Problem Transformed outermost 08 ex5.6

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed outermost 08 ex5.6

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed outermost 08 ex5.6

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
     , f_1(x, i_0(g_1(x))) -> a_0()
     , f_1(x, y) -> x
     , f_1(x, i_0(x)) -> f_1(x, x)
     , g_1(x) -> i_0(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_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
              , 2: f_1^#(x, i_0(g_1(x))) -> c_1()
              , 3: f_1^#(x, y) -> c_2()
              , 4: f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))
              , 5: g_1^#(x) -> c_4()}

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

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

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



         Sub-problems:
         -------------
           * Path {1,4}: NA
             --------------

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                i_0(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [1]
                          [0 0 0]      [1]
                g_1(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [3]
                          [0 0 0]      [3]
                a_0() = [0]
                        [0]
                        [0]
                f_1^#(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                g_1^#(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(n^1))

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

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                i_0(x1) = [1 1 1] x1 + [0]
                          [0 1 1]      [1]
                          [0 0 1]      [1]
                g_1(x1) = [3 3 3] x1 + [3]
                          [0 3 3]      [3]
                          [0 0 3]      [3]
                a_0() = [0]
                        [0]
                        [0]
                f_1^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
                                [0 0 0]      [0 0 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]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                g_1^#(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(n^3))

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

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                i_0(x1) = [1 1 1] x1 + [0]
                          [0 1 1]      [1]
                          [0 0 1]      [1]
                g_1(x1) = [3 3 3] x1 + [3]
                          [0 3 3]      [3]
                          [0 0 3]      [3]
                a_0() = [0]
                        [0]
                        [0]
                f_1^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
                                [0 0 0]      [0 0 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]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                g_1^#(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(n^3))

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

           * 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(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {},
                 Uargs(f_1^#) = {}, Uargs(c_0) = {}, Uargs(c_3) = {},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                i_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a_0() = [0]
                        [0]
                        [0]
                f_1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g_1^#(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: {g_1^#(x) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g_1^#(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: f_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
              , 2: f_1^#(x, i_0(g_1(x))) -> c_1()
              , 3: f_1^#(x, y) -> c_2()
              , 4: f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))
              , 5: g_1^#(x) -> c_4()}

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

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

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



         Sub-problems:
         -------------
           * Path {1,4}: MAYBE
             -----------------

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                i_0(x1) = [1 0] x1 + [0]
                          [0 0]      [1]
                g_1(x1) = [1 0] x1 + [3]
                          [0 0]      [3]
                a_0() = [0]
                        [0]
                f_1^#(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                g_1^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(n^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:
                 {  f_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
                  , f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))}
               Weak Rules: {g_1(x) -> i_0(x)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                i_0(x1) = [1 1] x1 + [0]
                          [0 1]      [1]
                g_1(x1) = [3 3] x1 + [3]
                          [0 3]      [3]
                a_0() = [0]
                        [0]
                f_1^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                g_1^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                i_0(x1) = [1 1] x1 + [0]
                          [0 1]      [1]
                g_1(x1) = [3 3] x1 + [3]
                          [0 3]      [3]
                a_0() = [0]
                        [0]
                f_1^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                g_1^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_4() = [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * 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(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {},
                 Uargs(f_1^#) = {}, Uargs(c_0) = {}, Uargs(c_3) = {},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                i_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a_0() = [0]
                        [0]
                f_1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g_1^#(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: {g_1^#(x) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g_1^#(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: f_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
              , 2: f_1^#(x, i_0(g_1(x))) -> c_1()
              , 3: f_1^#(x, y) -> c_2()
              , 4: f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))
              , 5: g_1^#(x) -> c_4()}

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

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

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



         Sub-problems:
         -------------
           * Path {1,4}: MAYBE
             -----------------

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0] x1 + [0] x2 + [0]
                i_0(x1) = [1] x1 + [1]
                g_1(x1) = [1] x1 + [3]
                a_0() = [0]
                f_1^#(x1, x2) = [0] x1 + [1] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                g_1^#(x1) = [0] x1 + [0]
                c_4() = [0]
             Complexity induced by the adequate RMI: YES(?,O(n^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:
                 {  f_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
                  , f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))}
               Weak Rules: {g_1(x) -> i_0(x)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0] x1 + [0] x2 + [0]
                i_0(x1) = [1] x1 + [0]
                g_1(x1) = [3] x1 + [3]
                a_0() = [0]
                f_1^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                g_1^#(x1) = [0] x1 + [0]
                c_4() = [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_3) = {1},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0] x1 + [0] x2 + [0]
                i_0(x1) = [1] x1 + [0]
                g_1(x1) = [3] x1 + [3]
                a_0() = [0]
                f_1^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                g_1^#(x1) = [0] x1 + [0]
                c_4() = [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * 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(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {},
                 Uargs(f_1^#) = {}, Uargs(c_0) = {}, Uargs(c_3) = {},
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0] x1 + [0] x2 + [0]
                i_0(x1) = [0] x1 + [0]
                g_1(x1) = [0] x1 + [0]
                a_0() = [0]
                f_1^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                g_1^#(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: {g_1^#(x) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g_1^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g_1^#(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	Transformed outermost 08 ex5.6

