Problem Transformed CSR 04 Ex24 GM04 Z

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex24 GM04 Z

stdout:

MAYBE

Problem:
f(X,n__g(X),Y) -> f(activate(Y),activate(Y),activate(Y))
g(b()) -> c()
b() -> c()
g(X) -> n__g(X)
activate(n__g(X)) -> g(X)
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex24 GM04 Z

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex24 GM04 Z

stdout:

MAYBE

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

Proof Output:
  None of the processors succeeded.

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

             {  1: f^#(X, n__g(X), Y) ->
                   c_0(f^#(activate(Y), activate(Y), activate(Y)))
              , 2: g^#(b()) -> c_1()
              , 3: b^#() -> c_2()
              , 4: g^#(X) -> c_3()
              , 5: activate^#(n__g(X)) -> c_4(g^#(X))
              , 6: activate^#(X) -> c_5()}

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

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

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

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  activate(n__g(X)) -> g(X)
                , activate(X) -> X
                , g(b()) -> c()
                , g(X) -> n__g(X)}

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(X, n__g(X), Y) ->
                    c_0(f^#(activate(Y), activate(Y), activate(Y)))
                  , activate(n__g(X)) -> g(X)
                  , activate(X) -> X
                  , g(b()) -> c()
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [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: {b^#() -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b^#() = [7]
                        [7]
                        [7]
                c_2() = [0]
                        [3]
                        [3]

           * Path {5}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [1 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 1]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [3 0 0] x1 + [0]
                          [3 0 0]      [0]
                          [3 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [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: {activate^#(n__g(X)) -> c_4(g^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2 2] x1 + [2]
                           [0 1 2]      [2]
                           [0 0 0]      [2]
                g^#(x1) = [0 2 0] x1 + [2]
                          [0 2 2]      [2]
                          [0 0 2]      [2]
                activate^#(x1) = [3 2 0] x1 + [5]
                                 [2 0 2]      [6]
                                 [2 3 2]      [0]
                c_4(x1) = [2 2 2] x1 + [1]
                          [2 0 0]      [6]
                          [0 0 0]      [7]

           * Path {5}->{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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(b()) -> c_1()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 0 2] x1 + [0]
                           [0 0 2]      [2]
                           [0 0 0]      [2]
                b() = [2]
                      [2]
                      [2]
                g^#(x1) = [2 0 2] x1 + [0]
                          [2 0 2]      [2]
                          [2 2 0]      [0]
                c_1() = [1]
                        [0]
                        [0]
                activate^#(x1) = [4 2 0] x1 + [3]
                                 [4 0 2]      [7]
                                 [2 2 0]      [2]
                c_4(x1) = [2 0 0] x1 + [7]
                          [0 2 0]      [3]
                          [0 0 0]      [3]

           * Path {5}->{4}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5() = [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: {g^#(X) -> c_3()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2 2] x1 + [2]
                           [0 1 2]      [0]
                           [0 0 0]      [0]
                g^#(x1) = [0 0 2] x1 + [2]
                          [0 0 0]      [0]
                          [0 0 0]      [2]
                c_3() = [1]
                        [0]
                        [0]
                activate^#(x1) = [2 4 0] x1 + [7]
                                 [6 0 0]      [2]
                                 [2 2 0]      [6]
                c_4(x1) = [1 0 2] x1 + [1]
                          [0 0 2]      [7]
                          [2 0 2]      [2]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [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_5()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [0 0 0] x1 + [7]
                                 [0 0 0]      [7]
                                 [0 0 0]      [7]
                c_5() = [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^#(X, n__g(X), Y) ->
                   c_0(f^#(activate(Y), activate(Y), activate(Y)))
              , 2: g^#(b()) -> c_1()
              , 3: b^#() -> c_2()
              , 4: g^#(X) -> c_3()
              , 5: activate^#(n__g(X)) -> c_4(g^#(X))
              , 6: activate^#(X) -> c_5()}

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

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

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

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  activate(n__g(X)) -> g(X)
                , activate(X) -> X
                , g(b()) -> c()
                , g(X) -> n__g(X)}

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(X, n__g(X), Y) ->
                    c_0(f^#(activate(Y), activate(Y), activate(Y)))
                  , activate(n__g(X)) -> g(X)
                  , activate(X) -> X
                  , g(b()) -> c()
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [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: {b^#() -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b^#() = [7]
                        [7]
                c_2() = [0]
                        [1]

           * Path {5}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [1 1] x1 + [0]
                           [0 1]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [3 0] x1 + [0]
                          [3 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                activate^#(x1) = [1 3] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [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: {activate^#(n__g(X)) -> c_4(g^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2] x1 + [2]
                           [0 1]      [0]
                g^#(x1) = [0 2] x1 + [2]
                          [0 0]      [2]
                activate^#(x1) = [6 1] x1 + [3]
                                 [6 1]      [2]
                c_4(x1) = [2 0] x1 + [7]
                          [2 2]      [3]

