons(X, Z)) -> c_3()
            , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
            , 6: from^#(X) -> c_5()
            , 7: first^#(X1, X2) -> c_6()
            , 8: activate^#(n__from(X)) -> c_7(from^#(X))
            , 9: activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))
            , 10: activate^#(X) -> c_9()}

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

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

           ->{3,9}                                                     [   YES(?,O(n^1))    ]
              |
              |->{2}                                                   [   YES(?,O(n^1))    ]
              |
              |->{7}                                                   [   YES(?,O(n^1))    ]
              |
              |->{8}                                                   [   YES(?,O(n^1))    ]
              |   |
              |   |->{1}                                               [   YES(?,O(n^1))    ]
              |   |
              |   `->{6}                                               [   YES(?,O(n^1))    ]
              |
              `->{10}                                                  [   YES(?,O(n^1))    ]



       Sub-problems:
       -------------
         * Path {3,9}: 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(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [1] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [3] x1 + [1] x2 + [0]
              c_1() = [0]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [3] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3() = [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9() = [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:
               {  first^#(s(X), cons(Y, Z)) -> c_2(activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}
             Weak Rules: {}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [1] x2 + [4]
              s(x1) = [0] x1 + [0]
              n__first(x1, x2) = [1] x1 + [1] x2 + [1]
              first^#(x1, x2) = [0] x1 + [1] x2 + [0]
              c_2(x1) = [1] x1 + [1]
              activate^#(x1) = [1] x1 + [1]
              c_8(x1) = [1] x1 + [1]

         * Path {3,9}->{2}: YES(?,O(n^1))
           ------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3() = [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9() = [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: {first^#(0(), Z) -> c_1()}
             Weak Rules:
               {  first^#(s(X), cons(Y, Z)) -> c_2(activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [1] x2 + [2]
              s(x1) = [0] x1 + [0]
              0() = [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              first^#(x1, x2) = [2] x1 + [4] x2 + [0]
              c_1() = [1]
              c_2(x1) = [1] x1 + [6]
              activate^#(x1) = [4] x1 + [0]
              c_8(x1) = [1] x1 + [0]

         * Path {3,9}->{7}: YES(?,O(n^1))
           ------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3() = [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9() = [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: {first^#(X1, X2) -> c_6()}
             Weak Rules:
               {  first^#(s(X), cons(Y, Z)) -> c_2(activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [1] x2 + [0]
              s(x1) = [0] x1 + [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [2]
              first^#(x1, x2) = [2] x1 + [4] x2 + [4]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [4] x1 + [0]
              c_6() = [1]
              c_8(x1) = [1] x1 + [2]

         * Path {3,9}->{8}: 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(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [1] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [3] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3() = [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [3] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9() = [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_7(from^#(X))}
             Weak Rules:
               {  first^#(s(X), cons(Y, Z)) -> c_2(activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(n__first) = {}, Uargs(from^#) = {}, Uargs(first^#) = {},
               Uargs(c_2) = {1}, Uargs(activate^#) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [1] x2 + [0]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [0] x1 + [0]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              from^#(x1) = [7] x1 + [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [1]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [1]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]

         * Path {3,9}->{8}->{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(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(c_7) = {1},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3() = [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [1] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9() = [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_0()}
             Weak Rules:
               {  activate^#(n__from(X)) -> c_7(from^#(X))
                , first^#(s(X), cons(Y, Z)) -> c_2(activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(n__first) = {}, Uargs(from^#) = {}, Uargs(first^#) = {},
               Uargs(c_2) = {1}, Uargs(activate^#) = {}, Uargs(c_7) = {1},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [1] x2 + [2]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [0] x1 + [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [4]
              from^#(x1) = [1] x1 + [2]
              c_0() = [1]
              first^#(x1, x2) = [2] x1 + [2] x2 + [0]
              c_2(x1) = [1] x1 + [3]
              activate^#(x1) = [2] x1 + [4]
              c_7(x1) = [2] x1 + [0]
              c_8(x1) = [1] x1 + [2]

