Problem Strategy outermost added 08 4.13

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.13

stdout:

MAYBE

Problem:
f(0(),1(),x) -> f(x,x,x)
f(x,y,z) -> 2()
0() -> 2()
1() -> 2()
g(x,x,y) -> y
g(x,y,y) -> x

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.13

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.13

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(0(), 1(), x) -> f(x, x, x)
     , f(x, y, z) -> 2()
     , 0() -> 2()
     , 1() -> 2()
     , g(x, x, y) -> y
     , g(x, y, y) -> 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^#(0(), 1(), x) -> c_0(f^#(x, x, x))
              , 2: f^#(x, y, z) -> c_1()
              , 3: 0^#() -> c_2()
              , 4: 1^#() -> c_3()
              , 5: g^#(x, x, y) -> c_4()
              , 6: g^#(x, y, y) -> c_5()}

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

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

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

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

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

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



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] x3 + [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(0(), 1(), x) -> c_0(f^#(x, x, x))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]

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

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {0^#() -> c_2()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]

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

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

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1^#() = [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(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [7]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [7]
                c_4() = [0]
                        [3]
                        [3]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(x, y, y) -> c_5()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [7]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [7]
                c_5() = [0]
                        [3]
                        [3]

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

             {  1: f^#(0(), 1(), x) -> c_0(f^#(x, x, x))
              , 2: f^#(x, y, z) -> c_1()
              , 3: 0^#() -> c_2()
              , 4: 1^#() -> c_3()
              , 5: g^#(x, x, y) -> c_4()
              , 6: g^#(x, y, y) -> c_5()}

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

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

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

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

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

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



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
                                  [3 3]      [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(0(), 1(), x) -> c_0(f^#(x, x, x))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]

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

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {0^#() -> c_2()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]

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

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

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1^#() = [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(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(x, x, y) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [7]
                                  [0 0]      [0 0]      [0 0]      [7]
                c_4() = [0]
                        [1]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(x, y, y) -> c_5()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [7]
                                  [0 0]      [0 0]      [0 0]      [7]
                c_5() = [0]
                        [1]

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

             {  1: f^#(0(), 1(), x) -> c_0(f^#(x, x, x))
              , 2: f^#(x, y, z) -> c_1()
              , 3: 0^#() -> c_2()
              , 4: 1^#() -> c_3()
              , 5: g^#(x, x, y) -> c_4()
              , 6: g^#(x, y, y) -> c_5()}

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

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

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

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

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

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



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {f^#(0(), 1(), x) -> c_0(f^#(x, x, x))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]

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

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {0^#() -> c_2()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]

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

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

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1^#() = [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(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [7]
                c_4() = [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [7]
                c_5() = [0]

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.13

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.13

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(0(), 1(), x) -> f(x, x, x)
     , f(x, y, z) -> 2()
     , 0() -> 2()
     , 1() -> 2()
     , g(x, x, y) -> y
     , g(x, y, y) -> 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^#(0(), 1(), x) -> c_0(f^#(x, x, x))
              , 2: f^#(x, y, z) -> c_1()
              , 3: 0^#() -> c_2()
              , 4: 1^#() -> c_3()
              , 5: g^#(x, x, y) -> c_4(y)
              , 6: g^#(x, y, y) -> c_5(x)}

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

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

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

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

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

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



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [3 3 3] x3 + [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(0(), 1(), x) -> c_0(f^#(x, x, x))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {0^#() -> c_2()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

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

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1^#() = [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(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [3 3 3] x1 + [3 3 3] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(x, x, y) -> c_4(y)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [7 7 7] x3 + [7]
                                  [0 0 0]      [0 0 0]      [7 7 7]      [7]
                                  [0 0 0]      [0 0 0]      [7 7 7]      [7]
                c_4(x1) = [3 3 3] x1 + [0]
                          [3 1 3]      [1]
                          [1 1 1]      [1]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                1() = [0]
                      [0]
                      [0]
                2() = [0]
                      [0]
                      [0]
                g(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                0^#() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                1^#() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                g^#(x1, x2, x3) = [3 3 3] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(x, y, y) -> c_5(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [7 7 7] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
                                  [7 7 7]      [0 0 0]      [0 0 0]      [7]
                                  [7 7 7]      [0 0 0]      [0 0 0]      [7]
                c_5(x1) = [3 3 3] x1 + [0]
                          [3 1 3]      [1]
                          [1 1 1]      [1]

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

             {  1: f^#(0(), 1(), x) -> c_0(f^#(x, x, x))
              , 2: f^#(x, y, z) -> c_1()
              , 3: 0^#() -> c_2()
              , 4: 1^#() -> c_3()
              , 5: g^#(x, x, y) -> c_4(y)
              , 6: g^#(x, y, y) -> c_5(x)}

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

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

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

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

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

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



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [3 3] x3 + [0]
                                  [3 3]      [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(0(), 1(), x) -> c_0(f^#(x, x, x))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {0^#() -> c_2()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

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

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1^#() = [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(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [3 3] x1 + [3 3] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(x, x, y) -> c_4(y)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [7 7] x3 + [7]
                                  [0 0]      [0 0]      [7 7]      [7]
                c_4(x1) = [1 3] x1 + [0]
                          [3 1]      [3]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                1() = [0]
                      [0]
                2() = [0]
                      [0]
                g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                0^#() = [0]
                        [0]
                c_2() = [0]
                        [0]
                1^#() = [0]
                        [0]
                c_3() = [0]
                        [0]
                g^#(x1, x2, x3) = [3 3] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 1] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(x, y, y) -> c_5(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [7 7] x1 + [0 0] x2 + [0 0] x3 + [7]
                                  [7 7]      [0 0]      [0 0]      [7]
                c_5(x1) = [1 3] x1 + [0]
                          [3 1]      [3]

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

             {  1: f^#(0(), 1(), x) -> c_0(f^#(x, x, x))
              , 2: f^#(x, y, z) -> c_1()
              , 3: 0^#() -> c_2()
              , 4: 1^#() -> c_3()
              , 5: g^#(x, x, y) -> c_4(y)
              , 6: g^#(x, y, y) -> c_5(x)}

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

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

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

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

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

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



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(0(), 1(), x) -> c_0(f^#(x, x, x))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {0^#() -> c_2()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

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

             Proof Output:
               The following argument positions are usable:
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1^#() = [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(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [3] x1 + [3] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [0] x1 + [0] x2 + [7] x3 + [7]
                c_4(x1) = [1] x1 + [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                0() = [0]
                1() = [0]
                2() = [0]
                g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                0^#() = [0]
                c_2() = [0]
                1^#() = [0]
                c_3() = [0]
                g^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2, x3) = [7] x1 + [0] x2 + [0] x3 + [7]
                c_5(x1) = [1] x1 + [0]

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

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

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