Problem Strategy outermost added 08 Ex1 GM99 FR

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex1 GM99 FR

stdout:

MAYBE

Problem:
f(n__a(),n__b(),X) -> f(X,X,X)
c() -> a()
c() -> b()
a() -> n__a()
b() -> n__b()
activate(n__a()) -> a()
activate(n__b()) -> b()
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex1 GM99 FR

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex1 GM99 FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(n__a(), n__b(), X) -> f(X, X, X)
     , c() -> a()
     , c() -> b()
     , a() -> n__a()
     , b() -> n__b()
     , activate(n__a()) -> a()
     , activate(n__b()) -> b()
     , 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^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1(a^#())
              , 3: c^#() -> c_2(b^#())
              , 4: a^#() -> c_3()
              , 5: b^#() -> c_4()
              , 6: activate^#(n__a()) -> c_5(a^#())
              , 7: activate^#(n__b()) -> c_6(b^#())
              , 8: activate^#(X) -> c_7()}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [    YES(?,O(1))     ]
                |
                `->{5}                                                   [    YES(?,O(1))     ]

             ->{6}                                                       [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

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

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

             ->{1}                                                       [       MAYBE        ]



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

             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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] x3 + [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {c^#() -> c_1(a^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_1) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_1(x1) = [7 7 4] x1 + [3]
                          [5 4 4]      [7]
                          [5 4 4]      [7]
                a^#() = [0]
                        [0]
                        [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {a^#() -> c_3()}
               Weak Rules: {c^#() -> c_1(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_1(x1) = [4 4 7] x1 + [3]
                          [4 4 4]      [3]
                          [4 4 4]      [3]
                a^#() = [1]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {c^#() -> c_2(b^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_2(x1) = [7 7 4] x1 + [3]
                          [5 4 4]      [7]
                          [5 4 4]      [7]
                b^#() = [0]
                        [0]
                        [0]

           * Path {3}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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_4()}
               Weak Rules: {c^#() -> c_2(b^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_2(x1) = [4 4 7] x1 + [3]
                          [4 4 4]      [3]
                          [4 4 4]      [3]
                b^#() = [1]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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^#(n__a()) -> c_5(a^#())}
               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:
                n__a() = [2]
                         [2]
                         [2]
                a^#() = [2]
                        [2]
                        [0]
                activate^#(x1) = [0 2 2] x1 + [7]
                                 [2 2 0]      [7]
                                 [0 0 0]      [7]
                c_5(x1) = [2 2 0] x1 + [1]
                          [0 0 0]      [7]
                          [2 0 0]      [3]

           * Path {6}->{4}: YES(?,O(1))
             --------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {a^#() -> c_3()}
               Weak Rules: {activate^#(n__a()) -> c_5(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__a() = [2]
                         [6]
                         [2]
                a^#() = [2]
                        [2]
                        [2]
                c_3() = [1]
                        [0]
                        [0]
                activate^#(x1) = [2 0 2] x1 + [7]
                                 [0 2 0]      [3]
                                 [2 0 0]      [7]
                c_5(x1) = [2 2 0] x1 + [2]
                          [2 2 2]      [3]
                          [2 2 0]      [3]

           * Path {7}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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^#(n__b()) -> c_6(b^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                         [2]
                         [2]
                b^#() = [2]
                        [2]
                        [0]
                activate^#(x1) = [0 2 2] x1 + [7]
                                 [2 2 0]      [7]
                                 [0 0 0]      [7]
                c_6(x1) = [2 2 0] x1 + [1]
                          [0 0 0]      [7]
                          [2 0 0]      [3]

           * Path {7}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [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_4()}
               Weak Rules: {activate^#(n__b()) -> c_6(b^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                         [6]
                         [2]
                b^#() = [2]
                        [2]
                        [2]
                c_4() = [1]
                        [0]
                        [0]
                activate^#(x1) = [2 0 2] x1 + [7]
                                 [0 2 0]      [3]
                                 [2 0 0]      [7]
                c_6(x1) = [2 2 0] x1 + [2]
                          [2 2 2]      [3]
                          [2 2 0]      [3]

           * Path {8}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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_7()}
               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_7() = [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^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1(a^#())
              , 3: c^#() -> c_2(b^#())
              , 4: a^#() -> c_3()
              , 5: b^#() -> c_4()
              , 6: activate^#(n__a()) -> c_5(a^#())
              , 7: activate^#(n__b()) -> c_6(b^#())
              , 8: activate^#(X) -> c_7()}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [    YES(?,O(1))     ]
                |
                `->{5}                                                   [    YES(?,O(1))     ]

