Problem Maude 06 MYNAT nosorts-noand

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 MYNAT nosorts-noand

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 MYNAT nosorts-noand

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  U11(tt(), M, N) -> U12(tt(), M, N)
     , U12(tt(), M, N) -> s(plus(N, M))
     , U21(tt(), M, N) -> U22(tt(), M, N)
     , U22(tt(), M, N) -> plus(x(N, M), N)
     , plus(N, 0()) -> N
     , plus(N, s(M)) -> U11(tt(), M, N)
     , x(N, 0()) -> 0()
     , x(N, s(M)) -> U21(tt(), M, N)}

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: U11^#(tt(), M, N) -> c_0(U12^#(tt(), M, N))
              , 2: U12^#(tt(), M, N) -> c_1(plus^#(N, M))
              , 3: U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
              , 4: U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
              , 5: plus^#(N, 0()) -> c_4()
              , 6: plus^#(N, s(M)) -> c_5(U11^#(tt(), M, N))
              , 7: x^#(N, 0()) -> c_6()
              , 8: x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))}

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

             ->{8}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{4}                                               [     inherited      ]
                        |
                        |->{1,6,2}                                       [     inherited      ]
                        |   |
                        |   `->{5}                                       [         NA         ]
                        |
                        `->{5}                                           [       MAYBE        ]

             ->{7}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {7}: 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(U11) = {}, Uargs(U12) = {}, Uargs(s) = {}, Uargs(plus) = {},
                 Uargs(U21) = {}, Uargs(U22) = {}, Uargs(x) = {}, Uargs(U11^#) = {},
                 Uargs(c_0) = {}, Uargs(U12^#) = {}, Uargs(c_1) = {},
                 Uargs(plus^#) = {}, Uargs(U21^#) = {}, Uargs(c_2) = {},
                 Uargs(U22^#) = {}, Uargs(c_3) = {}, Uargs(c_5) = {},
                 Uargs(x^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                U12(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]
                s(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]
                U21(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]
                U22(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]
                x(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]
                U11^#(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]
                U12^#(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_1(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]
                U21^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U22^#(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_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                x^#(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]

             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: {x^#(N, 0()) -> c_6()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(x^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                x^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
                              [0 0 0]      [2 2 0]      [3]
                              [0 0 0]      [2 2 2]      [3]
                c_6() = [0]
                        [1]
                        [1]

           * Path {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}: inherited
             -----------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}->{5}: NA
             ------------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(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.

           * Path {8}->{3}->{4}->{5}: MAYBE
             ------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(N, M))}

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
                  , U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
                  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
                  , plus^#(N, 0()) -> c_4()
                  , x(N, 0()) -> 0()
                  , x(N, s(M)) -> U21(tt(), M, N)
                  , U21(tt(), M, N) -> U22(tt(), M, N)
                  , U22(tt(), M, N) -> plus(x(N, M), N)
                  , plus(N, 0()) -> N
                  , plus(N, s(M)) -> U11(tt(), M, N)
                  , U11(tt(), M, N) -> U12(tt(), M, N)
                  , U12(tt(), M, N) -> s(plus(N, M))}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: U11^#(tt(), M, N) -> c_0(U12^#(tt(), M, N))
              , 2: U12^#(tt(), M, N) -> c_1(plus^#(N, M))
              , 3: U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
              , 4: U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
              , 5: plus^#(N, 0()) -> c_4()
              , 6: plus^#(N, s(M)) -> c_5(U11^#(tt(), M, N))
              , 7: x^#(N, 0()) -> c_6()
              , 8: x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))}

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

             ->{8}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{4}                                               [     inherited      ]
                        |
                        |->{1,6,2}                                       [     inherited      ]
                        |   |
                        |   `->{5}                                       [         NA         ]
                        |
                        `->{5}                                           [       MAYBE        ]

