Problem Strategy outermost added 08 Ex14 Luc06 FR

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 Luc06 FR

stdout:

MAYBE

Problem:
h(X) -> g(X,X)
g(a(),X) -> f(b(),activate(X))
f(X,X) -> h(a())
a() -> b()
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 Luc06 FR

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 Luc06 FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  h(X) -> g(X, X)
     , g(a(), X) -> f(b(), activate(X))
     , f(X, X) -> h(a())
     , a() -> 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: h^#(X) -> c_0(g^#(X, X))
              , 2: g^#(a(), X) -> c_1(f^#(b(), activate(X)))
              , 3: f^#(X, X) -> c_2(h^#(a()))
              , 4: a^#() -> c_3()
              , 5: activate^#(X) -> c_4()}

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

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

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

             ->{1,3,2}                                                   [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1,3,2}: MAYBE
             -------------------

             The usable rules for this path are:

               {  a() -> b()
                , activate(X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(activate) = {},
                 Uargs(h^#) = {1}, Uargs(c_0) = {1}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(f^#) = {2}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                a() = [3]
                      [3]
                      [3]
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                b() = [1]
                      [1]
                      [1]
                activate(x1) = [1 0 0] x1 + [3]
                               [0 1 0]      [0]
                               [0 1 3]      [0]
                h^#(x1) = [3 3 3] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                g^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                              [3 3 3]      [3 3 3]      [0]
                              [3 3 3]      [3 3 3]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                f^#(x1, x2) = [0 0 0] x1 + [3 1 1] x2 + [0]
                              [0 0 0]      [3 3 3]      [0]
                              [0 0 0]      [3 3 3]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(1))

             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:
                 {  h^#(X) -> c_0(g^#(X, X))
                  , f^#(X, X) -> c_2(h^#(a()))
                  , g^#(a(), X) -> c_1(f^#(b(), activate(X)))}
               Weak Rules:
                 {  a() -> b()
                  , activate(X) -> X}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

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

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

           * Path {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(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(activate) = {},
                 Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {},
                 Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_4() = [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_4()}
               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_4() = [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: h^#(X) -> c_0(g^#(X, X))
              , 2: g^#(a(), X) -> c_1(f^#(b(), activate(X)))
              , 3: f^#(X, X) -> c_2(h^#(a()))
              , 4: a^#() -> c_3()
              , 5: activate^#(X) -> c_4()}

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

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

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

             ->{1,3,2}                                                   [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1,3,2}: MAYBE
             -------------------

             The usable rules for this path are:

               {  a() -> b()
                , activate(X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(activate) = {},
                 Uargs(h^#) = {1}, Uargs(c_0) = {1}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(f^#) = {2}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                a() = [3]
                      [3]
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                b() = [1]
                      [1]
                activate(x1) = [1 0] x1 + [3]
                               [0 3]      [0]
                h^#(x1) = [3 3] x1 + [0]
                          [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                g^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                              [3 3]      [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                f^#(x1, x2) = [0 0] x1 + [3 1] x2 + [0]
                              [0 0]      [3 3]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                a^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4() = [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(1))

             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:
                 {  h^#(X) -> c_0(g^#(X, X))
                  , f^#(X, X) -> c_2(h^#(a()))
                  , g^#(a(), X) -> c_1(f^#(b(), activate(X)))}
               Weak Rules:
                 {  a() -> b()
                  , activate(X) -> X}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

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

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

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

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

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

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

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

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

             ->{1,3,2}                                                   [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1,3,2}: MAYBE
             -------------------

             The usable rules for this path are:

               {  a() -> b()
                , activate(X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(activate) = {},
                 Uargs(h^#) = {1}, Uargs(c_0) = {1}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(f^#) = {2}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [0] x1 + [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                a() = [3]
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                b() = [1]
                activate(x1) = [3] x1 + [3]
                h^#(x1) = [3] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                g^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                f^#(x1, x2) = [0] x1 + [1] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                a^#() = [0]
                c_3() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_4() = [0]
             Complexity induced by the adequate RMI: YES(?,O(1))

             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:
                 {  h^#(X) -> c_0(g^#(X, X))
                  , f^#(X, X) -> c_2(h^#(a()))
                  , g^#(a(), X) -> c_1(f^#(b(), activate(X)))}
               Weak Rules:
                 {  a() -> b()
                  , activate(X) -> X}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

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

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

           * Path {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(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(activate) = {},
                 Uargs(h^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {},
                 Uargs(f^#) = {}, Uargs(c_2) = {}, Uargs(activate^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [0] x1 + [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                a() = [0]
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                b() = [0]
                activate(x1) = [0] x1 + [0]
                h^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                a^#() = [0]
                c_3() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_4() = [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_4()}
               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_4() = [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 Ex14 Luc06 FR