           * Path {5}->{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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5() = [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: {g^#(b()) -> c_1()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2] x1 + [0]
                           [0 0]      [2]
                b() = [2]
                      [2]
                g^#(x1) = [2 2] x1 + [2]
                          [2 2]      [0]
                c_1() = [1]
                        [0]
                activate^#(x1) = [2 0] x1 + [7]
                                 [4 2]      [6]
                c_4(x1) = [1 0] x1 + [1]
                          [2 0]      [3]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5() = [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: {g^#(X) -> c_3()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2] x1 + [0]
                           [0 0]      [2]
                g^#(x1) = [0 0] x1 + [2]
                          [0 0]      [0]
                c_3() = [1]
                        [0]
                activate^#(x1) = [6 2] x1 + [3]
                                 [4 2]      [6]
                c_4(x1) = [1 0] x1 + [0]
                          [2 0]      [5]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [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_5()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [0 0] x1 + [7]
                                 [0 0]      [7]
                c_5() = [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^#(X, n__g(X), Y) ->
                   c_0(f^#(activate(Y), activate(Y), activate(Y)))
              , 2: g^#(b()) -> c_1()
              , 3: b^#() -> c_2()
              , 4: g^#(X) -> c_3()
              , 5: activate^#(n__g(X)) -> c_4(g^#(X))
              , 6: activate^#(X) -> c_5()}

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

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

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

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  activate(n__g(X)) -> g(X)
                , activate(X) -> X
                , g(b()) -> c()
                , g(X) -> n__g(X)}

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(X, n__g(X), Y) ->
                    c_0(f^#(activate(Y), activate(Y), activate(Y)))
                  , activate(n__g(X)) -> g(X)
                  , activate(X) -> X
                  , g(b()) -> c()
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]

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

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

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b^#() = [7]
                c_2() = [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [1] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [1] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3() = [0]
                activate^#(x1) = [3] x1 + [0]
                c_4(x1) = [3] x1 + [0]
                c_5() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}
               Weak Rules: {}

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

           * Path {5}->{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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(b()) -> c_1()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1] x1 + [2]
                b() = [2]
                g^#(x1) = [2] x1 + [4]
                c_1() = [1]
                activate^#(x1) = [4] x1 + [7]
                c_4(x1) = [2] x1 + [7]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(X) -> c_3()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]

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

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

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

    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	Transformed CSR 04 Ex24 GM04 Z

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex24 GM04 Z

stdout:

MAYBE

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

Proof Output:
  None of the processors succeeded.

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

             {  1: f^#(X, n__g(X), Y) ->
                   c_0(f^#(activate(Y), activate(Y), activate(Y)))
              , 2: g^#(b()) -> c_1()
              , 3: b^#() -> c_2()
              , 4: g^#(X) -> c_3(X)
              , 5: activate^#(n__g(X)) -> c_4(g^#(X))
              , 6: activate^#(X) -> c_5(X)}

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

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

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

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  activate(n__g(X)) -> g(X)
                , activate(X) -> X
                , g(b()) -> c()
                , g(X) -> n__g(X)}

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(X, n__g(X), Y) ->
                    c_0(f^#(activate(Y), activate(Y), activate(Y)))
                  , activate(n__g(X)) -> g(X)
                  , activate(X) -> X
                  , g(b()) -> c()
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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: {b^#() -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b^#() = [7]
                        [7]
                        [7]
                c_2() = [0]
                        [3]
                        [3]

           * Path {5}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [1 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 1]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [3 0 0] x1 + [0]
                          [3 0 0]      [0]
                          [3 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2 2] x1 + [2]
                           [0 1 2]      [2]
                           [0 0 0]      [2]
                g^#(x1) = [0 2 0] x1 + [2]
                          [0 2 2]      [2]
                          [0 0 2]      [2]
                activate^#(x1) = [3 2 0] x1 + [5]
                                 [2 0 2]      [6]
                                 [2 3 2]      [0]
                c_4(x1) = [2 2 2] x1 + [1]
                          [2 0 0]      [6]
                          [0 0 0]      [7]