             ->{7}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {7}: 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(U11) = {}, Uargs(U12) = {}, Uargs(s) = {}, Uargs(plus) = {},
                 Uargs(U21) = {}, Uargs(U22) = {}, Uargs(x) = {}, Uargs(U11^#) = {},
                 Uargs(c_0) = {}, Uargs(U12^#) = {}, Uargs(c_1) = {},
                 Uargs(plus^#) = {}, Uargs(U21^#) = {}, Uargs(c_2) = {},
                 Uargs(U22^#) = {}, Uargs(c_3) = {}, Uargs(c_5) = {},
                 Uargs(x^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U21(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                U22(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                x(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11^#(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]
                U12^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                U21^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U22^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                x^#(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]

             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: {x^#(N, 0()) -> c_6()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(x^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                x^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
                              [0 0]      [2 2]      [7]
                c_6() = [0]
                        [1]

           * Path {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}: inherited
             -----------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}->{5}: NA
             ------------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(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.

           * Path {8}->{3}->{4}->{5}: MAYBE
             ------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(N, M))}

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
                  , U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
                  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
                  , plus^#(N, 0()) -> c_4()
                  , x(N, 0()) -> 0()
                  , x(N, s(M)) -> U21(tt(), M, N)
                  , U21(tt(), M, N) -> U22(tt(), M, N)
                  , U22(tt(), M, N) -> plus(x(N, M), N)
                  , plus(N, 0()) -> N
                  , plus(N, s(M)) -> U11(tt(), M, N)
                  , U11(tt(), M, N) -> U12(tt(), M, N)
                  , U12(tt(), M, N) -> s(plus(N, M))}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: U11^#(tt(), M, N) -> c_0(U12^#(tt(), M, N))
              , 2: U12^#(tt(), M, N) -> c_1(plus^#(N, M))
              , 3: U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
              , 4: U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
              , 5: plus^#(N, 0()) -> c_4()
              , 6: plus^#(N, s(M)) -> c_5(U11^#(tt(), M, N))
              , 7: x^#(N, 0()) -> c_6()
              , 8: x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))}

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

             ->{8}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{4}                                               [     inherited      ]
                        |
                        |->{1,6,2}                                       [     inherited      ]
                        |   |
                        |   `->{5}                                       [         NA         ]
                        |
                        `->{5}                                           [       MAYBE        ]

             ->{7}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {7}: 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(U11) = {}, Uargs(U12) = {}, Uargs(s) = {}, Uargs(plus) = {},
                 Uargs(U21) = {}, Uargs(U22) = {}, Uargs(x) = {}, Uargs(U11^#) = {},
                 Uargs(c_0) = {}, Uargs(U12^#) = {}, Uargs(c_1) = {},
                 Uargs(plus^#) = {}, Uargs(U21^#) = {}, Uargs(c_2) = {},
                 Uargs(U22^#) = {}, Uargs(c_3) = {}, Uargs(c_5) = {},
                 Uargs(x^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                U21(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                U22(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                x(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                U21^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                U22^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                x^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {x^#(N, 0()) -> c_6()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(x^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                x^#(x1, x2) = [0] x1 + [1] x2 + [7]
                c_6() = [1]

           * Path {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}: inherited
             -----------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}->{5}: NA
             ------------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(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.

           * Path {8}->{3}->{4}->{5}: MAYBE
             ------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(N, M))}

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
                  , U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
                  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
                  , plus^#(N, 0()) -> c_4()
                  , x(N, 0()) -> 0()
                  , x(N, s(M)) -> U21(tt(), M, N)
                  , U21(tt(), M, N) -> U22(tt(), M, N)
                  , U22(tt(), M, N) -> plus(x(N, M), N)
                  , plus(N, 0()) -> N
                  , plus(N, s(M)) -> U11(tt(), M, N)
                  , U11(tt(), M, N) -> U12(tt(), M, N)
                  , U12(tt(), M, N) -> s(plus(N, M))}

             Proof Output:
               The input cannot be shown compatible

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 MYNAT nosorts-noand

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 MYNAT nosorts-noand

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  U11(tt(), M, N) -> U12(tt(), M, N)
     , U12(tt(), M, N) -> s(plus(N, M))
     , U21(tt(), M, N) -> U22(tt(), M, N)
     , U22(tt(), M, N) -> plus(x(N, M), N)
     , plus(N, 0()) -> N
     , plus(N, s(M)) -> U11(tt(), M, N)
     , x(N, 0()) -> 0()
     , x(N, s(M)) -> U21(tt(), M, N)}

