Problem Transformed CSR 04 Ex4 4 Luc96b GM

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 4 Luc96b GM

stdout:

MAYBE

Problem:
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
a__f(X1,X2) -> f(X1,X2)

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 4 Luc96b GM

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^2))
Input	Transformed CSR 04 Ex4 4 Luc96b GM

stdout:

YES(?,O(n^2))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^2))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
     , mark(f(X1, X2)) -> a__f(mark(X1), X2)
     , mark(g(X)) -> g(mark(X))
     , a__f(X1, X2) -> f(X1, X2)}

Proof Output:
  'wdg' proved the best result:

  Details:
  --------
    'wdg' succeeded with the following output:
     'wdg'
     -----
     Answer:           YES(?,O(n^2))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
          , mark(f(X1, X2)) -> a__f(mark(X1), X2)
          , mark(g(X)) -> g(mark(X))
          , a__f(X1, X2) -> f(X1, X2)}

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

           {  1: a__f^#(g(X), Y) -> c_0(a__f^#(mark(X), f(g(X), Y)))
            , 2: mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
            , 3: mark^#(g(X)) -> c_2(mark^#(X))
            , 4: a__f^#(X1, X2) -> c_3()}

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

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



       Sub-problems:
       -------------
         * Path {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(a__f) = {}, Uargs(g) = {}, Uargs(mark) = {}, Uargs(f) = {},
               Uargs(a__f^#) = {}, Uargs(c_0) = {}, Uargs(mark^#) = {},
               Uargs(c_1) = {}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              a__f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
              g(x1) = [1 2] x1 + [0]
                      [0 0]      [0]
              mark(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
              f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                          [0 0]      [0 0]      [0]
              a__f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
              c_0(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              mark^#(x1) = [3 3] x1 + [0]
                           [3 3]      [0]
              c_1(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_3() = [0]
                      [0]

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

           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {mark^#(g(X)) -> c_2(mark^#(X))}
             Weak Rules: {}

           Proof Output:
             The following argument positions are usable:
               Uargs(g) = {}, Uargs(mark^#) = {}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              g(x1) = [1 0] x1 + [0]
                      [0 1]      [1]
              mark^#(x1) = [0 1] x1 + [1]
                           [0 0]      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 0]      [0]

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

           The usable rules for this path are:

             {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
              , mark(g(X)) -> g(mark(X))
              , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
              , a__f(X1, X2) -> f(X1, X2)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(a__f) = {1}, Uargs(g) = {1}, Uargs(mark) = {}, Uargs(f) = {},
               Uargs(a__f^#) = {1}, Uargs(c_0) = {}, Uargs(mark^#) = {},
               Uargs(c_1) = {1}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              a__f(x1, x2) = [1 2] x1 + [0 0] x2 + [1]
                             [0 1]      [0 0]      [3]
              g(x1) = [1 3] x1 + [0]
                      [0 1]      [2]
              mark(x1) = [1 3] x1 + [0]
                         [0 1]      [0]
              f(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
                          [0 1]      [0 0]      [3]
              a__f^#(x1, x2) = [2 2] x1 + [0 0] x2 + [0]
                               [3 3]      [3 3]      [0]
              c_0(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              mark^#(x1) = [3 3] x1 + [0]
                           [0 0]      [0]
              c_1(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_3() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))}
             Weak Rules:
               {  mark^#(g(X)) -> c_2(mark^#(X))
                , mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

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

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

           The usable rules for this path are:

             {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
              , mark(g(X)) -> g(mark(X))
              , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
              , a__f(X1, X2) -> f(X1, X2)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(a__f) = {1}, Uargs(g) = {1}, Uargs(mark) = {}, Uargs(f) = {},
               Uargs(a__f^#) = {1}, Uargs(c_0) = {1}, Uargs(mark^#) = {},
               Uargs(c_1) = {1}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              a__f(x1, x2) = [1 0] x1 + [0 0] x2 + [2]
                             [0 1]      [0 0]      [1]
              g(x1) = [1 2] x1 + [3]
                      [0 1]      [2]
              mark(x1) = [1 2] x1 + [1]
                         [0 1]      [0]
              f(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                          [0 1]      [0 0]      [1]
              a__f^#(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                               [3 3]      [3 3]      [0]
              c_0(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              mark^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
              c_1(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_3() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {a__f^#(g(X), Y) -> c_0(a__f^#(mark(X), f(g(X), Y)))}
             Weak Rules:
               {  mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
                , mark^#(g(X)) -> c_2(mark^#(X))
                , mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

