Problem Strategy outermost added 08 ExAppendixB AEL03

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 ExAppendixB AEL03

stdout:

MAYBE

Problem:
from(X) -> cons(X,from(s(X)))
2ndspos(0(),Z) -> rnil()
2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,Z))
2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
2ndsneg(0(),Z) -> rnil()
2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,Z))
2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,Z))
pi(X) -> 2ndspos(X,from(0()))
plus(0(),Y) -> Y
plus(s(X),Y) -> s(plus(X,Y))
times(0(),Y) -> 0()
times(s(X),Y) -> plus(Y,times(X,Y))
square(X) -> times(X,X)

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 ExAppendixB AEL03

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 ExAppendixB AEL03

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)))
     , 2ndspos(0(), Z) -> rnil()
     , 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
     , 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
       rcons(posrecip(Y), 2ndsneg(N, Z))
     , 2ndsneg(0(), Z) -> rnil()
     , 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
     , 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
       rcons(negrecip(Y), 2ndspos(N, Z))
     , pi(X) -> 2ndspos(X, from(0()))
     , plus(0(), Y) -> Y
     , plus(s(X), Y) -> s(plus(X, Y))
     , times(0(), Y) -> 0()
     , times(s(X), Y) -> plus(Y, times(X, Y))
     , square(X) -> times(X, 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: 2ndspos^#(0(), Z) -> c_1()
              , 3: 2ndspos^#(s(N), cons(X, Z)) ->
                   c_2(2ndspos^#(s(N), cons2(X, Z)))
              , 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_3(2ndsneg^#(N, Z))
              , 5: 2ndsneg^#(0(), Z) -> c_4()
              , 6: 2ndsneg^#(s(N), cons(X, Z)) ->
                   c_5(2ndsneg^#(s(N), cons2(X, Z)))
              , 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_6(2ndspos^#(N, Z))
              , 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
              , 9: plus^#(0(), Y) -> c_8()
              , 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
              , 11: times^#(0(), Y) -> c_10()
              , 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
              , 13: square^#(X) -> c_12(times^#(X, X))}

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

             ->{13}                                                      [     inherited      ]
                |
                |->{11}                                                  [    YES(?,O(1))     ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    |->{9}                                               [         NA         ]
                    |
                    `->{10}                                              [     inherited      ]
                        |
                        `->{9}                                           [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,7,6,4}                                             [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{5}                                               [         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(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
                 Uargs(square^#) = {}, Uargs(c_12) = {}
               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]
                2ndspos(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]
                rnil() = [0]
                         [0]
                         [0]
                cons2(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]
                rcons(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]
                posrecip(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                2ndsneg(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]
                negrecip(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pi(x1) = [0 0 0] x1 + [0]
                         [0 0 0]      [0]
                         [0 0 0]      [0]
                plus(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]
                times(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]
                square(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]
                2ndspos^#(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]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2ndsneg^#(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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                pi^#(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(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_8() = [0]
                        [0]
                        [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(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_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                square^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_12(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 {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{2}: NA
             -----------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}: inherited
             ------------------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{3,7,6,4}->{2}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}->{5}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {13}: inherited
             --------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{11}: YES(?,O(1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
                 Uargs(square^#) = {}, Uargs(c_12) = {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]
                2ndspos(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]
                rnil() = [0]
                         [0]
                         [0]
                cons2(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]
                rcons(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]
                posrecip(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                2ndsneg(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]
                negrecip(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pi(x1) = [0 0 0] x1 + [0]
                         [0 0 0]      [0]
                         [0 0 0]      [0]
                plus(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]
                times(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]
                square(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]
                2ndspos^#(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]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                2ndsneg^#(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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                pi^#(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(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_8() = [0]
                        [0]
                        [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(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_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                square^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_12(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(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), Y) -> c_10()}
               Weak Rules: {square^#(X) -> c_12(times^#(X, X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                times^#(x1, x2) = [2 0 2] x1 + [0 0 0] x2 + [0]
                                  [2 0 2]      [0 0 0]      [0]
                                  [2 2 2]      [0 0 0]      [0]
                c_10() = [1]
                         [0]
                         [0]
                square^#(x1) = [7 7 7] x1 + [7]
                               [7 7 7]      [7]
                               [7 7 7]      [7]
                c_12(x1) = [2 0 0] x1 + [7]
                           [0 0 0]      [7]
                           [0 0 0]      [7]

