Problem Mixed innermost innermost4

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost4

stdout:

MAYBE

Problem:
f(a(x),y,s(z),u) -> f(a(b()),y,z,g(x,y,s(z),u))
g(x,y,z,u) -> h(x,y,z,u)
h(b(),y,z,u) -> f(y,y,z,u)
a(b()) -> c()

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost4

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost4

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))
     , g(x, y, z, u) -> h(x, y, z, u)
     , h(b(), y, z, u) -> f(y, y, z, u)
     , a(b()) -> c()}

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: f^#(a(x), y, s(z), u) ->
                   c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
              , 2: g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
              , 3: h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
              , 4: a^#(b()) -> c_3()}

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

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{1}                                               [         NA         ]



         Sub-problems:
         -------------
           * Path {2}: inherited
             -------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}->{1}: NA
             ----------------------

             The usable rules for this path are:

               {  g(x, y, z, u) -> h(x, y, z, u)
                , a(b()) -> c()
                , h(b(), y, z, u) -> f(y, y, z, u)
                , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             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 {4}: 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(f) = {}, Uargs(a) = {}, Uargs(s) = {}, Uargs(g) = {},
                 Uargs(h) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}, Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                a(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                h(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 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]
                g^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 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]
                h^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 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]
                a^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [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: {a^#(b()) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [2]
                      [2]
                      [2]
                a^#(x1) = [0 2 0] x1 + [7]
                          [2 2 0]      [3]
                          [2 2 2]      [3]
                c_3() = [0]
                        [1]
                        [1]

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

             {  1: f^#(a(x), y, s(z), u) ->
                   c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
              , 2: g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
              , 3: h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
              , 4: a^#(b()) -> c_3()}

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

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{1}                                               [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {2}: inherited
             -------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}->{1}: MAYBE
             -------------------------

             The usable rules for this path are:

               {  g(x, y, z, u) -> h(x, y, z, u)
                , a(b()) -> c()
                , h(b(), y, z, u) -> f(y, y, z, u)
                , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             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:
                 {  h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
                  , g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
                  , f^#(a(x), y, s(z), u) -> c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
                  , g(x, y, z, u) -> h(x, y, z, u)
                  , a(b()) -> c()
                  , h(b(), y, z, u) -> f(y, y, z, u)
                  , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             Proof Output:
               The input cannot be shown compatible

           * Path {4}: 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(f) = {}, Uargs(a) = {}, Uargs(s) = {}, Uargs(g) = {},
                 Uargs(h) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}, Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 0]      [0 0]      [0 0]      [0 0]      [0]
                a(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 0]      [0 0]      [0 0]      [0 0]      [0]
                h(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                h^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [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: {a^#(b()) -> c_3()}
               Weak Rules: {}

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

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

             {  1: f^#(a(x), y, s(z), u) ->
                   c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
              , 2: g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
              , 3: h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
              , 4: a^#(b()) -> c_3()}

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

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{1}                                               [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {2}: inherited
             -------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}->{1}: MAYBE
             -------------------------

             The usable rules for this path are:

               {  g(x, y, z, u) -> h(x, y, z, u)
                , a(b()) -> c()
                , h(b(), y, z, u) -> f(y, y, z, u)
                , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             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:
                 {  h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
                  , g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
                  , f^#(a(x), y, s(z), u) -> c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
                  , g(x, y, z, u) -> h(x, y, z, u)
                  , a(b()) -> c()
                  , h(b(), y, z, u) -> f(y, y, z, u)
                  , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             Proof Output:
               The input cannot be shown compatible

           * Path {4}: 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(f) = {}, Uargs(a) = {}, Uargs(s) = {}, Uargs(g) = {},
                 Uargs(h) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}, Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                a(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                h(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c() = [0]
                f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_1(x1) = [0] x1 + [0]
                h^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_2(x1) = [0] x1 + [0]
                a^#(x1) = [0] x1 + [0]
                c_3() = [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: {a^#(b()) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [7]
                a^#(x1) = [1] x1 + [7]
                c_3() = [1]

    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	Mixed innermost innermost4

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Mixed innermost innermost4

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))
     , g(x, y, z, u) -> h(x, y, z, u)
     , h(b(), y, z, u) -> f(y, y, z, u)
     , a(b()) -> c()}

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: f^#(a(x), y, s(z), u) ->
                   c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
              , 2: g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
              , 3: h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
              , 4: a^#(b()) -> c_3()}

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

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{1}                                               [         NA         ]