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

         * Path {3}->{2}->{1}->{4}: YES(?,O(n^2))
           --------------------------------------

           The usable rules for this path are:

             {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
              , mark(g(X)) -> g(mark(X))
              , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
              , a__f(X1, X2) -> f(X1, X2)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(a__f) = {1}, Uargs(g) = {1}, Uargs(mark) = {}, Uargs(f) = {},
               Uargs(a__f^#) = {1}, Uargs(c_0) = {1}, Uargs(mark^#) = {},
               Uargs(c_1) = {1}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              a__f(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                             [0 1]      [0 0]      [3]
              g(x1) = [1 3] x1 + [2]
                      [0 1]      [2]
              mark(x1) = [1 3] x1 + [0]
                         [0 1]      [0]
              f(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                          [0 1]      [0 0]      [3]
              a__f^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
              c_0(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              mark^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
              c_1(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_3() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^2))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {a__f^#(X1, X2) -> c_3()}
             Weak Rules:
               {  a__f^#(g(X), Y) -> c_0(a__f^#(mark(X), f(g(X), Y)))
                , mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
                , mark^#(g(X)) -> c_2(mark^#(X))
                , mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

           Proof Output:
             The following argument positions are usable:
               Uargs(a__f) = {}, Uargs(g) = {}, Uargs(mark) = {}, Uargs(f) = {},
               Uargs(a__f^#) = {}, Uargs(c_0) = {1}, Uargs(mark^#) = {},
               Uargs(c_1) = {1}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              a__f(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                             [1 2]      [0 0]      [0]
              g(x1) = [1 3] x1 + [0]
                      [0 1]      [0]
              mark(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
              f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                          [0 0]      [0 0]      [0]
              a__f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                               [4 0]      [4 0]      [4]
              c_0(x1) = [1 0] x1 + [0]
                        [0 0]      [3]
              mark^#(x1) = [0 0] x1 + [2]
                           [4 0]      [0]
              c_1(x1) = [2 0] x1 + [0]
                        [0 0]      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
              c_3() = [0]
                      [0]

         * Path {3}->{2}->{4}: YES(?,O(n^2))
           ---------------------------------

           The usable rules for this path are:

             {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
              , mark(g(X)) -> g(mark(X))
              , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
              , a__f(X1, X2) -> f(X1, X2)}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(a__f) = {1}, Uargs(g) = {1}, Uargs(mark) = {}, Uargs(f) = {},
               Uargs(a__f^#) = {1}, Uargs(c_0) = {}, Uargs(mark^#) = {},
               Uargs(c_1) = {1}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              a__f(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                             [0 1]      [0 0]      [3]
              g(x1) = [1 3] x1 + [0]
                      [0 1]      [3]
              mark(x1) = [1 3] x1 + [0]
                         [0 1]      [0]
              f(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                          [0 1]      [0 0]      [3]
              a__f^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
              c_0(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              mark^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
              c_1(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_3() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {a__f^#(X1, X2) -> c_3()}
             Weak Rules:
               {  mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
                , mark^#(g(X)) -> c_2(mark^#(X))
                , mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

           Proof Output:
             The following argument positions are usable:
               Uargs(a__f) = {}, Uargs(g) = {}, Uargs(mark) = {}, Uargs(f) = {},
               Uargs(a__f^#) = {}, Uargs(mark^#) = {}, Uargs(c_1) = {1},
               Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              a__f(x1, x2) = [0 0] x1 + [0 0] x2 + [4]
                             [0 0]      [0 0]      [0]
              g(x1) = [0 0] x1 + [0]
                      [0 0]      [0]
              mark(x1) = [0 0] x1 + [4]
                         [3 0]      [2]
              f(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
                          [0 0]      [0 0]      [0]
              a__f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                               [0 0]      [0 0]      [0]
              mark^#(x1) = [0 0] x1 + [4]
                           [0 0]      [0]
              c_1(x1) = [1 0] x1 + [1]
                        [0 0]      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
              c_3() = [0]
                      [0]

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 4 Luc96b GM

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex4 4 Luc96b GM

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
     , mark(f(X1, X2)) -> a__f(mark(X1), X2)
     , mark(g(X)) -> g(mark(X))
     , a__f(X1, X2) -> f(X1, X2)}

Proof Output:
  None of the processors succeeded.

