Problem Mixed innermost innermost5

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost5

stdout:

MAYBE

Problem:
f(s(x),y,b()) -> f(g(h(x)),y,i(y))
g(h(x)) -> g(x)
g(s(x)) -> s(x)
g(0()) -> s(0())
h(0()) -> a()
i(0()) -> b()
i(s(y)) -> i(y)

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost5

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost5

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(s(x), y, b()) -> f(g(h(x)), y, i(y))
     , g(h(x)) -> g(x)
     , g(s(x)) -> s(x)
     , g(0()) -> s(0())
     , h(0()) -> a()
     , i(0()) -> b()
     , i(s(y)) -> i(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^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
              , 2: g^#(h(x)) -> c_1(g^#(x))
              , 3: g^#(s(x)) -> c_2()
              , 4: g^#(0()) -> c_3()
              , 5: h^#(0()) -> c_4()
              , 6: i^#(0()) -> c_5()
              , 7: i^#(s(y)) -> c_6(i^#(y))}

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

             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]

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

             ->{2}                                                       [    YES(?,O(1))     ]
                |
                |->{3}                                                   [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

             ->{1}                                                       [         NA         ]



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

             The usable rules for this path are:

               {  g(h(x)) -> g(x)
                , g(s(x)) -> s(x)
                , g(0()) -> s(0())
                , h(0()) -> a()
                , i(0()) -> b()
                , i(s(y)) -> i(y)}

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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^#(h(x)) -> c_1(g^#(x))}
               Weak Rules: {}

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

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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^#(s(x)) -> c_2()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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()) -> c_3()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {5}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [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() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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: {h^#(0()) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(h^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                h^#(x1) = [0 2 0] x1 + [7]
                          [2 2 0]      [3]
                          [2 2 2]      [3]
                c_4() = [0]
                        [1]
                        [1]

           * Path {7}: YES(?,O(n^2))
             -----------------------

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               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]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [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() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {i^#(s(y)) -> c_6(i^#(y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [0]
                i^#(x1) = [0 1 0] x1 + [2]
                          [6 0 0]      [0]
                          [2 3 0]      [2]
                c_6(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]

           * Path {7}->{6}: YES(?,O(n^2))
             ----------------------------

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [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() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {i^#(0()) -> c_5()}
               Weak Rules: {i^#(s(y)) -> c_6(i^#(y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                i^#(x1) = [2 2 2] x1 + [0]
                          [0 6 0]      [0]
                          [0 0 2]      [0]
                c_5() = [1]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]

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

             {  1: f^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
              , 2: g^#(h(x)) -> c_1(g^#(x))
              , 3: g^#(s(x)) -> c_2()
              , 4: g^#(0()) -> c_3()
              , 5: h^#(0()) -> c_4()
              , 6: i^#(0()) -> c_5()
              , 7: i^#(s(y)) -> c_6(i^#(y))}

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

             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]

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

             ->{2}                                                       [    YES(?,O(1))     ]
                |
                |->{3}                                                   [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(h(x)) -> g(x)
                , g(s(x)) -> s(x)
                , g(0()) -> s(0())
                , h(0()) -> a()
                , i(0()) -> b()
                , i(s(y)) -> i(y)}

             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^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
                  , g(h(x)) -> g(x)
                  , g(s(x)) -> s(x)
                  , g(0()) -> s(0())
                  , h(0()) -> a()
                  , i(0()) -> b()
                  , i(s(y)) -> i(y)}

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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^#(h(x)) -> c_1(g^#(x))}
               Weak Rules: {}

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

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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^#(s(x)) -> c_2()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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()) -> c_3()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {5}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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: {h^#(0()) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(h^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                h^#(x1) = [2 0] x1 + [7]
                          [2 2]      [7]
                c_4() = [0]
                        [1]

           * Path {7}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               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]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [3 3] x1 + [0]
                          [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6(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(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {i^#(s(y)) -> c_6(i^#(y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                i^#(x1) = [0 1] x1 + [1]
                          [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 0]      [0]

           * Path {7}->{6}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {i^#(0()) -> c_5()}
               Weak Rules: {i^#(s(y)) -> c_6(i^#(y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                i^#(x1) = [1 2] x1 + [2]
                          [6 1]      [0]
                c_5() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [5]
                          [2 0]      [3]

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

             {  1: f^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
              , 2: g^#(h(x)) -> c_1(g^#(x))
              , 3: g^#(s(x)) -> c_2()
              , 4: g^#(0()) -> c_3()
              , 5: h^#(0()) -> c_4()
              , 6: i^#(0()) -> c_5()
              , 7: i^#(s(y)) -> c_6(i^#(y))}

