Problem Strategy outermost added 08 Ex1 2 AEL03

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 2 AEL03

stdout:

MAYBE

Problem:
 from(X) -> cons(X,from(s(X)))
 2ndspos(0(),Z) -> rnil()
 2ndspos(s(N),cons(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,Z))
 2ndsneg(0(),Z) -> rnil()
 2ndsneg(s(N),cons(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 TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 2 AEL03

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 2 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, cons(Y, Z))) ->
       rcons(posrecip(Y), 2ndsneg(N, Z))
     , 2ndsneg(0(), Z) -> rnil()
     , 2ndsneg(s(N), cons(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, cons(Y, Z))) -> c_2(2ndsneg^#(N, Z))
              , 4: 2ndsneg^#(0(), Z) -> c_3()
              , 5: 2ndsneg^#(s(N), cons(X, cons(Y, Z))) -> c_4(2ndspos^#(N, Z))
              , 6: pi^#(X) -> c_5(2ndspos^#(X, from(0())))
              , 7: plus^#(0(), Y) -> c_6()
              , 8: plus^#(s(X), Y) -> c_7(plus^#(X, Y))
              , 9: times^#(0(), Y) -> c_8()
              , 10: times^#(s(X), Y) -> c_9(plus^#(Y, times(X, Y)))
              , 11: square^#(X) -> c_10(times^#(X, X))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                |->{9}                                                   [    YES(?,O(1))     ]
                |
                `->{10}                                                  [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,5}                                                 [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{4}                                               [         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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_7) = {}, Uargs(times^#) = {}, Uargs(c_9) = {},
                 Uargs(square^#) = {}, Uargs(c_10) = {}
               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]
                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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(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_5(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_6() = [0]
                        [0]
                        [0]
                c_7(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_8() = [0]
                        [0]
                        [0]
                c_9(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_10(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 {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{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 {6}->{3,5}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{3,5}->{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 {6}->{3,5}->{4}: 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 {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{9}: 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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_7) = {}, Uargs(times^#) = {}, Uargs(c_9) = {},
                 Uargs(square^#) = {}, Uargs(c_10) = {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]
                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]
                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_3() = [0]
                        [0]
                        [0]
                c_4(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_5(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_6() = [0]
                        [0]
                        [0]
                c_7(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_8() = [0]
                        [0]
                        [0]
                c_9(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_10(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_8()}
               Weak Rules: {square^#(X) -> c_10(times^#(X, X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_10) = {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_8() = [1]
                        [0]
                        [0]
                square^#(x1) = [7 7 7] x1 + [7]
                               [7 7 7]      [7]
                               [7 7 7]      [7]
                c_10(x1) = [2 0 0] x1 + [7]
                           [0 0 0]      [7]
                           [0 0 0]      [7]
           
           * Path {11}->{10}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{7}: 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 {11}->{10}->{8}: inherited
             -------------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{8}->{7}: 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, cons(Y, Z))) -> c_2(2ndsneg^#(N, Z))
              , 4: 2ndsneg^#(0(), Z) -> c_3()
              , 5: 2ndsneg^#(s(N), cons(X, cons(Y, Z))) -> c_4(2ndspos^#(N, Z))
              , 6: pi^#(X) -> c_5(2ndspos^#(X, from(0())))
              , 7: plus^#(0(), Y) -> c_6()
              , 8: plus^#(s(X), Y) -> c_7(plus^#(X, Y))
              , 9: times^#(0(), Y) -> c_8()
              , 10: times^#(s(X), Y) -> c_9(plus^#(Y, times(X, Y)))
              , 11: square^#(X) -> c_10(times^#(X, X))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                |->{9}                                                   [    YES(?,O(1))     ]
                |
                `->{10}                                                  [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,5}                                                 [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{4}                                               [         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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_7) = {}, Uargs(times^#) = {}, Uargs(c_9) = {},
                 Uargs(square^#) = {}, Uargs(c_10) = {}
               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]
                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]
                2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                pi^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(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]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_10(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 {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{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 {6}->{3,5}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{3,5}->{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 {6}->{3,5}->{4}: 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 {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{9}: 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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_7) = {}, Uargs(times^#) = {}, Uargs(c_9) = {},
                 Uargs(square^#) = {}, Uargs(c_10) = {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]
                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]
                2ndsneg^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                pi^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(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]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_10(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_8()}
               Weak Rules: {square^#(X) -> c_10(times^#(X, X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_10) = {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_8() = [1]
                        [0]
                square^#(x1) = [7 7] x1 + [7]
                               [7 7]      [7]
                c_10(x1) = [2 0] x1 + [7]
                           [0 0]      [7]
           