           * Path {13}->{12}: inherited
             --------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{9}: NA
             ------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

           * Path {13}->{12}->{10}: inherited
             --------------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{10}->{9}: NA
             ------------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

    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: 2ndspos^#(0(), Z) -> c_1()
              , 3: 2ndspos^#(s(N), cons(X, Z)) ->
                   c_2(2ndspos^#(s(N), cons2(X, Z)))
              , 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_3(2ndsneg^#(N, Z))
              , 5: 2ndsneg^#(0(), Z) -> c_4()
              , 6: 2ndsneg^#(s(N), cons(X, Z)) ->
                   c_5(2ndsneg^#(s(N), cons2(X, Z)))
              , 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_6(2ndspos^#(N, Z))
              , 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
              , 9: plus^#(0(), Y) -> c_8()
              , 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
              , 11: times^#(0(), Y) -> c_10()
              , 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
              , 13: square^#(X) -> c_12(times^#(X, X))}

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

             ->{13}                                                      [     inherited      ]
                |
                |->{11}                                                  [    YES(?,O(1))     ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    |->{9}                                               [         NA         ]
                    |
                    `->{10}                                              [     inherited      ]
                        |
                        `->{9}                                           [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,7,6,4}                                             [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{5}                                               [         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(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
                 Uargs(square^#) = {}, Uargs(c_12) = {}
               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]
                2ndspos(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                rnil() = [0]
                         [0]
                cons2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                rcons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                posrecip(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                2ndsneg(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                negrecip(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pi(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                square(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]
                2ndspos^#(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]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                pi^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_12(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 {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{2}: NA
             -----------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}: inherited
             ------------------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{3,7,6,4}->{2}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}->{5}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {13}: inherited
             --------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{11}: YES(?,O(1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
                 Uargs(square^#) = {}, Uargs(c_12) = {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]
                2ndspos(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                rnil() = [0]
                         [0]
                cons2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                rcons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                posrecip(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                2ndsneg(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                negrecip(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pi(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                square(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]
                2ndspos^#(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]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                pi^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), Y) -> c_10()}
               Weak Rules: {square^#(X) -> c_12(times^#(X, X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                times^#(x1, x2) = [0 2] x1 + [0 0] x2 + [0]
                                  [2 0]      [0 0]      [0]
                c_10() = [1]
                         [0]
                square^#(x1) = [7 7] x1 + [7]
                               [7 7]      [7]
                c_12(x1) = [2 0] x1 + [7]
                           [0 0]      [7]

           * Path {13}->{12}: inherited
             --------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{9}: NA
             ------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

           * Path {13}->{12}->{10}: inherited
             --------------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{10}->{9}: NA
             ------------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

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

    4) '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: 2ndspos^#(0(), Z) -> c_1()
              , 3: 2ndspos^#(s(N), cons(X, Z)) ->
                   c_2(2ndspos^#(s(N), cons2(X, Z)))
              , 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_3(2ndsneg^#(N, Z))
              , 5: 2ndsneg^#(0(), Z) -> c_4()
              , 6: 2ndsneg^#(s(N), cons(X, Z)) ->
                   c_5(2ndsneg^#(s(N), cons2(X, Z)))
              , 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_6(2ndspos^#(N, Z))
              , 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
              , 9: plus^#(0(), Y) -> c_8()
              , 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
              , 11: times^#(0(), Y) -> c_10()
              , 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
              , 13: square^#(X) -> c_12(times^#(X, X))}