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

             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]

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

             ->{2}                                                       [    YES(?,O(1))     ]
                |
                |->{3}                                                   [    YES(?,O(1))     ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(h(x)) -> g(x)
                , g(s(x)) -> s(x)
                , g(0()) -> s(0())
                , h(0()) -> a()
                , i(0()) -> b()
                , i(s(y)) -> i(y)}

             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^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
                  , g(h(x)) -> g(x)
                  , g(s(x)) -> s(x)
                  , g(0()) -> s(0())
                  , h(0()) -> a()
                  , i(0()) -> b()
                  , i(s(y)) -> i(y)}

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [2] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [3] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(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^#(h(x)) -> c_1(g^#(x))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(h) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [2] x1 + [4]
                g^#(x1) = [2] x1 + [0]
                c_1(x1) = [2] x1 + [7]

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(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^#(s(x)) -> c_2()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(h) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [1]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(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()) -> c_3()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {5}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(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: {h^#(0()) -> c_4()}
               Weak Rules: {}

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

           * Path {7}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [1] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [3] x1 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {i^#(s(y)) -> c_6(i^#(y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                i^#(x1) = [2] x1 + [0]
                c_6(x1) = [1] x1 + [7]

           * Path {7}->{6}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {i^#(0()) -> c_5()}
               Weak Rules: {i^#(s(y)) -> c_6(i^#(y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                i^#(x1) = [2] x1 + [0]
                c_5() = [1]
                c_6(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.

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost5

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost5

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(s(x), y, b()) -> f(g(h(x)), y, i(y))
     , g(h(x)) -> g(x)
     , g(s(x)) -> s(x)
     , g(0()) -> s(0())
     , h(0()) -> a()
     , i(0()) -> b()
     , i(s(y)) -> i(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^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
              , 2: g^#(h(x)) -> c_1(g^#(x))
              , 3: g^#(s(x)) -> c_2(x)
              , 4: g^#(0()) -> c_3()
              , 5: h^#(0()) -> c_4()
              , 6: i^#(0()) -> c_5()
              , 7: i^#(s(y)) -> c_6(i^#(y))}

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

             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]

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

             ->{2}                                                       [    YES(?,O(1))     ]
                |
                |->{3}                                                   [   YES(?,O(n^3))    ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

             ->{1}                                                       [         NA         ]



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

             The usable rules for this path are:

               {  g(h(x)) -> g(x)
                , g(s(x)) -> s(x)
                , g(0()) -> s(0())
                , h(0()) -> a()
                , i(0()) -> b()
                , i(s(y)) -> i(y)}

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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^#(h(x)) -> c_1(g^#(x))}
               Weak Rules: {}

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

           * Path {2}->{3}: YES(?,O(n^3))
             ----------------------------

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [1 1 1] x1 + [0]
                        [0 1 3]      [0]
                        [0 0 1]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [3 1 3] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(s(x)) -> c_2(x)}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(h) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1},
                 Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [0]
                        [0 0 2]      [2]
                        [0 0 0]      [2]
                h(x1) = [2 2 5] x1 + [4]
                        [2 2 0]      [0]
                        [2 2 0]      [0]
                g^#(x1) = [2 3 0] x1 + [2]
                          [2 2 2]      [0]
                          [2 1 3]      [0]
                c_1(x1) = [2 0 0] x1 + [3]
                          [0 0 0]      [7]
                          [2 2 2]      [3]
                c_2(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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()) -> c_3()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {5}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [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]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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: {h^#(0()) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(h^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                h^#(x1) = [0 2 0] x1 + [7]
                          [2 2 0]      [3]
                          [2 2 2]      [3]
                c_4() = [0]
                        [1]
                        [1]

           * Path {7}: YES(?,O(n^2))
             -----------------------

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {1}
               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]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [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]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {i^#(s(y)) -> c_6(i^#(y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [0]
                i^#(x1) = [0 1 0] x1 + [2]
                          [6 0 0]      [0]
                          [2 3 0]      [2]
                c_6(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]

           * Path {7}->{6}: YES(?,O(n^2))
             ----------------------------

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {1}
               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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                h(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                i(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                a() = [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) = [0 0 0] x1 + [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]
                c_3() = [0]
                        [0]
                        [0]
                h^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                i^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(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(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {i^#(0()) -> c_5()}
               Weak Rules: {i^#(s(y)) -> c_6(i^#(y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                i^#(x1) = [2 2 2] x1 + [0]
                          [0 6 0]      [0]
                          [0 0 2]      [0]
                c_5() = [1]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]