           * Path {11}->{10}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{7}: 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 {11}->{10}->{8}: inherited
             -------------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{8}->{7}: 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, cons(Y, Z))) -> c_2(2ndsneg^#(N, Z))
              , 4: 2ndsneg^#(0(), Z) -> c_3()
              , 5: 2ndsneg^#(s(N), cons(X, cons(Y, Z))) -> c_4(2ndspos^#(N, Z))
              , 6: pi^#(X) -> c_5(2ndspos^#(X, from(0())))
              , 7: plus^#(0(), Y) -> c_6()
              , 8: plus^#(s(X), Y) -> c_7(plus^#(X, Y))
              , 9: times^#(0(), Y) -> c_8()
              , 10: times^#(s(X), Y) -> c_9(plus^#(Y, times(X, Y)))
              , 11: square^#(X) -> c_10(times^#(X, X))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                |->{9}                                                   [         NA         ]
                |
                `->{10}                                                  [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,5}                                                 [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{4}                                               [         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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_7) = {}, Uargs(times^#) = {}, Uargs(c_9) = {},
                 Uargs(square^#) = {}, Uargs(c_10) = {}
               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]
                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]
                2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                pi^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_10(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 {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{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 {6}->{3,5}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{3,5}->{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 {6}->{3,5}->{4}: 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 {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{9}: 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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_7) = {}, Uargs(times^#) = {}, Uargs(c_9) = {},
                 Uargs(square^#) = {}, Uargs(c_10) = {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]
                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]
                2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                pi^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_10(x1) = [1] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {11}->{10}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{7}: 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 {11}->{10}->{8}: inherited
             -------------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{8}->{7}: 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 TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 2 AEL03

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 2 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, cons(Y, Z))) ->
       rcons(posrecip(Y), 2ndsneg(N, Z))
     , 2ndsneg(0(), Z) -> rnil()
     , 2ndsneg(s(N), cons(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, cons(Y, Z))) ->
                   c_2(Y, 2ndsneg^#(N, Z))
              , 4: 2ndsneg^#(0(), Z) -> c_3()
              , 5: 2ndsneg^#(s(N), cons(X, cons(Y, Z))) ->
                   c_4(Y, 2ndspos^#(N, Z))
              , 6: pi^#(X) -> c_5(2ndspos^#(X, from(0())))
              , 7: plus^#(0(), Y) -> c_6(Y)
              , 8: plus^#(s(X), Y) -> c_7(plus^#(X, Y))
              , 9: times^#(0(), Y) -> c_8()
              , 10: times^#(s(X), Y) -> c_9(plus^#(Y, times(X, Y)))
              , 11: square^#(X) -> c_10(times^#(X, X))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                |->{9}                                                   [    YES(?,O(1))     ]
                |
                `->{10}                                                  [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,5}                                                 [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{4}                                               [         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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(times^#) = {},
                 Uargs(c_9) = {}, Uargs(square^#) = {}, Uargs(c_10) = {}
               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]
                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, 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_3() = [0]
                        [0]
                        [0]
                c_4(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_5(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_6(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]
                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_8() = [0]
                        [0]
                        [0]
                c_9(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_10(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 {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{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 {6}->{3,5}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{3,5}->{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 {6}->{3,5}->{4}: 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 {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{9}: 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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(times^#) = {},
                 Uargs(c_9) = {}, Uargs(square^#) = {}, Uargs(c_10) = {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]
                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, 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_3() = [0]
                        [0]
                        [0]
                c_4(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_5(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_6(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]
                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_8() = [0]
                        [0]
                        [0]
                c_9(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_10(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_8()}
               Weak Rules: {square^#(X) -> c_10(times^#(X, X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_10) = {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_8() = [1]
                        [0]
                        [0]
                square^#(x1) = [7 7 7] x1 + [7]
                               [7 7 7]      [7]
                               [7 7 7]      [7]
                c_10(x1) = [2 0 0] x1 + [7]
                           [0 0 0]      [7]
                           [0 0 0]      [7]
           
