Problem Strategy outermost added 08 toyama

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 toyama

stdout:

MAYBE

Problem:
f(0(),1(),X) -> f(X,X,X)
g(X,Y) -> X
g(X,Y) -> Y

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 toyama

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 toyama

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)
     , g(X, Y) -> X
     , g(X, Y) -> Y}

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: g^#(X, Y) -> c_1()
              , 3: g^#(X, Y) -> c_2()}

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(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]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                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]
                g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [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 {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(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]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [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) -> c_1()}
               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) = [0 0 0] x1 + [0 0 0] x2 + [7]
                              [0 0 0]      [0 0 0]      [7]
                              [0 0 0]      [0 0 0]      [7]
                c_1() = [0]
                        [3]
                        [3]

           * 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]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [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) -> c_2()}
               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) = [0 0 0] x1 + [0 0 0] x2 + [7]
                              [0 0 0]      [0 0 0]      [7]
                              [0 0 0]      [0 0 0]      [7]
                c_2() = [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: g^#(X, Y) -> c_1()
              , 3: g^#(X, Y) -> c_2()}

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(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]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
                g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [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 {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(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]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
                g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [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) -> c_1()}
               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) = [0 0] x1 + [0 0] x2 + [7]
                              [0 0]      [0 0]      [7]
                c_1() = [0]
                        [1]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(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]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
                g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [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) -> c_2()}
               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) = [0 0] x1 + [0 0] x2 + [7]
                              [0 0]      [0 0]      [7]
                c_2() = [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: g^#(X, Y) -> c_1()
              , 3: g^#(X, Y) -> c_2()}

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(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]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [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 {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(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]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [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) -> c_1()}
               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) = [0] x1 + [0] x2 + [7]
                c_1() = [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(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]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [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) -> c_2()}
               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) = [0] x1 + [0] x2 + [7]
                c_2() = [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 toyama

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 toyama

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)
     , g(X, Y) -> X
     , g(X, Y) -> Y}

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: g^#(X, Y) -> c_1(X)
              , 3: g^#(X, Y) -> c_2(Y)}

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                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]
                g^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(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 {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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) -> c_1(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_1) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2) = [7 7 7] x1 + [0 0 0] x2 + [7]
                              [7 7 7]      [0 0 0]      [7]
                              [7 7 7]      [0 0 0]      [7]
                c_1(x1) = [3 3 3] x1 + [0]
                          [3 1 3]      [1]
                          [1 1 1]      [1]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(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) -> c_2(Y)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2) = [0 0 0] x1 + [7 7 7] x2 + [7]
                              [0 0 0]      [7 7 7]      [7]
                              [0 0 0]      [7 7 7]      [7]
                c_2(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: g^#(X, Y) -> c_1(X)
              , 3: g^#(X, Y) -> c_2(Y)}

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
                g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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 {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
                g^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_2(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, Y) -> c_1(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_1) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2) = [7 7] x1 + [0 0] x2 + [7]
                              [7 7]      [0 0]      [7]
                c_1(x1) = [1 3] x1 + [0]
                          [3 1]      [3]

           * 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_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [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]
                g^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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) -> c_2(Y)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2) = [0 0] x1 + [7 7] x2 + [7]
                              [0 0]      [7 7]      [7]
                c_2(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: g^#(X, Y) -> c_1(X)
              , 3: g^#(X, Y) -> c_2(Y)}

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
                 Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(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 {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2(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, Y) -> c_1(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_1) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2) = [7] x1 + [0] x2 + [7]
                c_1(x1) = [1] x1 + [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {},
                 Uargs(g^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
               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]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(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) -> c_2(Y)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g^#(x1, x2) = [0] x1 + [7] x2 + [7]
                c_2(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.