         * Path {3,9}->{8}->{6}: YES(?,O(n^1))
           -----------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(c_7) = {1},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3() = [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [1] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9() = [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_5()}
             Weak Rules:
               {  activate^#(n__from(X)) -> c_7(from^#(X))
                , first^#(s(X), cons(Y, Z)) -> c_2(activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(n__first) = {}, Uargs(from^#) = {}, Uargs(first^#) = {},
               Uargs(c_2) = {1}, Uargs(activate^#) = {}, Uargs(c_7) = {1},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [1] x2 + [2]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [0] x1 + [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [4]
              from^#(x1) = [1] x1 + [2]
              first^#(x1, x2) = [2] x1 + [2] x2 + [0]
              c_2(x1) = [1] x1 + [3]
              activate^#(x1) = [2] x1 + [4]
              c_5() = [1]
              c_7(x1) = [2] x1 + [0]
              c_8(x1) = [1] x1 + [2]

         * Path {3,9}->{10}: 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(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(sel^#) = {}, Uargs(c_4) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3() = [0]
              c_4(x1) = [0] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9() = [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_9()}
             Weak Rules:
               {  first^#(s(X), cons(Y, Z)) -> c_2(activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {1}, Uargs(activate^#) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [0] x1 + [1] x2 + [2]
              s(x1) = [0] x1 + [0]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              first^#(x1, x2) = [0] x1 + [2] x2 + [0]
              c_2(x1) = [1] x1 + [0]
              activate^#(x1) = [2] x1 + [2]
              c_8(x1) = [1] x1 + [2]
              c_9() = [1]

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

           The usable rules for this path are:

             {  activate(n__from(X)) -> from(X)
              , activate(n__first(X1, X2)) -> first(X1, X2)
              , activate(X) -> X
              , from(X) -> cons(X, n__from(s(X)))
              , first(0(), Z) -> nil()
              , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
              , from(X) -> n__from(X)
              , first(X1, X2) -> n__first(X1, X2)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {2},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {}, Uargs(activate^#) = {},
               Uargs(sel^#) = {2}, Uargs(c_4) = {1}, Uargs(c_7) = {},
               Uargs(c_8) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [1]
              cons(x1, x2) = [0] x1 + [1] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [1] x1 + [2]
              first(x1, x2) = [1] x1 + [1] x2 + [3]
              0() = [3]
              nil() = [1]
              n__first(x1, x2) = [1] x1 + [1] x2 + [1]
              activate(x1) = [1] x1 + [3]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1) = [0] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
              c_3() = [0]
              c_4(x1) = [1] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [0] x1 + [0]
              c_9() = [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:
               {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))}
             Weak Rules:
               {  activate(n__from(X)) -> from(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(s(X)))
                , first(0(), Z) -> nil()
                , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
                , from(X) -> n__from(X)
                , first(X1, X2) -> n__first(X1, X2)}

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

         * Path {5}->{4}: YES(?,O(n^1))
           ----------------------------

           The usable rules for this path are:

             {  activate(n__from(X)) -> from(X)
              , activate(n__first(X1, X2)) -> first(X1, X2)
              , activate(X) -> X
              , from(X) -> cons(X, n__from(s(X)))
              , first(0(), Z) -> nil()
              , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
              , from(X) -> n__from(X)
              , first(X1, X2) -> n__first(X1, X2)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {2},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {}, Uargs(activate^#) = {},
               Uargs(sel^#) = {2}, Uargs(c_4) = {1}, Uargs(c_7) = {},
               Uargs(c_8) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [3]
              cons(x1, x2) = [0] x1 + [1] x2 + [0]
              n__from(x1) = [0] x1 + [1]
              s(x1) = [1] x1 + [2]
              first(x1, x2) = [1] x1 + [1] x2 + [2]
              0() = [3]
              nil() = [0]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [1] x1 + [3]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0() = [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1) = [0] x1 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [3] x2 + [0]
              c_3() = [0]
              c_4(x1) = [1] x1 + [0]
              c_5() = [0]
              c_6() = [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [0] x1 + [0]
              c_9() = [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: {sel^#(0(), cons(X, Z)) -> c_3()}
             Weak Rules:
               {  sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
                , activate(n__from(X)) -> from(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(s(X)))
                , first(0(), Z) -> nil()
                , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
                , from(X) -> n__from(X)
                , first(X1, X2) -> n__first(X1, X2)}