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

             ->{13}                                                      [     inherited      ]
                |
                |->{11}                                                  [         NA         ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    |->{9}                                               [         NA         ]
                    |
                    `->{10}                                              [     inherited      ]
                        |
                        `->{9}                                           [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,7,6,4}                                             [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{5}                                               [         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(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {1},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
                 Uargs(square^#) = {}, Uargs(c_12) = {}
               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]
                2ndspos(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                rnil() = [0]
                cons2(x1, x2) = [0] x1 + [0] x2 + [0]
                rcons(x1, x2) = [0] x1 + [0] x2 + [0]
                posrecip(x1) = [0] x1 + [0]
                2ndsneg(x1, x2) = [0] x1 + [0] x2 + [0]
                negrecip(x1) = [0] x1 + [0]
                pi(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                square(x1) = [0] x1 + [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                2ndspos^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                pi^#(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_12(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 {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{2}: NA
             -----------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}: inherited
             ------------------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{3,7,6,4}->{2}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}->{5}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {13}: inherited
             --------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{11}: 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(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_9) = {}, Uargs(times^#) = {}, Uargs(c_11) = {},
                 Uargs(square^#) = {}, Uargs(c_12) = {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]
                2ndspos(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                rnil() = [0]
                cons2(x1, x2) = [0] x1 + [0] x2 + [0]
                rcons(x1, x2) = [0] x1 + [0] x2 + [0]
                posrecip(x1) = [0] x1 + [0]
                2ndsneg(x1, x2) = [0] x1 + [0] x2 + [0]
                negrecip(x1) = [0] x1 + [0]
                pi(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                square(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                2ndspos^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                pi^#(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]

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

           * Path {13}->{12}: inherited
             --------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{9}: NA
             ------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

           * Path {13}->{12}->{10}: inherited
             --------------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{10}->{9}: NA
             ------------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

    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 ExAppendixB AEL03

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 ExAppendixB AEL03

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)))
     , 2ndspos(0(), Z) -> rnil()
     , 2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
     , 2ndspos(s(N), cons2(X, cons(Y, Z))) ->
       rcons(posrecip(Y), 2ndsneg(N, Z))
     , 2ndsneg(0(), Z) -> rnil()
     , 2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
     , 2ndsneg(s(N), cons2(X, cons(Y, Z))) ->
       rcons(negrecip(Y), 2ndspos(N, Z))
     , pi(X) -> 2ndspos(X, from(0()))
     , plus(0(), Y) -> Y
     , plus(s(X), Y) -> s(plus(X, Y))
     , times(0(), Y) -> 0()
     , times(s(X), Y) -> plus(Y, times(X, Y))
     , square(X) -> times(X, 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: 2ndspos^#(0(), Z) -> c_1()
              , 3: 2ndspos^#(s(N), cons(X, Z)) ->
                   c_2(2ndspos^#(s(N), cons2(X, Z)))
              , 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) ->
                   c_3(Y, 2ndsneg^#(N, Z))
              , 5: 2ndsneg^#(0(), Z) -> c_4()
              , 6: 2ndsneg^#(s(N), cons(X, Z)) ->
                   c_5(2ndsneg^#(s(N), cons2(X, Z)))
              , 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) ->
                   c_6(Y, 2ndspos^#(N, Z))
              , 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
              , 9: plus^#(0(), Y) -> c_8(Y)
              , 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
              , 11: times^#(0(), Y) -> c_10()
              , 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
              , 13: square^#(X) -> c_12(times^#(X, X))}

