Problem Strategy outermost added 08 Ex4 Zan97

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 Zan97

stdout:

MAYBE

Problem:
f(X) -> cons(X,f(g(X)))
g(0()) -> s(0())
g(s(X)) -> s(s(g(X)))
sel(0(),cons(X,Y)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,Z)

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 Zan97

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 Zan97

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(X) -> cons(X, f(g(X)))
     , g(0()) -> s(0())
     , g(s(X)) -> s(s(g(X)))
     , sel(0(), cons(X, Y)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, Z)}

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^#(X) -> c_0(f^#(g(X)))
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3()
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

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

             ->{5}                                                       [   YES(?,O(n^3))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))}

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(X) -> c_0(f^#(g(X)))
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))}

             Proof Output:
               The input cannot be shown compatible

           * Path {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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [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]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [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() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {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]
                g^#(x1) = [0 1 0] x1 + [2]
                          [6 0 0]      [0]
                          [2 3 0]      [2]
                c_2(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]

           * Path {3}->{2}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [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]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [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() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}

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

           * Path {5}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [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]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [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() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(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^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                s(x1) = [1 4 4] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [2]
                sel^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
                                [2 0 2]      [2 0 0]      [0]
                                [0 2 2]      [1 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [5]
                          [2 0 2]      [3]
                          [0 0 0]      [7]

           * Path {5}->{4}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [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]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [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() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(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: {sel^#(0(), cons(X, Y)) -> c_3()}
               Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 3 0] x2 + [2]
                               [0 0 0]      [0 1 4]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [2]
                sel^#(x1, x2) = [2 2 0] x1 + [2 0 0] x2 + [0]
                                [1 0 2]      [4 2 0]      [0]
                                [4 0 2]      [1 1 0]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [4]
                          [0 0 0]      [7]
                          [0 0 0]      [3]

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

             {  1: f^#(X) -> c_0(f^#(g(X)))
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3()
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

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

             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))}

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

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

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

             Proof Output:
               The input cannot be shown compatible

           * Path {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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1) = [3 3] x1 + [0]
                          [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
               Weak Rules: {}

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

           * Path {3}->{2}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                f^#(x1) = [0 0] x1 + [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() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                g^#(x1) = [1 2] x1 + [2]
                          [6 1]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [5]
                          [2 0]      [3]

           * Path {5}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                f^#(x1) = [0 0] x1 + [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() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(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: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [4]
                s(x1) = [1 2] x1 + [2]
                        [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
                                [4 0]      [4 0]      [0]
                c_4(x1) = [1 0] x1 + [5]
                          [0 0]      [7]

           * Path {5}->{4}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                f^#(x1) = [0 0] x1 + [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() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sel^#(0(), cons(X, Y)) -> c_3()}
               Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 3] x2 + [2]
                               [0 0]      [0 1]      [2]
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                sel^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
                                [2 1]      [2 2]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [7]
                          [0 0]      [6]

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

             {  1: f^#(X) -> c_0(f^#(g(X)))
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3()
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

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

             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [         NA         ]

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))}

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

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

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

             Proof Output:
               The input cannot be shown compatible

           * Path {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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(s(X)) -> c_2(g^#(X))}
               Weak Rules: {}

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

           * Path {3}->{2}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {g^#(0()) -> c_1()}
               Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}

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

           * Path {5}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [1] x1 + [2]
                sel^#(x1, x2) = [3] x1 + [0] x2 + [2]
                c_4(x1) = [1] x1 + [5]

           * Path {5}->{4}: NA
             -----------------

             The usable rules of this path are empty.

