Problem Strategy outermost added 08 Ex14 AEGL02

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 AEGL02

stdout:

MAYBE

Problem:
from(X) -> cons(X,from(s(X)))
length(nil()) -> 0()
length(cons(X,Y)) -> s(length1(Y))
length1(X) -> length(X)

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 AEGL02

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 AEGL02

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)))
     , length(nil()) -> 0()
     , length(cons(X, Y)) -> s(length1(Y))
     , length1(X) -> length(X)}

Proof Output:
  None of the processors succeeded.

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

             {  1: from^#(X) -> c_0(from^#(s(X)))
              , 2: length^#(nil()) -> c_1()
              , 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
              , 4: length1^#(X) -> c_3(length^#(X))}

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

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

             ->{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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {1}, Uargs(length^#) = {}, Uargs(c_2) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}
               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]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                0() = [0]
                      [0]
                      [0]
                length1(x1) = [0 0 0] x1 + [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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length1^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           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 {3,4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                0() = [0]
                      [0]
                      [0]
                length1(x1) = [0 0 0] x1 + [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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [3 3 3]      [0]
                               [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                length1^#(x1) = [0 0 0] x1 + [0]
                                [3 3 3]      [0]
                                [3 3 3]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules:
                 {  length^#(cons(X, Y)) -> c_2(length1^#(Y))
                  , length1^#(X) -> c_3(length^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [2 0 0] x1 + [0]
                               [6 0 0]      [0]
                               [6 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [1]
                          [2 2 2]      [0]
                          [2 0 0]      [7]
                length1^#(x1) = [2 2 2] x1 + [2]
                                [0 0 0]      [2]
                                [0 0 2]      [2]
                c_3(x1) = [1 0 0] x1 + [1]
                          [0 0 0]      [2]
                          [0 0 0]      [2]

           * Path {3,4}->{2}: YES(?,O(n^1))
             ------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                0() = [0]
                      [0]
                      [0]
                length1(x1) = [0 0 0] x1 + [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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                length1^#(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]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {length^#(nil()) -> c_1()}
               Weak Rules:
                 {  length^#(cons(X, Y)) -> c_2(length1^#(Y))
                  , length1^#(X) -> c_3(length^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 3 3] x2 + [0]
                               [0 0 0]      [0 0 0]      [2]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [2]
                        [0]
                        [2]
                length^#(x1) = [2 0 2] x1 + [0]
                               [0 0 0]      [0]
                               [4 2 0]      [4]
                c_1() = [1]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 2 2]      [0]
                length1^#(x1) = [2 0 2] x1 + [0]
                                [0 2 1]      [2]
                                [0 4 3]      [2]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [1]
                          [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: from^#(X) -> c_0(from^#(s(X)))
              , 2: length^#(nil()) -> c_1()
              , 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
              , 4: length1^#(X) -> c_3(length^#(X))}

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

             ->{3,4}                                                     [         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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {1}, Uargs(length^#) = {}, Uargs(c_2) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}
               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]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                nil() = [0]
                        [0]
                0() = [0]
                      [0]
                length1(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [3 3] x1 + [0]
                             [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length1^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           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 {3,4}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                nil() = [0]
                        [0]
                0() = [0]
                      [0]
                length1(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [3 3] x1 + [0]
                               [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length1^#(x1) = [3 3] x1 + [0]
                                [3 3]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]

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

           * Path {3,4}->{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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                nil() = [0]
                        [0]
                0() = [0]
                      [0]
                length1(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length1^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_3(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: length^#(nil()) -> c_1()
              , 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
              , 4: length1^#(X) -> c_3(length^#(X))}

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

             ->{3,4}                                                     [         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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {1}, Uargs(length^#) = {}, Uargs(c_2) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}
               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]
                length(x1) = [0] x1 + [0]
                nil() = [0]
                0() = [0]
                length1(x1) = [0] x1 + [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                length1^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           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 {3,4}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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]
                length(x1) = [0] x1 + [0]
                nil() = [0]
                0() = [0]
                length1(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                length^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                length1^#(x1) = [3] x1 + [0]
                c_3(x1) = [1] x1 + [0]

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

           * Path {3,4}->{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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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]
                length(x1) = [0] x1 + [0]
                nil() = [0]
                0() = [0]
                length1(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                length1^#(x1) = [0] x1 + [0]
                c_3(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 Ex14 AEGL02

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex14 AEGL02

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)))
     , length(nil()) -> 0()
     , length(cons(X, Y)) -> s(length1(Y))
     , length1(X) -> length(X)}

Proof Output:
  None of the processors succeeded.