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: U11^#(tt(), M, N) -> c_0(U12^#(tt(), M, N))
              , 2: U12^#(tt(), M, N) -> c_1(plus^#(N, M))
              , 3: U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
              , 4: U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
              , 5: plus^#(N, 0()) -> c_4(N)
              , 6: plus^#(N, s(M)) -> c_5(U11^#(tt(), M, N))
              , 7: x^#(N, 0()) -> c_6()
              , 8: x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))}

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

             ->{8}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{4}                                               [     inherited      ]
                        |
                        |->{1,6,2}                                       [     inherited      ]
                        |   |
                        |   `->{5}                                       [         NA         ]
                        |
                        `->{5}                                           [       MAYBE        ]

             ->{7}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {7}: 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(U11) = {}, Uargs(U12) = {}, Uargs(s) = {}, Uargs(plus) = {},
                 Uargs(U21) = {}, Uargs(U22) = {}, Uargs(x) = {}, Uargs(U11^#) = {},
                 Uargs(c_0) = {}, Uargs(U12^#) = {}, Uargs(c_1) = {},
                 Uargs(plus^#) = {}, Uargs(U21^#) = {}, Uargs(c_2) = {},
                 Uargs(U22^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(x^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                U12(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]
                s(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]
                U21(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]
                U22(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]
                x(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]
                U11^#(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]
                U12^#(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_1(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]
                U21^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U22^#(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_3(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]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                x^#(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]

             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: {x^#(N, 0()) -> c_6()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(x^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                x^#(x1, x2) = [0 0 0] x1 + [0 2 0] x2 + [7]
                              [0 0 0]      [2 2 0]      [3]
                              [0 0 0]      [2 2 2]      [3]
                c_6() = [0]
                        [1]
                        [1]

           * Path {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}: inherited
             -----------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}->{5}: NA
             ------------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(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.

           * Path {8}->{3}->{4}->{5}: MAYBE
             ------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(N, M))}

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
                  , U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
                  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
                  , plus^#(N, 0()) -> c_4(N)
                  , x(N, 0()) -> 0()
                  , x(N, s(M)) -> U21(tt(), M, N)
                  , U21(tt(), M, N) -> U22(tt(), M, N)
                  , U22(tt(), M, N) -> plus(x(N, M), N)
                  , plus(N, 0()) -> N
                  , plus(N, s(M)) -> U11(tt(), M, N)
                  , U11(tt(), M, N) -> U12(tt(), M, N)
                  , U12(tt(), M, N) -> s(plus(N, M))}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: U11^#(tt(), M, N) -> c_0(U12^#(tt(), M, N))
              , 2: U12^#(tt(), M, N) -> c_1(plus^#(N, M))
              , 3: U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
              , 4: U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
              , 5: plus^#(N, 0()) -> c_4(N)
              , 6: plus^#(N, s(M)) -> c_5(U11^#(tt(), M, N))
              , 7: x^#(N, 0()) -> c_6()
              , 8: x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))}

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

             ->{8}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{4}                                               [     inherited      ]
                        |
                        |->{1,6,2}                                       [     inherited      ]
                        |   |
                        |   `->{5}                                       [         NA         ]
                        |
                        `->{5}                                           [       MAYBE        ]

             ->{7}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {7}: 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(U11) = {}, Uargs(U12) = {}, Uargs(s) = {}, Uargs(plus) = {},
                 Uargs(U21) = {}, Uargs(U22) = {}, Uargs(x) = {}, Uargs(U11^#) = {},
                 Uargs(c_0) = {}, Uargs(U12^#) = {}, Uargs(c_1) = {},
                 Uargs(plus^#) = {}, Uargs(U21^#) = {}, Uargs(c_2) = {},
                 Uargs(U22^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(x^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                U12(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                U21(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                U22(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                x(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11^#(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]
                U12^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                U21^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U22^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                x^#(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]

             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: {x^#(N, 0()) -> c_6()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(x^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                x^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
                              [0 0]      [2 2]      [7]
                c_6() = [0]
                        [1]

           * Path {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}: inherited
             -----------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}->{5}: NA
             ------------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(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.