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed outermost 08 ex5.6

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f_1(x, x) -> f_1(i_0(x), g_1(g_1(x)))
     , f_1(x, i_0(g_1(x))) -> a_0()
     , f_1(x, y) -> x
     , f_1(x, i_0(x)) -> f_1(x, x)
     , g_1(x) -> i_0(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_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
              , 2: f_1^#(x, i_0(g_1(x))) -> c_1()
              , 3: f_1^#(x, y) -> c_2(x)
              , 4: f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))
              , 5: g_1^#(x) -> c_4(x)}

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

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

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



         Sub-problems:
         -------------
           * Path {1,4}: NA
             --------------

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                i_0(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [1]
                          [0 0 0]      [1]
                g_1(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [3]
                          [0 0 0]      [3]
                a_0() = [0]
                        [0]
                        [0]
                f_1^#(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                g_1^#(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(n^1))

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

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                i_0(x1) = [1 1 1] x1 + [0]
                          [0 1 1]      [1]
                          [0 0 1]      [1]
                g_1(x1) = [3 3 3] x1 + [3]
                          [0 3 3]      [3]
                          [0 0 3]      [3]
                a_0() = [0]
                        [0]
                        [0]
                f_1^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
                                [0 0 0]      [0 0 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]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                g_1^#(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(n^3))

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

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                i_0(x1) = [1 1 1] x1 + [0]
                          [0 1 1]      [1]
                          [0 0 1]      [1]
                g_1(x1) = [3 3 3] x1 + [3]
                          [0 3 3]      [3]
                          [0 0 3]      [3]
                a_0() = [0]
                        [0]
                        [0]
                f_1^#(x1, x2) = [3 3 3] x1 + [3 0 0] x2 + [0]
                                [0 0 0]      [0 0 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]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 3 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                g_1^#(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(n^3))

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

           * 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(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {},
                 Uargs(f_1^#) = {}, Uargs(c_0) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                i_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                a_0() = [0]
                        [0]
                        [0]
                f_1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g_1^#(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: {g_1^#(x) -> c_4(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g_1^#(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: f_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
              , 2: f_1^#(x, i_0(g_1(x))) -> c_1()
              , 3: f_1^#(x, y) -> c_2(x)
              , 4: f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))
              , 5: g_1^#(x) -> c_4(x)}

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

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

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



         Sub-problems:
         -------------
           * Path {1,4}: MAYBE
             -----------------

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                i_0(x1) = [1 0] x1 + [0]
                          [0 0]      [1]
                g_1(x1) = [1 0] x1 + [3]
                          [0 0]      [3]
                a_0() = [0]
                        [0]
                f_1^#(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                g_1^#(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(n^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:
                 {  f_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
                  , f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))}
               Weak Rules: {g_1(x) -> i_0(x)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                i_0(x1) = [1 1] x1 + [0]
                          [0 1]      [1]
                g_1(x1) = [3 3] x1 + [3]
                          [0 3]      [3]
                a_0() = [0]
                        [0]
                f_1^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                g_1^#(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(n^2))

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

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                i_0(x1) = [1 1] x1 + [0]
                          [0 1]      [1]
                g_1(x1) = [3 3] x1 + [3]
                          [0 3]      [3]
                a_0() = [0]
                        [0]
                f_1^#(x1, x2) = [3 3] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 3] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                g_1^#(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(n^2))

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

           * 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(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {},
                 Uargs(f_1^#) = {}, Uargs(c_0) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                i_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a_0() = [0]
                        [0]
                f_1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g_1^#(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: {g_1^#(x) -> c_4(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g_1^#(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: f_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
              , 2: f_1^#(x, i_0(g_1(x))) -> c_1()
              , 3: f_1^#(x, y) -> c_2(x)
              , 4: f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))
              , 5: g_1^#(x) -> c_4(x)}

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

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

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



         Sub-problems:
         -------------
           * Path {1,4}: MAYBE
             -----------------

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0] x1 + [0] x2 + [0]
                i_0(x1) = [1] x1 + [1]
                g_1(x1) = [1] x1 + [3]
                a_0() = [0]
                f_1^#(x1, x2) = [0] x1 + [1] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                g_1^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^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:
                 {  f_1^#(x, x) -> c_0(f_1^#(i_0(x), g_1(g_1(x))))
                  , f_1^#(x, i_0(x)) -> c_3(f_1^#(x, x))}
               Weak Rules: {g_1(x) -> i_0(x)}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0] x1 + [0] x2 + [0]
                i_0(x1) = [1] x1 + [0]
                g_1(x1) = [3] x1 + [3]
                a_0() = [0]
                f_1^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                g_1^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

             The usable rules for this path are:

               {g_1(x) -> i_0(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f_1) = {}, Uargs(i_0) = {1}, Uargs(g_1) = {1},
                 Uargs(f_1^#) = {2}, Uargs(c_0) = {1}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0] x1 + [0] x2 + [0]
                i_0(x1) = [1] x1 + [2]
                g_1(x1) = [3] x1 + [3]
                a_0() = [0]
                f_1^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                g_1^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * 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(f_1) = {}, Uargs(i_0) = {}, Uargs(g_1) = {},
                 Uargs(f_1^#) = {}, Uargs(c_0) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f_1(x1, x2) = [0] x1 + [0] x2 + [0]
                i_0(x1) = [0] x1 + [0]
                g_1(x1) = [0] x1 + [0]
                a_0() = [0]
                f_1^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                g_1^#(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: {g_1^#(x) -> c_4(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g_1^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g_1^#(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.