Problem Strategy outermost added 08 Ex3 12 Luc96a

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex3 12 Luc96a

stdout:

MAYBE

Problem:
from(X) -> cons(X,from(s(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 Ex3 12 Luc96a

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex3 12 Luc96a

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, from(s(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: from^#(X) -> c_0(from^#(s(X)))
              , 2: sel^#(0(), cons(X, Y)) -> c_1()
              , 3: sel^#(s(X), cons(Y, Z)) -> c_2(sel^#(X, Z))}

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {1}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(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]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 1]      [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]
                0() = [0]
                      [0]
                      [0]
                from^#(x1) = [3 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_0(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_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

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

             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_2(sel^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {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_2(x1) = [1 0 0] x1 + [5]
                          [2 0 2]      [3]
                          [0 0 0]      [7]

           * 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(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]
                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]
                0() = [0]
                      [0]
                      [0]
                from^#(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]
                c_1() = [0]
                        [0]
                        [0]
                c_2(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_1()}
               Weak Rules: {sel^#(s(X), cons(Y, Z)) -> c_2(sel^#(X, Z))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {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]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [2]
                0() = [0]
                      [2]
                      [0]
                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_1() = [1]
                        [0]
                        [0]
                c_2(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: from^#(X) -> c_0(from^#(s(X)))
              , 2: sel^#(0(), cons(X, Y)) -> c_1()
              , 3: sel^#(s(X), cons(Y, Z)) -> c_2(sel^#(X, Z))}

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

             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

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

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

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

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

             {  1: from^#(X) -> c_0(from^#(s(X)))
              , 2: sel^#(0(), cons(X, Y)) -> c_1()
              , 3: sel^#(s(X), cons(Y, Z)) -> c_2(sel^#(X, Z))}

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

             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

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

           * Path {3}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(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 Ex3 12 Luc96a

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex3 12 Luc96a

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, from(s(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: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: sel^#(0(), cons(X, Y)) -> c_1(X)
              , 3: sel^#(s(X), cons(Y, Z)) -> c_2(sel^#(X, Z))}

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

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

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {2}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(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]
                s(x1) = [0 1 0] x1 + [0]
                        [0 0 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]
                0() = [0]
                      [0]
                      [0]
                from^#(x1) = [1 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_0(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

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

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(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]
                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]
                0() = [0]
                      [0]
                      [0]
                from^#(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]
                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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(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_2(sel^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {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_2(x1) = [1 0 0] x1 + [5]
                          [2 0 2]      [3]
                          [0 0 0]      [7]

           * Path {3}->{2}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(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]
                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]
                0() = [0]
                      [0]
                      [0]
                from^#(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]
                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_1(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]

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

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

             {  1: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: sel^#(0(), cons(X, Y)) -> c_1(X)
              , 3: sel^#(s(X), cons(Y, Z)) -> c_2(sel^#(X, Z))}

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

             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {2}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                from^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_0(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

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

             Proof Output:
               The input cannot be shown compatible

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

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

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

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

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

             {  1: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: sel^#(0(), cons(X, Y)) -> c_1(X)
              , 3: sel^#(s(X), cons(Y, Z)) -> c_2(sel^#(X, Z))}

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

             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {2}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1, x2) = [2] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]

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

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

             Proof Output:
               The input cannot be shown compatible

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

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

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