Problem Strategy outermost added 08 Ex4 4 Luc96b Z

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 4 Luc96b Z

stdout:

MAYBE

Problem:
f(g(X),Y) -> f(X,n__f(g(X),activate(Y)))
f(X1,X2) -> n__f(X1,X2)
activate(n__f(X1,X2)) -> f(X1,X2)
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 4 Luc96b Z

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 4 Luc96b Z

stdout:

MAYBE

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

Proof Output:
  None of the processors succeeded.

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

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

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

             ->{4}                                                       [    YES(?,O(1))     ]

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



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

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

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

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

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

             The usable rules for this path are:

               {  activate(n__f(X1, X2)) -> f(X1, X2)
                , activate(X) -> X
                , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                , f(X1, X2) -> n__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}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(activate^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                n__f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_3() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(X1, X2) -> c_1()}
               Weak Rules: {activate^#(n__f(X1, X2)) -> c_2(f^#(X1, X2))}

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

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(activate^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                n__f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [0 0 0] x1 + [7]
                                 [0 0 0]      [7]
                                 [0 0 0]      [7]
                c_3() = [0]
                        [3]
                        [3]

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

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

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

             ->{4}                                                       [    YES(?,O(1))     ]

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



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

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

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

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

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

             The usable rules for this path are:

               {  activate(n__f(X1, X2)) -> f(X1, X2)
                , activate(X) -> X
                , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                , f(X1, X2) -> n__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:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(g(X), Y) -> c_0(f^#(X, n__f(g(X), activate(Y))))
                  , activate^#(n__f(X1, X2)) -> c_2(f^#(X1, X2))
                  , f^#(X1, X2) -> c_1()
                  , activate(n__f(X1, X2)) -> f(X1, X2)
                  , activate(X) -> X
                  , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                  , f(X1, X2) -> n__f(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(activate^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                n__f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                activate^#(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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(X1, X2) -> c_1()}
               Weak Rules: {activate^#(n__f(X1, X2)) -> c_2(f^#(X1, X2))}

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

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(activate^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                n__f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [0]
                        [0]

             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: {activate^#(X) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [0 0] x1 + [7]
                                 [0 0]      [7]
                c_3() = [0]
                        [1]

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

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

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

             ->{4}                                                       [    YES(?,O(1))     ]

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



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

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

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

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

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

             The usable rules for this path are:

               {  activate(n__f(X1, X2)) -> f(X1, X2)
                , activate(X) -> X
                , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                , f(X1, X2) -> n__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:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(g(X), Y) -> c_0(f^#(X, n__f(g(X), activate(Y))))
                  , activate^#(n__f(X1, X2)) -> c_2(f^#(X1, X2))
                  , f^#(X1, X2) -> c_1()
                  , activate(n__f(X1, X2)) -> f(X1, X2)
                  , activate(X) -> X
                  , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                  , f(X1, X2) -> n__f(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(activate^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                n__f(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                c_3() = [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: {f^#(X1, X2) -> c_1()}
               Weak Rules: {activate^#(n__f(X1, X2)) -> c_2(f^#(X1, X2))}

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

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(activate^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                n__f(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [0] x1 + [7]
                c_3() = [0]

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 4 Luc96b Z

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 4 Luc96b Z

stdout:

MAYBE

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

Proof Output:
  None of the processors succeeded.

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

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

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

             ->{4}                                                       [    YES(?,O(1))     ]

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



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

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

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

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

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

             The usable rules for this path are:

               {  activate(n__f(X1, X2)) -> f(X1, X2)
                , activate(X) -> X
                , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                , f(X1, X2) -> n__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}: YES(?,O(n^3))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(activate^#) = {}, Uargs(c_2) = {1},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                n__f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                f^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1, x2) = [1 1 1] x1 + [1 1 1] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(X1, X2) -> c_1(X1, X2)}
               Weak Rules: {activate^#(n__f(X1, X2)) -> c_2(f^#(X1, X2))}

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

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(activate^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                n__f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                activate^#(x1) = [3 3 3] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_3(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7 7] x1 + [7]
                                 [7 7 7]      [7]
                                 [7 7 7]      [7]
                c_3(x1) = [3 3 3] x1 + [0]
                          [3 1 3]      [1]
                          [1 1 1]      [1]

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

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

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

             ->{4}                                                       [    YES(?,O(1))     ]

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



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

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

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

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

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

             The usable rules for this path are:

               {  activate(n__f(X1, X2)) -> f(X1, X2)
                , activate(X) -> X
                , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                , f(X1, X2) -> n__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:
                 {  f^#(g(X), Y) -> c_0(f^#(X, n__f(g(X), activate(Y))))
                  , activate^#(n__f(X1, X2)) -> c_2(f^#(X1, X2))
                  , f^#(X1, X2) -> c_1(X1, X2)
                  , activate(n__f(X1, X2)) -> f(X1, X2)
                  , activate(X) -> X
                  , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                  , f(X1, X2) -> n__f(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(activate^#) = {}, Uargs(c_2) = {1},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                n__f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                f^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
                              [0 0]      [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

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

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

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(activate^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                n__f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 1] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_3(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7] x1 + [7]
                                 [7 7]      [7]
                c_3(x1) = [1 3] x1 + [0]
                          [3 1]      [3]

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

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

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

             ->{4}                                                       [    YES(?,O(1))     ]

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



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

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

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

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

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

             The usable rules for this path are:

               {  activate(n__f(X1, X2)) -> f(X1, X2)
                , activate(X) -> X
                , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                , f(X1, X2) -> n__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:
                 {  f^#(g(X), Y) -> c_0(f^#(X, n__f(g(X), activate(Y))))
                  , activate^#(n__f(X1, X2)) -> c_2(f^#(X1, X2))
                  , f^#(X1, X2) -> c_1(X1, X2)
                  , activate(n__f(X1, X2)) -> f(X1, X2)
                  , activate(X) -> X
                  , f(g(X), Y) -> f(X, n__f(g(X), activate(Y)))
                  , f(X1, X2) -> n__f(X1, X2)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(activate^#) = {}, Uargs(c_2) = {1},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                n__f(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                c_3(x1) = [0] x1 + [0]

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

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

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

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(n__f) = {},
                 Uargs(activate) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(activate^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                n__f(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1, x2) = [0] x1 + [0] x2 + [0]
                activate^#(x1) = [3] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_3(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7] x1 + [7]
                c_3(x1) = [1] x1 + [0]

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

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

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