             ->{6}                                                       [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

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

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

             ->{1}                                                       [       MAYBE        ]



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

             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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
                                  [3 3]      [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {c^#() -> c_1(a^#())}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {a^#() -> c_3()}
               Weak Rules: {c^#() -> c_1(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                c_1(x1) = [2 0] x1 + [1]
                          [2 0]      [1]
                a^#() = [3]
                        [7]
                c_3() = [0]
                        [1]

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {c^#() -> c_2(b^#())}
               Weak Rules: {}

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

           * Path {3}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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_4()}
               Weak Rules: {c^#() -> c_2(b^#())}

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

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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^#(n__a()) -> c_5(a^#())}
               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:
                n__a() = [2]
                         [2]
                a^#() = [0]
                        [2]
                activate^#(x1) = [2 2] x1 + [7]
                                 [2 0]      [7]
                c_5(x1) = [0 2] x1 + [3]
                          [0 2]      [3]

           * Path {6}->{4}: YES(?,O(1))
             --------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {a^#() -> c_3()}
               Weak Rules: {activate^#(n__a()) -> c_5(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__a() = [2]
                         [2]
                a^#() = [2]
                        [2]
                c_3() = [1]
                        [0]
                activate^#(x1) = [0 0] x1 + [7]
                                 [1 2]      [7]
                c_5(x1) = [2 0] x1 + [2]
                          [2 2]      [3]

           * Path {7}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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^#(n__b()) -> c_6(b^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                         [2]
                b^#() = [0]
                        [2]
                activate^#(x1) = [2 2] x1 + [7]
                                 [2 0]      [7]
                c_6(x1) = [0 2] x1 + [3]
                          [0 2]      [3]

           * Path {7}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [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_4()}
               Weak Rules: {activate^#(n__b()) -> c_6(b^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                         [2]
                b^#() = [2]
                        [2]
                c_4() = [1]
                        [0]
                activate^#(x1) = [0 0] x1 + [7]
                                 [1 2]      [7]
                c_6(x1) = [2 0] x1 + [2]
                          [2 2]      [3]

           * Path {8}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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_7()}
               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_7() = [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^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1(a^#())
              , 3: c^#() -> c_2(b^#())
              , 4: a^#() -> c_3()
              , 5: b^#() -> c_4()
              , 6: activate^#(n__a()) -> c_5(a^#())
              , 7: activate^#(n__b()) -> c_6(b^#())
              , 8: activate^#(X) -> c_7()}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [    YES(?,O(1))     ]
                |
                `->{5}                                                   [    YES(?,O(1))     ]

             ->{6}                                                       [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

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

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

             ->{1}                                                       [       MAYBE        ]



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

             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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {c^#() -> c_1(a^#())}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [1] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {a^#() -> c_3()}
               Weak Rules: {c^#() -> c_1(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                c_1(x1) = [7] x1 + [0]
                a^#() = [1]
                c_3() = [0]

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {c^#() -> c_2(b^#())}
               Weak Rules: {}

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

           * Path {3}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [1] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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_4()}
               Weak Rules: {c^#() -> c_2(b^#())}

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

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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^#(n__a()) -> c_5(a^#())}
               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:
                n__a() = [2]
                a^#() = [2]
                activate^#(x1) = [2] x1 + [5]
                c_5(x1) = [2] x1 + [3]

           * Path {6}->{4}: YES(?,O(1))
             --------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {a^#() -> c_3()}
               Weak Rules: {activate^#(n__a()) -> c_5(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__a() = [0]
                a^#() = [2]
                c_3() = [1]
                activate^#(x1) = [0] x1 + [6]
                c_5(x1) = [2] x1 + [2]

           * Path {7}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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^#(n__b()) -> c_6(b^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                b^#() = [2]
                activate^#(x1) = [2] x1 + [5]
                c_6(x1) = [2] x1 + [3]

           * Path {7}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [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_4()}
               Weak Rules: {activate^#(n__b()) -> c_6(b^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [0]
                b^#() = [2]
                c_4() = [1]
                activate^#(x1) = [0] x1 + [6]
                c_6(x1) = [2] x1 + [2]

           * Path {8}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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_7()}
               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_7() = [0]

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex1 GM99 FR

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex1 GM99 FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(n__a(), n__b(), X) -> f(X, X, X)
     , c() -> a()
     , c() -> b()
     , a() -> n__a()
     , b() -> n__b()
     , activate(n__a()) -> a()
     , activate(n__b()) -> b()
     , 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^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1(a^#())
              , 3: c^#() -> c_2(b^#())
              , 4: a^#() -> c_3()
              , 5: b^#() -> c_4()
              , 6: activate^#(n__a()) -> c_5(a^#())
              , 7: activate^#(n__b()) -> c_6(b^#())
              , 8: activate^#(X) -> c_7(X)}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [    YES(?,O(1))     ]
                |
                `->{5}                                                   [    YES(?,O(1))     ]