           * Path {11}->{10}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{7}: 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 {11}->{10}->{8}: inherited
             -------------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{8}->{7}: 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, cons(Y, Z))) ->
                   c_2(Y, 2ndsneg^#(N, Z))
              , 4: 2ndsneg^#(0(), Z) -> c_3()
              , 5: 2ndsneg^#(s(N), cons(X, cons(Y, Z))) ->
                   c_4(Y, 2ndspos^#(N, Z))
              , 6: pi^#(X) -> c_5(2ndspos^#(X, from(0())))
              , 7: plus^#(0(), Y) -> c_6(Y)
              , 8: plus^#(s(X), Y) -> c_7(plus^#(X, Y))
              , 9: times^#(0(), Y) -> c_8()
              , 10: times^#(s(X), Y) -> c_9(plus^#(Y, times(X, Y)))
              , 11: square^#(X) -> c_10(times^#(X, X))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                |->{9}                                                   [    YES(?,O(1))     ]
                |
                `->{10}                                                  [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,5}                                                 [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{4}                                               [         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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(times^#) = {},
                 Uargs(c_9) = {}, Uargs(square^#) = {}, Uargs(c_10) = {}
               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]
                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, 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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pi^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(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]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_10(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 {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{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 {6}->{3,5}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{3,5}->{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 {6}->{3,5}->{4}: 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 {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{9}: 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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(times^#) = {},
                 Uargs(c_9) = {}, Uargs(square^#) = {}, Uargs(c_10) = {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]
                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, 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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pi^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(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]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_10(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_8()}
               Weak Rules: {square^#(X) -> c_10(times^#(X, X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}, Uargs(square^#) = {}, Uargs(c_10) = {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_8() = [1]
                        [0]
                square^#(x1) = [7 7] x1 + [7]
                               [7 7]      [7]
                c_10(x1) = [2 0] x1 + [7]
                           [0 0]      [7]
           
           * Path {11}->{10}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{7}: 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 {11}->{10}->{8}: inherited
             -------------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{8}->{7}: 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, cons(Y, Z))) ->
                   c_2(Y, 2ndsneg^#(N, Z))
              , 4: 2ndsneg^#(0(), Z) -> c_3()
              , 5: 2ndsneg^#(s(N), cons(X, cons(Y, Z))) ->
                   c_4(Y, 2ndspos^#(N, Z))
              , 6: pi^#(X) -> c_5(2ndspos^#(X, from(0())))
              , 7: plus^#(0(), Y) -> c_6(Y)
              , 8: plus^#(s(X), Y) -> c_7(plus^#(X, Y))
              , 9: times^#(0(), Y) -> c_8()
              , 10: times^#(s(X), Y) -> c_9(plus^#(Y, times(X, Y)))
              , 11: square^#(X) -> c_10(times^#(X, X))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [     inherited      ]
                |
                |->{9}                                                   [         NA         ]
                |
                `->{10}                                                  [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3,5}                                                 [     inherited      ]
                    |
                    |->{2}                                               [         NA         ]
                    |
                    `->{4}                                               [         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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(times^#) = {},
                 Uargs(c_9) = {}, Uargs(square^#) = {}, Uargs(c_10) = {}
               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]
                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, x2) = [0] x1 + [0] x2 + [0]
                2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                pi^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_10(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 {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{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 {6}->{3,5}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {6}->{3,5}->{2}.
           
           * Path {6}->{3,5}->{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 {6}->{3,5}->{4}: 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 {11}: inherited
             --------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{9}: 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(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(2ndsneg^#) = {}, Uargs(c_4) = {},
                 Uargs(pi^#) = {}, Uargs(c_5) = {}, Uargs(plus^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(times^#) = {},
                 Uargs(c_9) = {}, Uargs(square^#) = {}, Uargs(c_10) = {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]
                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, x2) = [0] x1 + [0] x2 + [0]
                2ndsneg^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                pi^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_10(x1) = [1] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {11}->{10}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{7}: 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 {11}->{10}->{8}: inherited
             -------------------------------
             
             This path is subsumed by the proof of path {11}->{10}->{8}->{7}.
           
           * Path {11}->{10}->{8}->{7}: 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.