Problem Strategy outermost added 08 Ex8 BLR02

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex8 BLR02

stdout:

MAYBE

Problem:
fib(N) -> sel(N,fib1(s(0()),s(0())))
fib1(X,Y) -> cons(X,fib1(Y,add(X,Y)))
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
sel(0(),cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,XS)

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex8 BLR02

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex8 BLR02

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  fib(N) -> sel(N, fib1(s(0()), s(0())))
     , fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(add(X, Y))
     , sel(0(), cons(X, XS)) -> X
     , sel(s(N), cons(X, XS)) -> sel(N, XS)}

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: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
              , 2: fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
              , 3: add^#(0(), X) -> c_2()
              , 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
              , 5: sel^#(0(), cons(X, XS)) -> c_4()
              , 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}

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

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

             ->{2}                                                       [       MAYBE        ]

             ->{1}                                                       [     inherited      ]
                |
                |->{5}                                                   [         NA         ]
                |
                `->{6}                                                   [     inherited      ]
                    |
                    `->{5}                                               [         NA         ]



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

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{5}: NA
             -----------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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 {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{6}->{5}: NA
             ----------------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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}: MAYBE
             ---------------

             The usable rules for this path are:

               {  add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(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]
                fib1(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]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [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]
                add(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]
                fib^#(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]
                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]
                fib1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
               Weak Rules: {}

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

           * Path {4}->{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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(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]
                fib1(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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [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]
                add(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]
                fib^#(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]
                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]
                fib1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(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_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {add^#(0(), X) -> c_2()}
               Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 4]      [0]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                add^#(x1, x2) = [1 3 1] x1 + [0 0 0] x2 + [0]
                                [3 2 2]      [0 0 4]      [0]
                                [0 2 2]      [0 0 2]      [2]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 2]      [3]
                          [0 0 0]      [2]

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

             {  1: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
              , 2: fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
              , 3: add^#(0(), X) -> c_2()
              , 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
              , 5: sel^#(0(), cons(X, XS)) -> c_4()
              , 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}

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

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

             ->{2}                                                       [       MAYBE        ]

             ->{1}                                                       [     inherited      ]
                |
                |->{5}                                                   [         NA         ]
                |
                `->{6}                                                   [     inherited      ]
                    |
                    `->{5}                                               [         NA         ]



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

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{5}: NA
             -----------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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 {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{6}->{5}: NA
             ----------------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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}: MAYBE
             ---------------

             The usable rules for this path are:

               {  add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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:
                 {  fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {4}: 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fib1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fib^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                fib1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

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

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

           * Path {4}->{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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fib1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fib^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                fib1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {add^#(0(), X) -> c_2()}
               Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [4]
                        [0 1]      [0]
                0() = [2]
                      [2]
                add^#(x1, x2) = [1 3] x1 + [0 0] x2 + [4]
                                [2 2]      [4 4]      [0]
                c_2() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [3]
                          [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: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
              , 2: fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
              , 3: add^#(0(), X) -> c_2()
              , 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
              , 5: sel^#(0(), cons(X, XS)) -> c_4()
              , 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}

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

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

             ->{2}                                                       [       MAYBE        ]

             ->{1}                                                       [     inherited      ]
                |
                |->{5}                                                   [         NA         ]
                |
                `->{6}                                                   [     inherited      ]
                    |
                    `->{5}                                               [         NA         ]



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

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{5}: NA
             -----------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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 {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{6}->{5}: NA
             ----------------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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}: MAYBE
             ---------------

             The usable rules for this path are:

               {  add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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:
                 {  fib1^#(X, Y) -> c_1(fib1^#(Y, add(X, Y)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {4}: 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                fib1(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                fib^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                fib1^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                add^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]

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

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

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

           * Path {4}->{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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                fib1(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                fib^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                fib1^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {add^#(0(), X) -> c_2()}
               Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                add^#(x1, x2) = [6] x1 + [7] x2 + [0]
                c_2() = [1]
                c_3(x1) = [1] x1 + [7]

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex8 BLR02

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex8 BLR02

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  fib(N) -> sel(N, fib1(s(0()), s(0())))
     , fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(add(X, Y))
     , sel(0(), cons(X, XS)) -> X
     , sel(s(N), cons(X, XS)) -> sel(N, XS)}

