Problem Strategy outermost added 08 Ex9 BLR02

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex9 BLR02

stdout:

MAYBE

Problem:
filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M))
filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M))
sieve(cons(0(),Y)) -> cons(0(),sieve(Y))
sieve(cons(s(N),Y)) -> cons(s(N),sieve(filter(Y,N,N)))
nats(N) -> cons(N,nats(s(N)))
zprimes() -> sieve(nats(s(s(0()))))

Proof:
Open

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex9 BLR02

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
     , filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
     , sieve(cons(0(), Y)) -> cons(0(), sieve(Y))
     , sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
     , nats(N) -> cons(N, nats(s(N)))
     , zprimes() -> sieve(nats(s(s(0()))))}

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: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
              , 2: filter^#(cons(X, Y), s(N), M) -> c_1(filter^#(Y, N, M))
              , 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
              , 4: sieve^#(cons(s(N), Y)) -> c_3(sieve^#(filter(Y, N, N)))
              , 5: nats^#(N) -> c_4(nats^#(s(N)))
              , 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}

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

             ->{6}                                                       [     inherited      ]
                |
                `->{3,4}                                                 [         NA         ]

             ->{5}                                                       [       MAYBE        ]

             ->{1,2}                                                     [         NA         ]



         Sub-problems:
         -------------
           * Path {1,2}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {1}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [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]
                sieve(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                nats(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                zprimes() = [0]
                            [0]
                            [0]
                filter^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                       [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                       [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sieve^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                nats^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zprimes^#() = [0]
                              [0]
                              [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

           * Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 0]      [0]
                sieve(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                nats(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                zprimes() = [0]
                            [0]
                            [0]
                filter^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 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]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sieve^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                nats^#(x1) = [3 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                zprimes^#() = [0]
                              [0]
                              [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {nats^#(N) -> c_4(nats^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {6}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  nats(N) -> cons(N, nats(s(N)))
                , filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
                , filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}

             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: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
              , 2: filter^#(cons(X, Y), s(N), M) -> c_1(filter^#(Y, N, M))
              , 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
              , 4: sieve^#(cons(s(N), Y)) -> c_3(sieve^#(filter(Y, N, N)))
              , 5: nats^#(N) -> c_4(nats^#(s(N)))
              , 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}

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

             ->{6}                                                       [     inherited      ]
                |
                `->{3,4}                                                 [         NA         ]

             ->{5}                                                       [       MAYBE        ]

             ->{1,2}                                                     [         NA         ]



         Sub-problems:
         -------------
           * Path {1,2}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {1}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [1 1] x2 + [0]
                               [0 1]      [0 1]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                nats(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                zprimes() = [0]
                            [0]
                filter^#(x1, x2, x3) = [1 3] x1 + [0 0] x2 + [1 0] x3 + [0]
                                       [3 3]      [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                nats^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zprimes^#() = [0]
                              [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                nats(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                zprimes() = [0]
                            [0]
                filter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                       [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                nats^#(x1) = [3 3] x1 + [0]
                             [3 3]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                zprimes^#() = [0]
                              [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {nats^#(N) -> c_4(nats^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {6}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  nats(N) -> cons(N, nats(s(N)))
                , filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
                , filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}

             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: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
              , 2: filter^#(cons(X, Y), s(N), M) -> c_1(filter^#(Y, N, M))
              , 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
              , 4: sieve^#(cons(s(N), Y)) -> c_3(sieve^#(filter(Y, N, N)))
              , 5: nats^#(N) -> c_4(nats^#(s(N)))
              , 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}

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

             ->{6}                                                       [     inherited      ]
                |
                `->{3,4}                                                 [         NA         ]

             ->{5}                                                       [       MAYBE        ]

             ->{1,2}                                                     [         NA         ]



         Sub-problems:
         -------------
           * Path {1,2}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {1}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                sieve(x1) = [0] x1 + [0]
                nats(x1) = [0] x1 + [0]
                zprimes() = [0]
                filter^#(x1, x2, x3) = [3] x1 + [1] x2 + [1] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                nats^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                zprimes^#() = [0]
                c_5(x1) = [0] x1 + [0]

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

           * Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {1}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                sieve(x1) = [0] x1 + [0]
                nats(x1) = [0] x1 + [0]
                zprimes() = [0]
                filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                nats^#(x1) = [3] x1 + [0]
                c_4(x1) = [1] x1 + [0]
                zprimes^#() = [0]
                c_5(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: {nats^#(N) -> c_4(nats^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {6}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  nats(N) -> cons(N, nats(s(N)))
                , filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
                , filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}

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

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex9 BLR02

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex9 BLR02

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
     , filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
     , sieve(cons(0(), Y)) -> cons(0(), sieve(Y))
     , sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
     , nats(N) -> cons(N, nats(s(N)))
     , zprimes() -> sieve(nats(s(s(0()))))}