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

             ->{13}                                                      [     inherited      ]
                |
                |->{11}                                                  [    YES(?,O(1))     ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    |->{9}                                               [         NA         ]
                    |
                    `->{10}                                              [     inherited      ]
                        |
                        `->{9}                                           [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,7,6,4}                                             [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{5}                                               [         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(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
                 Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {}
               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]
                2ndspos(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]
                rnil() = [0]
                         [0]
                         [0]
                cons2(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]
                rcons(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]
                posrecip(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                2ndsneg(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]
                negrecip(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pi(x1) = [0 0 0] x1 + [0]
                         [0 0 0]      [0]
                         [0 0 0]      [0]
                plus(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]
                times(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]
                square(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]
                2ndspos^#(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]
                c_3(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]
                2ndsneg^#(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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(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]
                pi^#(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(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_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(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_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                square^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_12(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 {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{2}: NA
             -----------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}: inherited
             ------------------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{3,7,6,4}->{2}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}->{5}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {13}: inherited
             --------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{11}: YES(?,O(1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
                 Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {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]
                2ndspos(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]
                rnil() = [0]
                         [0]
                         [0]
                cons2(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]
                rcons(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]
                posrecip(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                2ndsneg(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]
                negrecip(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pi(x1) = [0 0 0] x1 + [0]
                         [0 0 0]      [0]
                         [0 0 0]      [0]
                plus(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]
                times(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]
                square(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]
                2ndspos^#(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]
                c_3(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]
                2ndsneg^#(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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(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]
                pi^#(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(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_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(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_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                square^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_12(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(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), Y) -> c_10()}
               Weak Rules: {square^#(X) -> c_12(times^#(X, X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                times^#(x1, x2) = [2 0 2] x1 + [0 0 0] x2 + [0]
                                  [2 0 2]      [0 0 0]      [0]
                                  [2 2 2]      [0 0 0]      [0]
                c_10() = [1]
                         [0]
                         [0]
                square^#(x1) = [7 7 7] x1 + [7]
                               [7 7 7]      [7]
                               [7 7 7]      [7]
                c_12(x1) = [2 0 0] x1 + [7]
                           [0 0 0]      [7]
                           [0 0 0]      [7]

           * Path {13}->{12}: inherited
             --------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{9}: NA
             ------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

           * Path {13}->{12}->{10}: inherited
             --------------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{10}->{9}: NA
             ------------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

    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: 2ndspos^#(0(), Z) -> c_1()
              , 3: 2ndspos^#(s(N), cons(X, Z)) ->
                   c_2(2ndspos^#(s(N), cons2(X, Z)))
              , 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) ->
                   c_3(Y, 2ndsneg^#(N, Z))
              , 5: 2ndsneg^#(0(), Z) -> c_4()
              , 6: 2ndsneg^#(s(N), cons(X, Z)) ->
                   c_5(2ndsneg^#(s(N), cons2(X, Z)))
              , 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) ->
                   c_6(Y, 2ndspos^#(N, Z))
              , 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
              , 9: plus^#(0(), Y) -> c_8(Y)
              , 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
              , 11: times^#(0(), Y) -> c_10()
              , 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
              , 13: square^#(X) -> c_12(times^#(X, X))}

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

             ->{13}                                                      [     inherited      ]
                |
                |->{11}                                                  [    YES(?,O(1))     ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    |->{9}                                               [         NA         ]
                    |
                    `->{10}                                              [     inherited      ]
                        |
                        `->{9}                                           [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,7,6,4}                                             [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{5}                                               [         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(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
                 Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {}
               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]
                2ndspos(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                rnil() = [0]
                         [0]
                cons2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                rcons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                posrecip(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                2ndsneg(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                negrecip(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pi(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                square(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]
                2ndspos^#(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]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pi^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_12(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 {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{2}: NA
             -----------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}: inherited
             ------------------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{3,7,6,4}->{2}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}->{5}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {13}: inherited
             --------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{11}: YES(?,O(1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
                 Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {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]
                2ndspos(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                rnil() = [0]
                         [0]
                cons2(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                rcons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                posrecip(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                2ndsneg(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                negrecip(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pi(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                square(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]
                2ndspos^#(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]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pi^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {times^#(0(), Y) -> c_10()}
               Weak Rules: {square^#(X) -> c_12(times^#(X, X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                times^#(x1, x2) = [0 2] x1 + [0 0] x2 + [0]
                                  [2 0]      [0 0]      [0]
                c_10() = [1]
                         [0]
                square^#(x1) = [7 7] x1 + [7]
                               [7 7]      [7]
                c_12(x1) = [2 0] x1 + [7]
                           [0 0]      [7]

           * Path {13}->{12}: inherited
             --------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{9}: NA
             ------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

           * Path {13}->{12}->{10}: inherited
             --------------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{10}->{9}: NA
             ------------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