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 Luc06 FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  h(X) -> g(X, X)
     , g(a(), X) -> f(b(), activate(X))
     , f(X, X) -> h(a())
     , a() -> 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: h^#(X) -> c_0(g^#(X, X))
              , 2: g^#(a(), X) -> c_1(f^#(b(), activate(X)))
              , 3: f^#(X, X) -> c_2(h^#(a()))
              , 4: a^#() -> c_3()
              , 5: activate^#(X) -> c_4(X)}

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

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

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

             ->{1,3,2}                                                   [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1,3,2}: MAYBE
             -------------------

             The usable rules for this path are:

               {  a() -> b()
                , activate(X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(activate) = {1},
                 Uargs(h^#) = {1}, Uargs(c_0) = {1}, Uargs(g^#) = {1, 2},
                 Uargs(c_1) = {1}, Uargs(f^#) = {2}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                a() = [3]
                      [3]
                      [3]
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                b() = [1]
                      [1]
                      [1]
                activate(x1) = [1 0 0] x1 + [3]
                               [3 3 3]      [0]
                               [1 3 3]      [0]
                h^#(x1) = [3 3 3] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                g^#(x1, x2) = [1 0 0] x1 + [2 3 3] x2 + [0]
                              [3 3 3]      [3 3 3]      [0]
                              [3 3 3]      [3 3 3]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                f^#(x1, x2) = [0 0 0] x1 + [1 0 1] x2 + [0]
                              [0 0 0]      [3 3 3]      [0]
                              [0 0 0]      [3 3 3]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                a^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(1))

             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:
                 {  h^#(X) -> c_0(g^#(X, X))
                  , f^#(X, X) -> c_2(h^#(a()))
                  , g^#(a(), X) -> c_1(f^#(b(), activate(X)))}
               Weak Rules:
                 {  a() -> b()
                  , activate(X) -> X}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

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

             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: {}

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               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_4(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: h^#(X) -> c_0(g^#(X, X))
              , 2: g^#(a(), X) -> c_1(f^#(b(), activate(X)))
              , 3: f^#(X, X) -> c_2(h^#(a()))
              , 4: a^#() -> c_3()
              , 5: activate^#(X) -> c_4(X)}

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

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

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

             ->{1,3,2}                                                   [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1,3,2}: MAYBE
             -------------------

             The usable rules for this path are:

               {  a() -> b()
                , activate(X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(activate) = {1},
                 Uargs(h^#) = {1}, Uargs(c_0) = {1}, Uargs(g^#) = {1, 2},
                 Uargs(c_1) = {1}, Uargs(f^#) = {2}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                a() = [3]
                      [3]
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                b() = [1]
                      [1]
                activate(x1) = [1 3] x1 + [3]
                               [3 3]      [0]
                h^#(x1) = [3 3] x1 + [0]
                          [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                g^#(x1, x2) = [1 0] x1 + [2 3] x2 + [0]
                              [3 3]      [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                f^#(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [3 3]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                a^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(1))

             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:
                 {  h^#(X) -> c_0(g^#(X, X))
                  , f^#(X, X) -> c_2(h^#(a()))
                  , g^#(a(), X) -> c_1(f^#(b(), activate(X)))}
               Weak Rules:
                 {  a() -> b()
                  , activate(X) -> X}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

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

             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: {}

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

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

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

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

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

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

             ->{1,3,2}                                                   [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1,3,2}: MAYBE
             -------------------

             The usable rules for this path are:

               {  a() -> b()
                , activate(X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(h) = {}, Uargs(g) = {}, Uargs(f) = {}, Uargs(activate) = {1},
                 Uargs(h^#) = {1}, Uargs(c_0) = {1}, Uargs(g^#) = {1, 2},
                 Uargs(c_1) = {1}, Uargs(f^#) = {2}, Uargs(c_2) = {1},
                 Uargs(activate^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [0] x1 + [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                a() = [3]
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                b() = [1]
                activate(x1) = [1] x1 + [3]
                h^#(x1) = [3] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                g^#(x1, x2) = [1] x1 + [2] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                f^#(x1, x2) = [0] x1 + [2] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                a^#() = [0]
                c_3() = [0]
                activate^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(1))

             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:
                 {  h^#(X) -> c_0(g^#(X, X))
                  , f^#(X, X) -> c_2(h^#(a()))
                  , g^#(a(), X) -> c_1(f^#(b(), activate(X)))}
               Weak Rules:
                 {  a() -> b()
                  , activate(X) -> X}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

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

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

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

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