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

             {  1: f^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
              , 2: g^#(h(x)) -> c_1(g^#(x))
              , 3: g^#(s(x)) -> c_2(x)
              , 4: g^#(0()) -> c_3()
              , 5: h^#(0()) -> c_4()
              , 6: i^#(0()) -> c_5()
              , 7: i^#(s(y)) -> c_6(i^#(y))}

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

             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]

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

             ->{2}                                                       [    YES(?,O(1))     ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(h(x)) -> g(x)
                , g(s(x)) -> s(x)
                , g(0()) -> s(0())
                , h(0()) -> a()
                , i(0()) -> b()
                , i(s(y)) -> i(y)}

             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^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
                  , g(h(x)) -> g(x)
                  , g(s(x)) -> s(x)
                  , g(0()) -> s(0())
                  , h(0()) -> a()
                  , i(0()) -> b()
                  , i(s(y)) -> i(y)}

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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^#(h(x)) -> c_1(g^#(x))}
               Weak Rules: {}

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

           * Path {2}->{3}: YES(?,O(n^2))
             ----------------------------

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [1 3] x1 + [0]
                        [0 1]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [1 3] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(s(x)) -> c_2(x)}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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()) -> c_3()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {5}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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: {h^#(0()) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(h^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                h^#(x1) = [2 0] x1 + [7]
                          [2 2]      [7]
                c_4() = [0]
                        [1]

           * Path {7}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {1}
               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]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [3 3] x1 + [0]
                          [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6(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(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {i^#(s(y)) -> c_6(i^#(y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                i^#(x1) = [0 1] x1 + [1]
                          [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 0]      [0]

           * Path {7}->{6}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {1}
               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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                h(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                i(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                a() = [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) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [0]
                        [0]
                h^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                i^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(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(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {i^#(0()) -> c_5()}
               Weak Rules: {i^#(s(y)) -> c_6(i^#(y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                i^#(x1) = [1 2] x1 + [2]
                          [6 1]      [0]
                c_5() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [5]
                          [2 0]      [3]

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

             {  1: f^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
              , 2: g^#(h(x)) -> c_1(g^#(x))
              , 3: g^#(s(x)) -> c_2(x)
              , 4: g^#(0()) -> c_3()
              , 5: h^#(0()) -> c_4()
              , 6: i^#(0()) -> c_5()
              , 7: i^#(s(y)) -> c_6(i^#(y))}

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

             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]

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

             ->{2}                                                       [    YES(?,O(1))     ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(h(x)) -> g(x)
                , g(s(x)) -> s(x)
                , g(0()) -> s(0())
                , h(0()) -> a()
                , i(0()) -> b()
                , i(s(y)) -> i(y)}

             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^#(s(x), y, b()) -> c_0(f^#(g(h(x)), y, i(y)))
                  , g(h(x)) -> g(x)
                  , g(s(x)) -> s(x)
                  , g(0()) -> s(0())
                  , h(0()) -> a()
                  , i(0()) -> b()
                  , i(s(y)) -> i(y)}

             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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [2] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [3] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(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^#(h(x)) -> c_1(g^#(x))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(h) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                h(x1) = [2] x1 + [4]
                g^#(x1) = [2] x1 + [0]
                c_1(x1) = [2] x1 + [7]

           * Path {2}->{3}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [1] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [3] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {g^#(s(x)) -> c_2(x)}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {2}->{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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {1}, Uargs(c_2) = {}, Uargs(h^#) = {},
                 Uargs(i^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(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()) -> c_3()}
               Weak Rules: {g^#(h(x)) -> c_1(g^#(x))}

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

           * Path {5}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(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: {h^#(0()) -> c_4()}
               Weak Rules: {}

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

           * Path {7}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [1] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [3] x1 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {i^#(s(y)) -> c_6(i^#(y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                i^#(x1) = [2] x1 + [0]
                c_6(x1) = [1] x1 + [7]

           * Path {7}->{6}: YES(?,O(n^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(s) = {}, Uargs(g) = {}, Uargs(h) = {},
                 Uargs(i) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(h^#) = {}, Uargs(i^#) = {},
                 Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1) = [0] x1 + [0]
                h(x1) = [0] x1 + [0]
                i(x1) = [0] x1 + [0]
                0() = [0]
                a() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3() = [0]
                h^#(x1) = [0] x1 + [0]
                c_4() = [0]
                i^#(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {i^#(0()) -> c_5()}
               Weak Rules: {i^#(s(y)) -> c_6(i^#(y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(i^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                i^#(x1) = [2] x1 + [0]
                c_5() = [1]
                c_6(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.