         Sub-problems:
         -------------
           * Path {2}: inherited
             -------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}->{1}: NA
             ----------------------

             The usable rules for this path are:

               {  g(x, y, z, u) -> h(x, y, z, u)
                , a(b()) -> c()
                , h(b(), y, z, u) -> f(y, y, z, u)
                , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             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 {4}: 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(f) = {}, Uargs(a) = {}, Uargs(s) = {}, Uargs(g) = {},
                 Uargs(h) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}, Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                a(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                b() = [0]
                      [0]
                      [0]
                g(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                h(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                    [0 0 0]      [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c() = [0]
                      [0]
                      [0]
                f^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 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]
                g^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 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]
                h^#(x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 0] x4 + [0]
                                      [0 0 0]      [0 0 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]
                a^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [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: {a^#(b()) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [2]
                      [2]
                      [2]
                a^#(x1) = [0 2 0] x1 + [7]
                          [2 2 0]      [3]
                          [2 2 2]      [3]
                c_3() = [0]
                        [1]
                        [1]

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

             {  1: f^#(a(x), y, s(z), u) ->
                   c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
              , 2: g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
              , 3: h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
              , 4: a^#(b()) -> c_3()}

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

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{1}                                               [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {2}: inherited
             -------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}->{1}: MAYBE
             -------------------------

             The usable rules for this path are:

               {  g(x, y, z, u) -> h(x, y, z, u)
                , a(b()) -> c()
                , h(b(), y, z, u) -> f(y, y, z, u)
                , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             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:
                 {  h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
                  , g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
                  , f^#(a(x), y, s(z), u) -> c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
                  , g(x, y, z, u) -> h(x, y, z, u)
                  , a(b()) -> c()
                  , h(b(), y, z, u) -> f(y, y, z, u)
                  , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             Proof Output:
               The input cannot be shown compatible

           * Path {4}: 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(f) = {}, Uargs(a) = {}, Uargs(s) = {}, Uargs(g) = {},
                 Uargs(h) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}, Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 0]      [0 0]      [0 0]      [0 0]      [0]
                a(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                b() = [0]
                      [0]
                g(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 0]      [0 0]      [0 0]      [0 0]      [0]
                h(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                    [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c() = [0]
                      [0]
                f^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                h^#(x1, x2, x3, x4) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0]
                                      [0 0]      [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                a^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [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: {a^#(b()) -> c_3()}
               Weak Rules: {}

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

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

             {  1: f^#(a(x), y, s(z), u) ->
                   c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
              , 2: g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
              , 3: h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
              , 4: a^#(b()) -> c_3()}

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

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    `->{1}                                               [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {2}: inherited
             -------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}: inherited
             ------------------------

             This path is subsumed by the proof of path {2}->{3}->{1}.

           * Path {2}->{3}->{1}: MAYBE
             -------------------------

             The usable rules for this path are:

               {  g(x, y, z, u) -> h(x, y, z, u)
                , a(b()) -> c()
                , h(b(), y, z, u) -> f(y, y, z, u)
                , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             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:
                 {  h^#(b(), y, z, u) -> c_2(f^#(y, y, z, u))
                  , g^#(x, y, z, u) -> c_1(h^#(x, y, z, u))
                  , f^#(a(x), y, s(z), u) -> c_0(f^#(a(b()), y, z, g(x, y, s(z), u)))
                  , g(x, y, z, u) -> h(x, y, z, u)
                  , a(b()) -> c()
                  , h(b(), y, z, u) -> f(y, y, z, u)
                  , f(a(x), y, s(z), u) -> f(a(b()), y, z, g(x, y, s(z), u))}

             Proof Output:
               The input cannot be shown compatible

           * Path {4}: 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(f) = {}, Uargs(a) = {}, Uargs(s) = {}, Uargs(g) = {},
                 Uargs(h) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(g^#) = {},
                 Uargs(c_1) = {}, Uargs(h^#) = {}, Uargs(c_2) = {}, Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                a(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                b() = [0]
                g(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                h(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c() = [0]
                f^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_0(x1) = [0] x1 + [0]
                g^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_1(x1) = [0] x1 + [0]
                h^#(x1, x2, x3, x4) = [0] x1 + [0] x2 + [0] x3 + [0] x4 + [0]
                c_2(x1) = [0] x1 + [0]
                a^#(x1) = [0] x1 + [0]
                c_3() = [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: {a^#(b()) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(a^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                b() = [7]
                a^#(x1) = [1] x1 + [7]
                c_3() = [1]

    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.