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex1 Luc02b Z

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex1 Luc02b 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:
    {  from(X) -> cons(X, n__from(s(X)))
     , first(0(), Z) -> nil()
     , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
     , sel(0(), cons(X, Z)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
     , from(X) -> n__from(X)
     , first(X1, X2) -> n__first(X1, X2)
     , activate(n__from(X)) -> from(X)
     , activate(n__first(X1, X2)) -> first(X1, X2)
     , 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:
         {  from(X) -> cons(X, n__from(s(X)))
          , first(0(), Z) -> nil()
          , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
          , sel(0(), cons(X, Z)) -> X
          , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
          , from(X) -> n__from(X)
          , first(X1, X2) -> n__first(X1, X2)
          , activate(n__from(X)) -> from(X)
          , activate(n__first(X1, X2)) -> first(X1, X2)
          , activate(X) -> X}

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

           {  1: from^#(X) -> c_0(X, X)
            , 2: first^#(0(), Z) -> c_1()
            , 3: first^#(s(X), cons(Y, Z)) -> c_2(Y, X, activate^#(Z))
            , 4: sel^#(0(), cons(X, Z)) -> c_3(X)
            , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
            , 6: from^#(X) -> c_5(X)
            , 7: first^#(X1, X2) -> c_6(X1, X2)
            , 8: activate^#(n__from(X)) -> c_7(from^#(X))
            , 9: activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))
            , 10: activate^#(X) -> c_9(X)}

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

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

           ->{3,9}                                                     [   YES(?,O(n^1))    ]
              |
              |->{2}                                                   [   YES(?,O(n^1))    ]
              |
              |->{7}                                                   [   YES(?,O(n^1))    ]
              |
              |->{8}                                                   [   YES(?,O(n^1))    ]
              |   |
              |   |->{1}                                               [   YES(?,O(n^1))    ]
              |   |
              |   `->{6}                                               [   YES(?,O(n^1))    ]
              |
              `->{10}                                                  [   YES(?,O(n^1))    ]



       Sub-problems:
       -------------
         * Path {3,9}: 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(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {3},
               Uargs(activate^#) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
               Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [1] x1 + [1] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [1] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0(x1, x2) = [0] x1 + [0] x2 + [0]
              first^#(x1, x2) = [1] x1 + [3] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
              activate^#(x1) = [3] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9(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:
               {  first^#(s(X), cons(Y, Z)) -> c_2(Y, X, activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}
             Weak Rules: {}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {3}, Uargs(activate^#) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [0]
              s(x1) = [1] x1 + [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [2]
              first^#(x1, x2) = [2] x1 + [4] x2 + [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [3]
              activate^#(x1) = [4] x1 + [0]
              c_8(x1) = [1] x1 + [7]

         * Path {3,9}->{2}: YES(?,O(n^1))
           ------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {3},
               Uargs(activate^#) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
               Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0(x1, x2) = [0] x1 + [0] x2 + [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9(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: {first^#(0(), Z) -> c_1()}
             Weak Rules:
               {  first^#(s(X), cons(Y, Z)) -> c_2(Y, X, activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {3}, Uargs(activate^#) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [2]
              s(x1) = [1] x1 + [2]
              0() = [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [4]
              first^#(x1, x2) = [2] x1 + [2] x2 + [4]
              c_1() = [1]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [7]
              activate^#(x1) = [2] x1 + [0]
              c_8(x1) = [1] x1 + [3]

         * Path {3,9}->{7}: YES(?,O(n^1))
           ------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {3},
               Uargs(activate^#) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
               Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0(x1, x2) = [0] x1 + [0] x2 + [0]
              first^#(x1, x2) = [3] x1 + [3] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1, x2) = [1] x1 + [1] x2 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9(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: {first^#(X1, X2) -> c_6(X1, X2)}
             Weak Rules:
               {  first^#(s(X), cons(Y, Z)) -> c_2(Y, X, activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {3}, Uargs(activate^#) = {},
               Uargs(c_6) = {}, Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [2]
              s(x1) = [1] x1 + [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [4]
              first^#(x1, x2) = [2] x1 + [2] x2 + [4]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [2] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [1]
              c_8(x1) = [1] x1 + [2]

         * Path {3,9}->{8}: 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(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {3},
               Uargs(activate^#) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
               Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [1] x1 + [0]
              c_0(x1, x2) = [0] x1 + [0] x2 + [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [3] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [0]
              c_7(x1) = [3] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9(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_7(from^#(X))}
             Weak Rules:
               {  first^#(s(X), cons(Y, Z)) -> c_2(Y, X, activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(n__first) = {}, Uargs(from^#) = {}, Uargs(first^#) = {},
               Uargs(c_2) = {3}, Uargs(activate^#) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [2]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [1] x1 + [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [1]
              from^#(x1) = [1] x1 + [2]
              first^#(x1, x2) = [2] x1 + [2] x2 + [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [2] x1 + [6]
              c_7(x1) = [2] x1 + [1]
              c_8(x1) = [1] x1 + [7]