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

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

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 Zan97

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex4 Zan97

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(X) -> cons(X, f(g(X)))
     , g(0()) -> s(0())
     , g(s(X)) -> s(s(g(X)))
     , sel(0(), cons(X, Y)) -> X
     , sel(s(X), cons(Y, Z)) -> sel(X, Z)}

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^#(X) -> c_0(X, f^#(g(X)))
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3(X)
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

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

             ->{5}                                                       [   YES(?,O(n^3))    ]
                |
                `->{4}                                                   [   YES(?,O(n^3))    ]

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))}

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(X) -> c_0(X, f^#(g(X)))
                  , g(0()) -> s(0())
                  , g(s(X)) -> s(s(g(X)))}

             Proof Output:
               The input cannot be shown compatible

           * Path {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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [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]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                g^#(x1) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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: {g^#(s(X)) -> c_2(g^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {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]
                g^#(x1) = [0 1 0] x1 + [2]
                          [6 0 0]      [0]
                          [2 3 0]      [2]
                c_2(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]

           * Path {3}->{2}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [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]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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: {g^#(0()) -> c_1()}
               Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}

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

           * Path {5}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [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]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(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^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                s(x1) = [1 4 4] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [2]
                sel^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
                                [2 0 2]      [2 0 0]      [0]
                                [0 2 2]      [1 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [5]
                          [2 0 2]      [3]
                          [0 0 0]      [7]

           * Path {5}->{4}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [1 1 1] x1 + [0 0 0] x2 + [0]
                               [0 1 3]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                g(x1) = [0 0 0] x1 + [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]
                sel(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                g^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [3 1 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(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: {sel^#(0(), cons(X, Y)) -> c_3(X)}
               Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 2 2] x2 + [0]
                               [0 0 2]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [2]
                      [3]
                      [0]
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [2]
                sel^#(x1, x2) = [2 2 0] x1 + [2 0 0] x2 + [0]
                                [2 2 2]      [2 2 0]      [0]
                                [3 3 0]      [2 2 0]      [0]
                c_3(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [1]
                c_4(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [6]

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

             {  1: f^#(X) -> c_0(X, f^#(g(X)))
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3(X)
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

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

             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))}

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

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

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

             Proof Output:
               The input cannot be shown compatible

           * Path {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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                g^#(x1) = [3 3] x1 + [0]
                          [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(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(g^#(X))}
               Weak Rules: {}

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

           * Path {3}->{2}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(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^#(0()) -> c_1()}
               Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                g^#(x1) = [1 2] x1 + [2]
                          [6 1]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [5]
                          [2 0]      [3]

           * Path {5}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(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: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [4]
                s(x1) = [1 2] x1 + [2]
                        [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
                                [4 0]      [4 0]      [0]
                c_4(x1) = [1 0] x1 + [5]
                          [0 0]      [7]

           * Path {5}->{4}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                g(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                g^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_4(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^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sel^#(0(), cons(X, Y)) -> c_3(X)}
               Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_3) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [0]
                0() = [2]
                      [2]
                s(x1) = [1 0] x1 + [3]
                        [0 1]      [0]
                sel^#(x1, x2) = [2 2] x1 + [3 3] x2 + [0]
                                [2 2]      [2 5]      [0]
                c_3(x1) = [1 0] x1 + [1]
                          [0 0]      [0]
                c_4(x1) = [1 0] x1 + [7]
                          [0 0]      [7]

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

             {  1: f^#(X) -> c_0(X, f^#(g(X)))
              , 2: g^#(0()) -> c_1()
              , 3: g^#(s(X)) -> c_2(g^#(X))
              , 4: sel^#(0(), cons(X, Y)) -> c_3(X)
              , 5: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}

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

             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [         NA         ]

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules for this path are:

               {  g(0()) -> s(0())
                , g(s(X)) -> s(s(g(X)))}

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

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

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

             Proof Output:
               The input cannot be shown compatible

           * Path {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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                g^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(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(g^#(X))}
               Weak Rules: {}

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

           * Path {3}->{2}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {1}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(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^#(0()) -> c_1()}
               Weak Rules: {g^#(s(X)) -> c_2(g^#(X))}

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

           * Path {5}: 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(cons) = {}, Uargs(g) = {}, Uargs(s) = {},
                 Uargs(sel) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_2) = {}, Uargs(sel^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                g(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                f^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                g^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [1] x1 + [2]
                sel^#(x1, x2) = [3] x1 + [0] x2 + [2]
                c_4(x1) = [1] x1 + [5]

           * Path {5}->{4}: NA
             -----------------

             The usable rules of this path are empty.

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

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

    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.