           * Path {8}->{3}->{4}->{5}: MAYBE
             ------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(N, M))}

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
                  , U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
                  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
                  , plus^#(N, 0()) -> c_4(N)
                  , x(N, 0()) -> 0()
                  , x(N, s(M)) -> U21(tt(), M, N)
                  , U21(tt(), M, N) -> U22(tt(), M, N)
                  , U22(tt(), M, N) -> plus(x(N, M), N)
                  , plus(N, 0()) -> N
                  , plus(N, s(M)) -> U11(tt(), M, N)
                  , U11(tt(), M, N) -> U12(tt(), M, N)
                  , U12(tt(), M, N) -> s(plus(N, M))}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: U11^#(tt(), M, N) -> c_0(U12^#(tt(), M, N))
              , 2: U12^#(tt(), M, N) -> c_1(plus^#(N, M))
              , 3: U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
              , 4: U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
              , 5: plus^#(N, 0()) -> c_4(N)
              , 6: plus^#(N, s(M)) -> c_5(U11^#(tt(), M, N))
              , 7: x^#(N, 0()) -> c_6()
              , 8: x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))}

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

             ->{8}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{4}                                               [     inherited      ]
                        |
                        |->{1,6,2}                                       [     inherited      ]
                        |   |
                        |   `->{5}                                       [         NA         ]
                        |
                        `->{5}                                           [       MAYBE        ]

             ->{7}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {7}: 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(U11) = {}, Uargs(U12) = {}, Uargs(s) = {}, Uargs(plus) = {},
                 Uargs(U21) = {}, Uargs(U22) = {}, Uargs(x) = {}, Uargs(U11^#) = {},
                 Uargs(c_0) = {}, Uargs(U12^#) = {}, Uargs(c_1) = {},
                 Uargs(plus^#) = {}, Uargs(U21^#) = {}, Uargs(c_2) = {},
                 Uargs(U22^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(x^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                U11(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                U12(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                U21(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                U22(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                x(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                U12^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                U21^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                U22^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                x^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {x^#(N, 0()) -> c_6()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(x^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                x^#(x1, x2) = [0] x1 + [1] x2 + [7]
                c_6() = [1]

           * Path {8}: inherited
             -------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}: inherited
             -----------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {8}->{3}->{4}->{1,6,2}->{5}.

           * Path {8}->{3}->{4}->{1,6,2}->{5}: NA
             ------------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(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.

           * Path {8}->{3}->{4}->{5}: MAYBE
             ------------------------------

             The usable rules for this path are:

               {  x(N, 0()) -> 0()
                , x(N, s(M)) -> U21(tt(), M, N)
                , U21(tt(), M, N) -> U22(tt(), M, N)
                , U22(tt(), M, N) -> plus(x(N, M), N)
                , plus(N, 0()) -> N
                , plus(N, s(M)) -> U11(tt(), M, N)
                , U11(tt(), M, N) -> U12(tt(), M, N)
                , U12(tt(), M, N) -> s(plus(N, M))}

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  U22^#(tt(), M, N) -> c_3(plus^#(x(N, M), N))
                  , U21^#(tt(), M, N) -> c_2(U22^#(tt(), M, N))
                  , x^#(N, s(M)) -> c_7(U21^#(tt(), M, N))
                  , plus^#(N, 0()) -> c_4(N)
                  , x(N, 0()) -> 0()
                  , x(N, s(M)) -> U21(tt(), M, N)
                  , U21(tt(), M, N) -> U22(tt(), M, N)
                  , U22(tt(), M, N) -> plus(x(N, M), N)
                  , plus(N, 0()) -> N
                  , plus(N, s(M)) -> U11(tt(), M, N)
                  , U11(tt(), M, N) -> U12(tt(), M, N)
                  , U12(tt(), M, N) -> s(plus(N, M))}

             Proof Output:
               The input cannot be shown compatible

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

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

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