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

             {  1: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: length^#(nil()) -> c_1()
              , 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
              , 4: length1^#(X) -> c_3(length^#(X))}

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

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

             ->{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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {2}, Uargs(length^#) = {}, Uargs(c_2) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}
               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]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                0() = [0]
                      [0]
                      [0]
                length1(x1) = [0 0 0] x1 + [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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length1^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           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 {3,4}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                0() = [0]
                      [0]
                      [0]
                length1(x1) = [0 0 0] x1 + [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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [3 3 3]      [0]
                               [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                length1^#(x1) = [0 0 0] x1 + [0]
                                [3 3 3]      [0]
                                [3 3 3]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {  length^#(cons(X, Y)) -> c_2(length1^#(Y))
                  , length1^#(X) -> c_3(length^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [2 0 0] x1 + [0]
                               [6 0 0]      [0]
                               [6 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [1]
                          [2 2 2]      [0]
                          [2 0 0]      [7]
                length1^#(x1) = [2 2 2] x1 + [2]
                                [0 0 0]      [2]
                                [0 0 2]      [2]
                c_3(x1) = [1 0 0] x1 + [1]
                          [0 0 0]      [2]
                          [0 0 0]      [2]

           * Path {3,4}->{2}: YES(?,O(n^1))
             ------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                0() = [0]
                      [0]
                      [0]
                length1(x1) = [0 0 0] x1 + [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]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                length1^#(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]

             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: {length^#(nil()) -> c_1()}
               Weak Rules:
                 {  length^#(cons(X, Y)) -> c_2(length1^#(Y))
                  , length1^#(X) -> c_3(length^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 3 3] x2 + [0]
                               [0 0 0]      [0 0 0]      [2]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [2]
                        [0]
                        [2]
                length^#(x1) = [2 0 2] x1 + [0]
                               [0 0 0]      [0]
                               [4 2 0]      [4]
                c_1() = [1]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 2 2]      [0]
                length1^#(x1) = [2 0 2] x1 + [0]
                                [0 2 1]      [2]
                                [0 4 3]      [2]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [1]
                          [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: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: length^#(nil()) -> c_1()
              , 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
              , 4: length1^#(X) -> c_3(length^#(X))}

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

             ->{3,4}                                                     [         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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {2}, Uargs(length^#) = {}, Uargs(c_2) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}
               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]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                nil() = [0]
                        [0]
                0() = [0]
                      [0]
                length1(x1) = [0 0] x1 + [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]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length1^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           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 {3,4}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                nil() = [0]
                        [0]
                0() = [0]
                      [0]
                length1(x1) = [0 0] x1 + [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]
                length^#(x1) = [3 3] x1 + [0]
                               [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length1^#(x1) = [3 3] x1 + [0]
                                [3 3]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]

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

           * Path {3,4}->{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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                nil() = [0]
                        [0]
                0() = [0]
                      [0]
                length1(x1) = [0 0] x1 + [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]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length1^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_3(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: length^#(nil()) -> c_1()
              , 3: length^#(cons(X, Y)) -> c_2(length1^#(Y))
              , 4: length1^#(X) -> c_3(length^#(X))}

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

             ->{3,4}                                                     [         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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {2}, Uargs(length^#) = {}, Uargs(c_2) = {},
                 Uargs(length1^#) = {}, Uargs(c_3) = {}
               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]
                length(x1) = [0] x1 + [0]
                nil() = [0]
                0() = [0]
                length1(x1) = [0] x1 + [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1, x2) = [2] x1 + [1] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                length1^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           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 {3,4}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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]
                length(x1) = [0] x1 + [0]
                nil() = [0]
                0() = [0]
                length1(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                length1^#(x1) = [3] x1 + [0]
                c_3(x1) = [1] x1 + [0]

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

           * Path {3,4}->{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(length) = {}, Uargs(length1) = {}, Uargs(from^#) = {},
                 Uargs(c_0) = {}, Uargs(length^#) = {}, Uargs(c_2) = {1},
                 Uargs(length1^#) = {}, Uargs(c_3) = {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]
                length(x1) = [0] x1 + [0]
                nil() = [0]
                0() = [0]
                length1(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                length1^#(x1) = [0] x1 + [0]
                c_3(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.