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: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
              , 2: filter^#(cons(X, Y), s(N), M) -> c_1(X, filter^#(Y, N, M))
              , 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
              , 4: sieve^#(cons(s(N), Y)) -> c_3(N, sieve^#(filter(Y, N, N)))
              , 5: nats^#(N) -> c_4(N, nats^#(s(N)))
              , 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}

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

             ->{6}                                                       [     inherited      ]
                |
                `->{3,4}                                                 [         NA         ]

             ->{5}                                                       [       MAYBE        ]

             ->{1,2}                                                     [         NA         ]



         Sub-problems:
         -------------
           * Path {1,2}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {2}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                sieve(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                nats(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                zprimes() = [0]
                            [0]
                            [0]
                filter^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                       [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                       [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_1(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                sieve^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                nats^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 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]
                zprimes^#() = [0]
                              [0]
                              [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

           * Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {2}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 1 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [0]
                sieve(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                nats(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                zprimes() = [0]
                            [0]
                            [0]
                filter^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                       [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                       [0 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]
                c_1(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sieve^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                nats^#(x1) = [1 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_4(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]
                zprimes^#() = [0]
                              [0]
                              [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {nats^#(N) -> c_4(N, nats^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {6}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  nats(N) -> cons(N, nats(s(N)))
                , filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
                , filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}

             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: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
              , 2: filter^#(cons(X, Y), s(N), M) -> c_1(X, filter^#(Y, N, M))
              , 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
              , 4: sieve^#(cons(s(N), Y)) -> c_3(N, sieve^#(filter(Y, N, N)))
              , 5: nats^#(N) -> c_4(N, nats^#(s(N)))
              , 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}

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

             ->{6}                                                       [     inherited      ]
                |
                `->{3,4}                                                 [         NA         ]

             ->{5}                                                       [       MAYBE        ]

             ->{1,2}                                                     [         NA         ]



         Sub-problems:
         -------------
           * Path {1,2}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {2}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
                               [0 1]      [0 1]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 1]      [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                nats(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                zprimes() = [0]
                            [0]
                filter^#(x1, x2, x3) = [3 3] x1 + [3 3] x2 + [0 3] x3 + [0]
                                       [3 3]      [3 3]      [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_1(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                nats^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                zprimes^#() = [0]
                              [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {2}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                nats(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                zprimes() = [0]
                            [0]
                filter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                       [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                nats^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_4(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                zprimes^#() = [0]
                              [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {nats^#(N) -> c_4(N, nats^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {6}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  nats(N) -> cons(N, nats(s(N)))
                , filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
                , filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}

             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: filter^#(cons(X, Y), 0(), M) -> c_0(filter^#(Y, M, M))
              , 2: filter^#(cons(X, Y), s(N), M) -> c_1(X, filter^#(Y, N, M))
              , 3: sieve^#(cons(0(), Y)) -> c_2(sieve^#(Y))
              , 4: sieve^#(cons(s(N), Y)) -> c_3(N, sieve^#(filter(Y, N, N)))
              , 5: nats^#(N) -> c_4(N, nats^#(s(N)))
              , 6: zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}

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

             ->{6}                                                       [     inherited      ]
                |
                `->{3,4}                                                 [         NA         ]

             ->{5}                                                       [       MAYBE        ]

             ->{1,2}                                                     [         NA         ]



         Sub-problems:
         -------------
           * Path {1,2}: NA
             --------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {1}, Uargs(c_1) = {2}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                sieve(x1) = [0] x1 + [0]
                nats(x1) = [0] x1 + [0]
                zprimes() = [0]
                filter^#(x1, x2, x3) = [3] x1 + [0] x2 + [3] x3 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1(x1, x2) = [1] x1 + [1] x2 + [0]
                sieve^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                nats^#(x1) = [0] x1 + [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                zprimes^#() = [0]
                c_5(x1) = [0] x1 + [0]

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

           * Path {5}: 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(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                 Uargs(sieve) = {}, Uargs(nats) = {}, Uargs(filter^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(sieve^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(nats^#) = {},
                 Uargs(c_4) = {2}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                sieve(x1) = [0] x1 + [0]
                nats(x1) = [0] x1 + [0]
                zprimes() = [0]
                filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1, x2) = [0] x1 + [0] x2 + [0]
                sieve^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                nats^#(x1) = [3] x1 + [0]
                c_4(x1, x2) = [2] x1 + [1] x2 + [0]
                zprimes^#() = [0]
                c_5(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: {nats^#(N) -> c_4(N, nats^#(s(N)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {6}: inherited
             -------------------

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

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

             The usable rules for this path are:

               {  nats(N) -> cons(N, nats(s(N)))
                , filter(cons(X, Y), 0(), M) -> cons(0(), filter(Y, M, M))
                , filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))}

             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.