           * Path {5}->{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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [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_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5(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^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(b()) -> c_1()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 0 2] x1 + [0]
                           [0 0 2]      [2]
                           [0 0 0]      [2]
                b() = [2]
                      [2]
                      [2]
                g^#(x1) = [2 0 2] x1 + [0]
                          [2 0 2]      [2]
                          [2 2 0]      [0]
                c_1() = [1]
                        [0]
                        [0]
                activate^#(x1) = [4 2 0] x1 + [3]
                                 [4 0 2]      [7]
                                 [2 2 0]      [2]
                c_4(x1) = [2 0 0] x1 + [7]
                          [0 2 0]      [3]
                          [0 0 0]      [3]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [3 3 3] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 1 1] x1 + [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_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5(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^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(X) -> c_3(X)}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(c_3) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2 1] x1 + [0]
                           [0 0 0]      [2]
                           [0 0 0]      [0]
                g^#(x1) = [0 0 0] x1 + [2]
                          [0 0 0]      [2]
                          [0 0 0]      [2]
                c_3(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                activate^#(x1) = [6 0 0] x1 + [7]
                                 [0 0 0]      [7]
                                 [4 2 0]      [7]
                c_4(x1) = [1 0 0] x1 + [1]
                          [0 2 0]      [3]
                          [2 0 2]      [3]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                n__g(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 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]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                b^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [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_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(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_5(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7 7] x1 + [7]
                                 [7 7 7]      [7]
                                 [7 7 7]      [7]
                c_5(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^#(X, n__g(X), Y) ->
                   c_0(f^#(activate(Y), activate(Y), activate(Y)))
              , 2: g^#(b()) -> c_1()
              , 3: b^#() -> c_2()
              , 4: g^#(X) -> c_3(X)
              , 5: activate^#(n__g(X)) -> c_4(g^#(X))
              , 6: activate^#(X) -> c_5(X)}

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

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

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

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  activate(n__g(X)) -> g(X)
                , activate(X) -> X
                , g(b()) -> c()
                , g(X) -> n__g(X)}

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(X, n__g(X), Y) ->
                    c_0(f^#(activate(Y), activate(Y), activate(Y)))
                  , activate(n__g(X)) -> g(X)
                  , activate(X) -> X
                  , g(b()) -> c()
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

             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: {b^#() -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b^#() = [7]
                        [7]
                c_2() = [0]
                        [1]

           * Path {5}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [1 1] x1 + [0]
                           [0 1]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [3 0] x1 + [0]
                          [3 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [1 3] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

             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: {activate^#(n__g(X)) -> c_4(g^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2] x1 + [2]
                           [0 1]      [0]
                g^#(x1) = [0 2] x1 + [2]
                          [0 0]      [2]
                activate^#(x1) = [6 1] x1 + [3]
                                 [6 1]      [2]
                c_4(x1) = [2 0] x1 + [7]
                          [2 2]      [3]

           * Path {5}->{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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5(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^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(b()) -> c_1()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2] x1 + [0]
                           [0 0]      [2]
                b() = [2]
                      [2]
                g^#(x1) = [2 2] x1 + [2]
                          [2 2]      [0]
                c_1() = [1]
                        [0]
                activate^#(x1) = [2 0] x1 + [7]
                                 [4 2]      [6]
                c_4(x1) = [1 0] x1 + [1]
                          [2 0]      [3]

           * Path {5}->{4}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [3 3] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5(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: {g^#(X) -> c_3(X)}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(c_3) = {},
                 Uargs(activate^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1 2] x1 + [2]
                           [0 1]      [1]
                g^#(x1) = [0 2] x1 + [2]
                          [0 4]      [0]
                c_3(x1) = [0 0] x1 + [1]
                          [0 0]      [0]
                activate^#(x1) = [2 5] x1 + [5]
                                 [6 0]      [2]
                c_4(x1) = [2 0] x1 + [6]
                          [2 1]      [3]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                n__g(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                b^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(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_5(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7] x1 + [7]
                                 [7 7]      [7]
                c_5(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^#(X, n__g(X), Y) ->
                   c_0(f^#(activate(Y), activate(Y), activate(Y)))
              , 2: g^#(b()) -> c_1()
              , 3: b^#() -> c_2()
              , 4: g^#(X) -> c_3(X)
              , 5: activate^#(n__g(X)) -> c_4(g^#(X))
              , 6: activate^#(X) -> c_5(X)}

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

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

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

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  activate(n__g(X)) -> g(X)
                , activate(X) -> X
                , g(b()) -> c()
                , g(X) -> n__g(X)}

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(X, n__g(X), Y) ->
                    c_0(f^#(activate(Y), activate(Y), activate(Y)))
                  , activate(n__g(X)) -> g(X)
                  , activate(X) -> X
                  , g(b()) -> c()
                  , g(X) -> n__g(X)}

             Proof Output:
               The input cannot be shown compatible

           * Path {3}: 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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

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

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b^#() = [7]
                c_2() = [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [1] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [1] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                activate^#(x1) = [3] x1 + [0]
                c_4(x1) = [3] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}
               Weak Rules: {}

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

           * Path {5}->{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(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(b()) -> c_1()}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(n__g) = {}, Uargs(g^#) = {}, Uargs(activate^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__g(x1) = [1] x1 + [2]
                b() = [2]
                g^#(x1) = [2] x1 + [4]
                c_1() = [1]
                activate^#(x1) = [4] x1 + [7]
                c_4(x1) = [2] x1 + [7]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [3] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(X) -> c_3(X)}
               Weak Rules: {activate^#(n__g(X)) -> c_4(g^#(X))}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(n__g) = {}, Uargs(activate) = {},
                 Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_3) = {}, Uargs(activate^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__g(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                g(x1) = [0] x1 + [0]
                b() = [0]
                c() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                b^#() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                activate^#(x1) = [3] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]

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

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

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

    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.