             ->{6}                                                       [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

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

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

             ->{1}                                                       [       MAYBE        ]



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

             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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] x3 + [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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: {c^#() -> c_1(a^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_1) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_1(x1) = [7 7 4] x1 + [3]
                          [5 4 4]      [7]
                          [5 4 4]      [7]
                a^#() = [0]
                        [0]
                        [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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: {a^#() -> c_3()}
               Weak Rules: {c^#() -> c_1(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_1(x1) = [4 4 7] x1 + [3]
                          [4 4 4]      [3]
                          [4 4 4]      [3]
                a^#() = [1]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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: {c^#() -> c_2(b^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_2(x1) = [7 7 4] x1 + [3]
                          [5 4 4]      [7]
                          [5 4 4]      [7]
                b^#() = [0]
                        [0]
                        [0]

           * Path {3}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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_4()}
               Weak Rules: {c^#() -> c_2(b^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                        [7]
                c_2(x1) = [4 4 7] x1 + [3]
                          [4 4 4]      [3]
                          [4 4 4]      [3]
                b^#() = [1]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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: {activate^#(n__a()) -> c_5(a^#())}
               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:
                n__a() = [2]
                         [2]
                         [2]
                a^#() = [2]
                        [2]
                        [0]
                activate^#(x1) = [0 2 2] x1 + [7]
                                 [2 2 0]      [7]
                                 [0 0 0]      [7]
                c_5(x1) = [2 2 0] x1 + [1]
                          [0 0 0]      [7]
                          [2 0 0]      [3]

           * Path {6}->{4}: YES(?,O(1))
             --------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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: {a^#() -> c_3()}
               Weak Rules: {activate^#(n__a()) -> c_5(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__a() = [2]
                         [6]
                         [2]
                a^#() = [2]
                        [2]
                        [2]
                c_3() = [1]
                        [0]
                        [0]
                activate^#(x1) = [2 0 2] x1 + [7]
                                 [0 2 0]      [3]
                                 [2 0 0]      [7]
                c_5(x1) = [2 2 0] x1 + [2]
                          [2 2 2]      [3]
                          [2 2 0]      [3]

           * Path {7}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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: {activate^#(n__b()) -> c_6(b^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                         [2]
                         [2]
                b^#() = [2]
                        [2]
                        [0]
                activate^#(x1) = [0 2 2] x1 + [7]
                                 [2 2 0]      [7]
                                 [0 0 0]      [7]
                c_6(x1) = [2 2 0] x1 + [1]
                          [0 0 0]      [7]
                          [2 0 0]      [3]

           * Path {7}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(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]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7(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_4()}
               Weak Rules: {activate^#(n__b()) -> c_6(b^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                         [6]
                         [2]
                b^#() = [2]
                        [2]
                        [2]
                c_4() = [1]
                        [0]
                        [0]
                activate^#(x1) = [2 0 2] x1 + [7]
                                 [0 2 0]      [3]
                                 [2 0 0]      [7]
                c_6(x1) = [2 2 0] x1 + [2]
                          [2 2 2]      [3]
                          [2 2 0]      [3]

           * Path {8}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                         [0]
                n__b() = [0]
                         [0]
                         [0]
                c() = [0]
                      [0]
                      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 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]
                c^#() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                b^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                activate^#(x1) = [3 3 3] 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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(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_7(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               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_7(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^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1(a^#())
              , 3: c^#() -> c_2(b^#())
              , 4: a^#() -> c_3()
              , 5: b^#() -> c_4()
              , 6: activate^#(n__a()) -> c_5(a^#())
              , 7: activate^#(n__b()) -> c_6(b^#())
              , 8: activate^#(X) -> c_7(X)}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [    YES(?,O(1))     ]
                |
                `->{5}                                                   [    YES(?,O(1))     ]

             ->{6}                                                       [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

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

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

             ->{1}                                                       [       MAYBE        ]



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

             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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
                                  [3 3]      [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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: {c^#() -> c_1(a^#())}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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: {a^#() -> c_3()}
               Weak Rules: {c^#() -> c_1(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                        [7]
                c_1(x1) = [2 0] x1 + [1]
                          [2 0]      [1]
                a^#() = [3]
                        [7]
                c_3() = [0]
                        [1]