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

    4) '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: 2ndspos^#(0(), Z) -> c_1()
              , 3: 2ndspos^#(s(N), cons(X, Z)) ->
                   c_2(2ndspos^#(s(N), cons2(X, Z)))
              , 4: 2ndspos^#(s(N), cons2(X, cons(Y, Z))) ->
                   c_3(Y, 2ndsneg^#(N, Z))
              , 5: 2ndsneg^#(0(), Z) -> c_4()
              , 6: 2ndsneg^#(s(N), cons(X, Z)) ->
                   c_5(2ndsneg^#(s(N), cons2(X, Z)))
              , 7: 2ndsneg^#(s(N), cons2(X, cons(Y, Z))) ->
                   c_6(Y, 2ndspos^#(N, Z))
              , 8: pi^#(X) -> c_7(2ndspos^#(X, from(0())))
              , 9: plus^#(0(), Y) -> c_8(Y)
              , 10: plus^#(s(X), Y) -> c_9(plus^#(X, Y))
              , 11: times^#(0(), Y) -> c_10()
              , 12: times^#(s(X), Y) -> c_11(plus^#(Y, times(X, Y)))
              , 13: square^#(X) -> c_12(times^#(X, X))}

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

             ->{13}                                                      [     inherited      ]
                |
                |->{11}                                                  [         NA         ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    |->{9}                                               [         NA         ]
                    |
                    `->{10}                                              [     inherited      ]
                        |
                        `->{9}                                           [         NA         ]

             ->{8}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,7,6,4}                                             [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{5}                                               [         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(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {2},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
                 Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {}
               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]
                2ndspos(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                rnil() = [0]
                cons2(x1, x2) = [0] x1 + [0] x2 + [0]
                rcons(x1, x2) = [0] x1 + [0] x2 + [0]
                posrecip(x1) = [0] x1 + [0]
                2ndsneg(x1, x2) = [0] x1 + [0] x2 + [0]
                negrecip(x1) = [0] x1 + [0]
                pi(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                square(x1) = [0] x1 + [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1, x2) = [2] x1 + [1] x2 + [0]
                2ndspos^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                pi^#(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_12(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 {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{2}: NA
             -----------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}: inherited
             ------------------------------

             This path is subsumed by the proof of path {8}->{3,7,6,4}->{2}.

           * Path {8}->{3,7,6,4}->{2}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {8}->{3,7,6,4}->{5}: NA
             ----------------------------

             The usable rules for this path are:

               {from(X) -> cons(X, from(s(X)))}

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

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

           * Path {13}: inherited
             --------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{11}: 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(2ndspos) = {}, Uargs(cons2) = {}, Uargs(rcons) = {},
                 Uargs(posrecip) = {}, Uargs(2ndsneg) = {}, Uargs(negrecip) = {},
                 Uargs(pi) = {}, Uargs(plus) = {}, Uargs(times) = {},
                 Uargs(square) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(2ndspos^#) = {}, Uargs(c_2) = {}, Uargs(c_3) = {},
                 Uargs(2ndsneg^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(pi^#) = {}, Uargs(c_7) = {}, Uargs(plus^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}, Uargs(times^#) = {},
                 Uargs(c_11) = {}, Uargs(square^#) = {}, Uargs(c_12) = {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]
                2ndspos(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                rnil() = [0]
                cons2(x1, x2) = [0] x1 + [0] x2 + [0]
                rcons(x1, x2) = [0] x1 + [0] x2 + [0]
                posrecip(x1) = [0] x1 + [0]
                2ndsneg(x1, x2) = [0] x1 + [0] x2 + [0]
                negrecip(x1) = [0] x1 + [0]
                pi(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                square(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                2ndspos^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1, x2) = [0] x1 + [0] x2 + [0]
                pi^#(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]

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

           * Path {13}->{12}: inherited
             --------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{9}: NA
             ------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

           * Path {13}->{12}->{10}: inherited
             --------------------------------

             This path is subsumed by the proof of path {13}->{12}->{10}->{9}.

           * Path {13}->{12}->{10}->{9}: NA
             ------------------------------

             The usable rules for this path are:

               {  times(0(), Y) -> 0()
                , times(s(X), Y) -> plus(Y, times(X, Y))
                , plus(0(), Y) -> Y
                , plus(s(X), Y) -> s(plus(X, Y))}

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

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

    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.