Problem Strategy outermost added 08 4.15

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.15

stdout:

MAYBE

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

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.15

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.15

stdout:

MAYBE

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

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

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

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

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

             ->{1}                                                       [         NA         ]



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

             The usable rules for this path are:

               {  g(0(), 1()) -> 0()
                , g(0(), 1()) -> 1()
                , h(g(x, y)) -> h(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * 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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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^#(0(), 1()) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                1() = [2]
                      [2]
                      [2]
                g^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [3]
                              [0 0 2]      [2 0 0]      [7]
                              [0 0 0]      [0 2 2]      [7]
                c_1() = [0]
                        [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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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^#(0(), 1()) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                1() = [2]
                      [2]
                      [2]
                g^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [3]
                              [0 0 2]      [2 0 0]      [7]
                              [0 0 0]      [0 2 2]      [7]
                c_2() = [0]
                        [1]
                        [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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h^#(x1) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [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: {h^#(g(x, y)) -> c_3(h^#(x))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g(x1, x2) = [3 4 0] x1 + [0 0 0] x2 + [1]
                            [2 0 0]      [0 0 0]      [3]
                            [0 0 0]      [0 0 0]      [0]
                h^#(x1) = [2 1 0] x1 + [0]
                          [2 1 0]      [2]
                          [3 0 0]      [6]
                c_3(x1) = [2 2 0] x1 + [0]
                          [0 0 0]      [3]
                          [0 2 0]      [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(), g(x, y), z) ->
                   c_0(f^#(g(x, y), g(x, y), g(x, y), h(x)))
              , 2: g^#(0(), 1()) -> c_1()
              , 3: g^#(0(), 1()) -> c_2()
              , 4: h^#(g(x, y)) -> c_3(h^#(x))}

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

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0(), 1()) -> 0()
                , g(0(), 1()) -> 1()
                , h(g(x, y)) -> h(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

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

             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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 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]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 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]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(0(), 1()) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                1() = [2]
                      [0]
                g^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
                              [0 0]      [0 0]      [3]
                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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 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]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 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]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(0(), 1()) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                1() = [2]
                      [0]
                g^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
                              [0 0]      [0 0]      [3]
                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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 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]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 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]
                h^#(x1) = [0 0] x1 + [0]
                          [3 3]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [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: {h^#(g(x, y)) -> c_3(h^#(x))}
               Weak Rules: {}

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

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

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0(), 1()) -> 0()
                , g(0(), 1()) -> 1()
                , h(g(x, y)) -> h(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

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

             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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                0() = [0]
                1() = [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                h(x1) = [0] x1 + [0]
                f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [0]
                h^#(x1) = [0] x1 + [0]
                c_3(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(0(), 1()) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                1() = [2]
                g^#(x1, x2) = [2] x1 + [2] 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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                0() = [0]
                1() = [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                h(x1) = [0] x1 + [0]
                f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [0]
                h^#(x1) = [0] x1 + [0]
                c_3(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(0(), 1()) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                1() = [2]
                g^#(x1, x2) = [2] x1 + [2] x2 + [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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                0() = [0]
                1() = [0]
                g(x1, x2) = [2] x1 + [0] x2 + [0]
                h(x1) = [0] x1 + [0]
                f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [0]
                h^#(x1) = [3] x1 + [0]
                c_3(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {h^#(g(x, y)) -> c_3(h^#(x))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g(x1, x2) = [2] x1 + [0] x2 + [4]
                h^#(x1) = [2] x1 + [0]
                c_3(x1) = [2] x1 + [7]

    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 4.15

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.15

stdout:

MAYBE

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

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

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

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

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

             ->{1}                                                       [         NA         ]



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

             The usable rules for this path are:

               {  g(0(), 1()) -> 0()
                , g(0(), 1()) -> 1()
                , h(g(x, y)) -> h(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * 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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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^#(0(), 1()) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                1() = [2]
                      [2]
                      [2]
                g^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [3]
                              [0 0 2]      [2 0 0]      [7]
                              [0 0 0]      [0 2 2]      [7]
                c_1() = [0]
                        [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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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^#(0(), 1()) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                1() = [2]
                      [2]
                      [2]
                g^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [3]
                              [0 0 2]      [2 0 0]      [7]
                              [0 0 0]      [0 2 2]      [7]
                c_2() = [0]
                        [1]
                        [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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [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 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]
                h^#(x1) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [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: {h^#(g(x, y)) -> c_3(h^#(x))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g(x1, x2) = [3 4 0] x1 + [0 0 0] x2 + [1]
                            [2 0 0]      [0 0 0]      [3]
                            [0 0 0]      [0 0 0]      [0]
                h^#(x1) = [2 1 0] x1 + [0]
                          [2 1 0]      [2]
                          [3 0 0]      [6]
                c_3(x1) = [2 2 0] x1 + [0]
                          [0 0 0]      [3]
                          [0 2 0]      [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(), g(x, y), z) ->
                   c_0(f^#(g(x, y), g(x, y), g(x, y), h(x)))
              , 2: g^#(0(), 1()) -> c_1()
              , 3: g^#(0(), 1()) -> c_2()
              , 4: h^#(g(x, y)) -> c_3(h^#(x))}

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

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0(), 1()) -> 0()
                , g(0(), 1()) -> 1()
                , h(g(x, y)) -> h(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

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

             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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 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]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 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]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(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^#(0(), 1()) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                1() = [2]
                      [0]
                g^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
                              [0 0]      [0 0]      [3]
                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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 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]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 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]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(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^#(0(), 1()) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                1() = [2]
                      [0]
                g^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
                              [0 0]      [0 0]      [3]
                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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 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]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 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]
                h^#(x1) = [0 0] x1 + [0]
                          [3 3]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [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: {h^#(g(x, y)) -> c_3(h^#(x))}
               Weak Rules: {}

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

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

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

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0(), 1()) -> 0()
                , g(0(), 1()) -> 1()
                , h(g(x, y)) -> h(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

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

             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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                0() = [0]
                1() = [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                h(x1) = [0] x1 + [0]
                f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [0]
                h^#(x1) = [0] x1 + [0]
                c_3(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^#(0(), 1()) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                1() = [2]
                g^#(x1, x2) = [2] x1 + [2] 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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                0() = [0]
                1() = [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                h(x1) = [0] x1 + [0]
                f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [0]
                h^#(x1) = [0] x1 + [0]
                c_3(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^#(0(), 1()) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                1() = [2]
                g^#(x1, x2) = [2] x1 + [2] x2 + [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(h) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                0() = [0]
                1() = [0]
                g(x1, x2) = [2] x1 + [0] x2 + [0]
                h(x1) = [0] x1 + [0]
                f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2() = [0]
                h^#(x1) = [3] x1 + [0]
                c_3(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: {h^#(g(x, y)) -> c_3(h^#(x))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(g) = {}, Uargs(h^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                g(x1, x2) = [2] x1 + [0] x2 + [4]
                h^#(x1) = [2] x1 + [0]
                c_3(x1) = [2] x1 + [7]

    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.