         * Path {3,9}->{8}->{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(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {3},
               Uargs(activate^#) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
               Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
               Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [3] x1 + [0]
              c_0(x1, x2) = [2] x1 + [1] x2 + [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [0]
              c_7(x1) = [1] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9(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_0(X, X)}
             Weak Rules:
               {  activate^#(n__from(X)) -> c_7(from^#(X))
                , first^#(s(X), cons(Y, Z)) -> c_2(Y, X, activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(n__first) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {3}, Uargs(activate^#) = {},
               Uargs(c_7) = {1}, Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [0]
              n__from(x1) = [1] x1 + [0]
              s(x1) = [1] x1 + [0]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              from^#(x1) = [0] x1 + [2]
              c_0(x1, x2) = [0] x1 + [0] x2 + [1]
              first^#(x1, x2) = [0] x1 + [0] x2 + [4]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [0] x1 + [4]
              c_7(x1) = [2] x1 + [0]
              c_8(x1) = [1] x1 + [0]

         * Path {3,9}->{8}->{6}: YES(?,O(n^1))
           -----------------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {3},
               Uargs(activate^#) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
               Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
               Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [3] x1 + [0]
              c_0(x1, x2) = [0] x1 + [0] x2 + [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [1] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [0]
              c_7(x1) = [1] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9(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_5(X)}
             Weak Rules:
               {  activate^#(n__from(X)) -> c_7(from^#(X))
                , first^#(s(X), cons(Y, Z)) -> c_2(Y, X, activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(n__from) = {}, Uargs(s) = {},
               Uargs(n__first) = {}, Uargs(from^#) = {}, Uargs(first^#) = {},
               Uargs(c_2) = {3}, Uargs(activate^#) = {}, Uargs(c_5) = {},
               Uargs(c_7) = {1}, Uargs(c_8) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [3]
              n__from(x1) = [1] x1 + [2]
              s(x1) = [1] x1 + [1]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              from^#(x1) = [0] x1 + [1]
              first^#(x1, x2) = [2] x1 + [2] x2 + [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [3]
              activate^#(x1) = [2] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_7(x1) = [4] x1 + [0]
              c_8(x1) = [1] x1 + [0]

         * Path {3,9}->{10}: 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(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {3},
               Uargs(activate^#) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
               Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
               Uargs(c_8) = {1}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [0]
              cons(x1, x2) = [0] x1 + [0] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              first(x1, x2) = [0] x1 + [0] x2 + [0]
              0() = [0]
              nil() = [0]
              n__first(x1, x2) = [0] x1 + [0] x2 + [0]
              activate(x1) = [0] x1 + [0]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0(x1, x2) = [0] x1 + [0] x2 + [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
              activate^#(x1) = [3] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [0] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [1] x1 + [0]
              c_9(x1) = [1] x1 + [0]

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

           'matrix-interpretation of dimension 1'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    DP runtime-complexity with respect to
             Strict Rules: {activate^#(X) -> c_9(X)}
             Weak Rules:
               {  first^#(s(X), cons(Y, Z)) -> c_2(Y, X, activate^#(Z))
                , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2))}

           Proof Output:
             The following argument positions are usable:
               Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
               Uargs(first^#) = {}, Uargs(c_2) = {3}, Uargs(activate^#) = {},
               Uargs(c_8) = {1}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              cons(x1, x2) = [1] x1 + [1] x2 + [2]
              s(x1) = [1] x1 + [2]
              n__first(x1, x2) = [1] x1 + [1] x2 + [2]
              first^#(x1, x2) = [2] x1 + [2] x2 + [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [3]
              activate^#(x1) = [2] x1 + [4]
              c_8(x1) = [1] x1 + [6]
              c_9(x1) = [0] x1 + [1]

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

           The usable rules for this path are:

             {  activate(n__from(X)) -> from(X)
              , activate(n__first(X1, X2)) -> first(X1, X2)
              , activate(X) -> X
              , from(X) -> cons(X, n__from(s(X)))
              , first(0(), Z) -> nil()
              , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
              , from(X) -> n__from(X)
              , first(X1, X2) -> n__first(X1, X2)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {2}, Uargs(n__first) = {2},
               Uargs(activate) = {1}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {},
               Uargs(activate^#) = {}, Uargs(sel^#) = {2}, Uargs(c_3) = {},
               Uargs(c_4) = {1}, Uargs(c_5) = {}, Uargs(c_6) = {},
               Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [1]
              cons(x1, x2) = [0] x1 + [1] x2 + [0]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [1] x1 + [2]
              first(x1, x2) = [1] x1 + [1] x2 + [1]
              0() = [3]
              nil() = [1]
              n__first(x1, x2) = [1] x1 + [1] x2 + [0]
              activate(x1) = [1] x1 + [2]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0(x1, x2) = [0] x1 + [0] x2 + [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [1] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [0] x1 + [0]
              c_9(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:
               {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))}
             Weak Rules:
               {  activate(n__from(X)) -> from(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(s(X)))
                , first(0(), Z) -> nil()
                , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
                , from(X) -> n__from(X)
                , first(X1, X2) -> n__first(X1, X2)}

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

         * Path {5}->{4}: YES(?,O(n^1))
           ----------------------------

           The usable rules for this path are:

             {  activate(n__from(X)) -> from(X)
              , activate(n__first(X1, X2)) -> first(X1, X2)
              , activate(X) -> X
              , from(X) -> cons(X, n__from(s(X)))
              , first(0(), Z) -> nil()
              , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
              , from(X) -> n__from(X)
              , first(X1, X2) -> n__first(X1, X2)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {2}, Uargs(n__first) = {2},
               Uargs(activate) = {1}, Uargs(sel) = {}, Uargs(from^#) = {},
               Uargs(c_0) = {}, Uargs(first^#) = {}, Uargs(c_2) = {},
               Uargs(activate^#) = {}, Uargs(sel^#) = {2}, Uargs(c_3) = {},
               Uargs(c_4) = {1}, Uargs(c_5) = {}, Uargs(c_6) = {},
               Uargs(c_7) = {}, Uargs(c_8) = {}, Uargs(c_9) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [3]
              cons(x1, x2) = [0] x1 + [1] x2 + [0]
              n__from(x1) = [0] x1 + [2]
              s(x1) = [1] x1 + [1]
              first(x1, x2) = [2] x1 + [2] x2 + [3]
              0() = [3]
              nil() = [0]
              n__first(x1, x2) = [1] x1 + [1] x2 + [2]
              activate(x1) = [2] x1 + [1]
              sel(x1, x2) = [0] x1 + [0] x2 + [0]
              from^#(x1) = [0] x1 + [0]
              c_0(x1, x2) = [0] x1 + [0] x2 + [0]
              first^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_1() = [0]
              c_2(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
              activate^#(x1) = [0] x1 + [0]
              sel^#(x1, x2) = [0] x1 + [3] x2 + [0]
              c_3(x1) = [0] x1 + [0]
              c_4(x1) = [1] x1 + [0]
              c_5(x1) = [0] x1 + [0]
              c_6(x1, x2) = [0] x1 + [0] x2 + [0]
              c_7(x1) = [0] x1 + [0]
              c_8(x1) = [0] x1 + [0]
              c_9(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: {sel^#(0(), cons(X, Z)) -> c_3(X)}
             Weak Rules:
               {  sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, activate(Z)))
                , activate(n__from(X)) -> from(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , from(X) -> cons(X, n__from(s(X)))
                , first(0(), Z) -> nil()
                , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
                , from(X) -> n__from(X)
                , first(X1, X2) -> n__first(X1, X2)}

           Proof Output:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {},
               Uargs(activate) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
               Uargs(c_4) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0] x1 + [2]
              cons(x1, x2) = [0] x1 + [1] x2 + [2]
              n__from(x1) = [0] x1 + [0]
              s(x1) = [1] x1 + [2]
              first(x1, x2) = [1] x1 + [0] x2 + [0]
              0() = [1]
              nil() = [1]
              n__first(x1, x2) = [1] x1 + [0] x2 + [0]
              activate(x1) = [1] x1 + [2]
              sel^#(x1, x2) = [4] x1 + [2] x2 + [0]
              c_3(x1) = [0] x1 + [1]
              c_4(x1) = [1] x1 + [7]