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: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
              , 2: fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
              , 3: add^#(0(), X) -> c_2(X)
              , 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
              , 5: sel^#(0(), cons(X, XS)) -> c_4(X)
              , 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}

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

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

             ->{2}                                                       [       MAYBE        ]

             ->{1}                                                       [     inherited      ]
                |
                |->{5}                                                   [         NA         ]
                |
                `->{6}                                                   [     inherited      ]
                    |
                    `->{5}                                               [         NA         ]



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

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{5}: NA
             -----------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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 {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{6}->{5}: NA
             ----------------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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}: MAYBE
             ---------------

             The usable rules for this path are:

               {  add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(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]
                fib1(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]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [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]
                add(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]
                fib^#(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]
                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]
                fib1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}
               Weak Rules: {}

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

           * Path {4}->{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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(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]
                fib1(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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [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]
                add(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]
                fib^#(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]
                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]
                fib1^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {add^#(0(), X) -> c_2(X)}
               Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                add^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [0]
                                [2 2 2]      [4 4 4]      [0]
                                [2 2 2]      [4 0 4]      [0]
                c_2(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [2]

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

             {  1: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
              , 2: fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
              , 3: add^#(0(), X) -> c_2(X)
              , 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
              , 5: sel^#(0(), cons(X, XS)) -> c_4(X)
              , 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}

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

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

             ->{2}                                                       [       MAYBE        ]

             ->{1}                                                       [     inherited      ]
                |
                |->{5}                                                   [         NA         ]
                |
                `->{6}                                                   [     inherited      ]
                    |
                    `->{5}                                               [         NA         ]



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

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{5}: NA
             -----------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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 {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{6}->{5}: NA
             ----------------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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}: MAYBE
             ---------------

             The usable rules for this path are:

               {  add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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:
                 {  fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {4}: 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fib1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fib^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                fib1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

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

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

           * Path {4}->{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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fib1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                fib^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                fib1^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {add^#(0(), X) -> c_2(X)}
               Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [2]
                      [0]
                add^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
                                [4 0]      [0 4]      [0]
                c_2(x1) = [0 0] x1 + [1]
                          [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [0]

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

             {  1: fib^#(N) -> c_0(sel^#(N, fib1(s(0()), s(0()))))
              , 2: fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
              , 3: add^#(0(), X) -> c_2(X)
              , 4: add^#(s(X), Y) -> c_3(add^#(X, Y))
              , 5: sel^#(0(), cons(X, XS)) -> c_4(X)
              , 6: sel^#(s(N), cons(X, XS)) -> c_5(sel^#(N, XS))}

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

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

             ->{2}                                                       [       MAYBE        ]

             ->{1}                                                       [     inherited      ]
                |
                |->{5}                                                   [         NA         ]
                |
                `->{6}                                                   [     inherited      ]
                    |
                    `->{5}                                               [         NA         ]



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

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{5}: NA
             -----------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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 {1}->{6}: inherited
             ------------------------

             This path is subsumed by the proof of path {1}->{6}->{5}.

           * Path {1}->{6}->{5}: NA
             ----------------------

             The usable rules for this path are:

               {  fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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}: MAYBE
             ---------------

             The usable rules for this path are:

               {  add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, 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:
                 {  fib1^#(X, Y) -> c_1(X, fib1^#(Y, add(X, Y)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {4}: 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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                fib1(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                fib^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                fib1^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2) = [0] x1 + [0] x2 + [0]
                add^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

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

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

           * Path {4}->{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(fib) = {}, Uargs(sel) = {}, Uargs(fib1) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(add) = {}, Uargs(fib^#) = {},
                 Uargs(c_0) = {}, Uargs(sel^#) = {}, Uargs(fib1^#) = {},
                 Uargs(c_1) = {}, Uargs(add^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                fib(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                fib1(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                fib^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                fib1^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1, x2) = [0] x1 + [0] x2 + [0]
                add^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {add^#(0(), X) -> c_2(X)}
               Weak Rules: {add^#(s(X), Y) -> c_3(add^#(X, Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                add^#(x1, x2) = [2] x1 + [7] x2 + [4]
                c_2(x1) = [1] x1 + [1]
                c_3(x1) = [1] x1 + [4]

    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.