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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: {c^#() -> c_2(b^#())}
               Weak Rules: {}

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

           * Path {3}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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_4()}
               Weak Rules: {c^#() -> c_2(b^#())}

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

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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: {activate^#(n__a()) -> c_5(a^#())}
               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:
                n__a() = [2]
                         [2]
                a^#() = [0]
                        [2]
                activate^#(x1) = [2 2] x1 + [7]
                                 [2 0]      [7]
                c_5(x1) = [0 2] x1 + [3]
                          [0 2]      [3]

           * Path {6}->{4}: YES(?,O(1))
             --------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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: {a^#() -> c_3()}
               Weak Rules: {activate^#(n__a()) -> c_5(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__a() = [2]
                         [2]
                a^#() = [2]
                        [2]
                c_3() = [1]
                        [0]
                activate^#(x1) = [0 0] x1 + [7]
                                 [1 2]      [7]
                c_5(x1) = [2 0] x1 + [2]
                          [2 2]      [3]

           * Path {7}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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: {activate^#(n__b()) -> c_6(b^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                         [2]
                b^#() = [0]
                        [2]
                activate^#(x1) = [2 2] x1 + [7]
                                 [2 0]      [7]
                c_6(x1) = [0 2] x1 + [3]
                          [0 2]      [3]

           * Path {7}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7(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_4()}
               Weak Rules: {activate^#(n__b()) -> c_6(b^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                         [2]
                b^#() = [2]
                        [2]
                c_4() = [1]
                        [0]
                activate^#(x1) = [0 0] x1 + [7]
                                 [1 2]      [7]
                c_6(x1) = [2 0] x1 + [2]
                          [2 2]      [3]

           * Path {8}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               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__a() = [0]
                         [0]
                n__b() = [0]
                         [0]
                c() = [0]
                      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 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]
                c^#() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                b^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(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_7(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7] x1 + [7]
                                 [7 7]      [7]
                c_7(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^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))
              , 2: c^#() -> c_1(a^#())
              , 3: c^#() -> c_2(b^#())
              , 4: a^#() -> c_3()
              , 5: b^#() -> c_4()
              , 6: activate^#(n__a()) -> c_5(a^#())
              , 7: activate^#(n__b()) -> c_6(b^#())
              , 8: activate^#(X) -> c_7(X)}

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

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [    YES(?,O(1))     ]
                |
                `->{5}                                                   [    YES(?,O(1))     ]

             ->{6}                                                       [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

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

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

             ->{1}                                                       [       MAYBE        ]



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

             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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(n__a(), n__b(), X) -> c_0(f^#(X, X, X))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {c^#() -> c_1(a^#())}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [1] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {a^#() -> c_3()}
               Weak Rules: {c^#() -> c_1(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c^#() = [7]
                c_1(x1) = [7] x1 + [0]
                a^#() = [1]
                c_3() = [0]

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {c^#() -> c_2(b^#())}
               Weak Rules: {}

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

           * Path {3}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [1] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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_4()}
               Weak Rules: {c^#() -> c_2(b^#())}

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

           * 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {activate^#(n__a()) -> c_5(a^#())}
               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:
                n__a() = [2]
                a^#() = [2]
                activate^#(x1) = [2] x1 + [5]
                c_5(x1) = [2] x1 + [3]

           * Path {6}->{4}: YES(?,O(1))
             --------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {a^#() -> c_3()}
               Weak Rules: {activate^#(n__a()) -> c_5(a^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__a() = [0]
                a^#() = [2]
                c_3() = [1]
                activate^#(x1) = [0] x1 + [6]
                c_5(x1) = [2] x1 + [2]

           * Path {7}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {activate^#(n__b()) -> c_6(b^#())}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [2]
                b^#() = [2]
                activate^#(x1) = [2] x1 + [5]
                c_6(x1) = [2] x1 + [3]

           * Path {7}->{5}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7(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_4()}
               Weak Rules: {activate^#(n__b()) -> c_6(b^#())}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                n__b() = [0]
                b^#() = [2]
                c_4() = [1]
                activate^#(x1) = [0] x1 + [6]
                c_6(x1) = [2] x1 + [2]

           * Path {8}: 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(activate) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                n__a() = [0]
                n__b() = [0]
                c() = [0]
                a() = [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c^#() = [0]
                c_1(x1) = [0] x1 + [0]
                a^#() = [0]
                c_2(x1) = [0] x1 + [0]
                b^#() = [0]
                c_3() = [0]
                c_4() = [0]
                activate^#(x1) = [3] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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_7(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7] x1 + [7]
                c_7(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.