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

             {  1: a__f^#(g(X), Y) -> c_0(a__f^#(mark(X), f(g(X), Y)))
              , 2: mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
              , 3: mark^#(g(X)) -> c_2(mark^#(X))
              , 4: a__f^#(X1, X2) -> c_3(X1, X2)}

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

             ->{3}                                                       [     inherited      ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    |->{1}                                               [     inherited      ]
                    |   |
                    |   `->{4}                                           [         NA         ]
                    |
                    `->{4}                                               [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}: inherited
             ------------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}->{1}: inherited
             -----------------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}->{1}->{4}: NA
             ---------------------------

             The usable rules for this path are:

               {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

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

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

           * Path {3}->{2}->{4}: MAYBE
             -------------------------

             The usable rules for this path are:

               {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
                  , mark^#(g(X)) -> c_2(mark^#(X))
                  , a__f^#(X1, X2) -> c_3(X1, X2)
                  , mark(f(X1, X2)) -> a__f(mark(X1), X2)
                  , mark(g(X)) -> g(mark(X))
                  , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                  , a__f(X1, X2) -> f(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: a__f^#(g(X), Y) -> c_0(a__f^#(mark(X), f(g(X), Y)))
              , 2: mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
              , 3: mark^#(g(X)) -> c_2(mark^#(X))
              , 4: a__f^#(X1, X2) -> c_3(X1, X2)}

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

             ->{3}                                                       [     inherited      ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    |->{1}                                               [     inherited      ]
                    |   |
                    |   `->{4}                                           [         NA         ]
                    |
                    `->{4}                                               [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}: inherited
             ------------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}->{1}: inherited
             -----------------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}->{1}->{4}: NA
             ---------------------------

             The usable rules for this path are:

               {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

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

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

           * Path {3}->{2}->{4}: MAYBE
             -------------------------

             The usable rules for this path are:

               {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
                  , mark^#(g(X)) -> c_2(mark^#(X))
                  , a__f^#(X1, X2) -> c_3(X1, X2)
                  , mark(f(X1, X2)) -> a__f(mark(X1), X2)
                  , mark(g(X)) -> g(mark(X))
                  , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                  , a__f(X1, X2) -> f(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: a__f^#(g(X), Y) -> c_0(a__f^#(mark(X), f(g(X), Y)))
              , 2: mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
              , 3: mark^#(g(X)) -> c_2(mark^#(X))
              , 4: a__f^#(X1, X2) -> c_3(X1, X2)}

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

             ->{3}                                                       [     inherited      ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    |->{1}                                               [     inherited      ]
                    |   |
                    |   `->{4}                                           [         NA         ]
                    |
                    `->{4}                                               [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}: inherited
             ------------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}->{1}: inherited
             -----------------------------

             This path is subsumed by the proof of path {3}->{2}->{1}->{4}.

           * Path {3}->{2}->{1}->{4}: NA
             ---------------------------

             The usable rules for this path are:

               {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

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

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

           * Path {3}->{2}->{4}: MAYBE
             -------------------------

             The usable rules for this path are:

               {  mark(f(X1, X2)) -> a__f(mark(X1), X2)
                , mark(g(X)) -> g(mark(X))
                , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                , a__f(X1, X2) -> f(X1, X2)}

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  mark^#(f(X1, X2)) -> c_1(a__f^#(mark(X1), X2))
                  , mark^#(g(X)) -> c_2(mark^#(X))
                  , a__f^#(X1, X2) -> c_3(X1, X2)
                  , mark(f(X1, X2)) -> a__f(mark(X1), X2)
                  , mark(g(X)) -> g(mark(X))
                  , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
                  , a__f(X1, X2) -> f(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible

    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.

    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.