'Weak Dependency Graph [60.0]'
------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  a(x1) -> g(d(x1))
     , b(b(b(x1))) -> c(d(c(x1)))
     , b(b(x1)) -> a(g(g(x1)))
     , c(d(x1)) -> g(g(x1))
     , g(g(g(x1))) -> b(b(x1))}

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  a^#(x1) -> c_0(g^#(d(x1)))
    , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
    , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
    , c^#(d(x1)) -> c_3(g^#(g(x1)))
    , g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
  
  The usable rules are:
   {  b(b(b(x1))) -> c(d(c(x1)))
    , b(b(x1)) -> a(g(g(x1)))
    , c(d(x1)) -> g(g(x1))
    , g(g(g(x1))) -> b(b(x1))
    , a(x1) -> g(d(x1))}
  
  The estimated dependency graph contains the following edges:
   {b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
     ==> {c^#(d(x1)) -> c_3(g^#(g(x1)))}
   {b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
     ==> {a^#(x1) -> c_0(g^#(d(x1)))}
   {c^#(d(x1)) -> c_3(g^#(g(x1)))}
     ==> {g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
   {g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
     ==> {b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
   {g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
     ==> {b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
  
  We consider the following path(s):
   1) {  b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
       , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
       , c^#(d(x1)) -> c_3(g^#(g(x1)))
       , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
       , a^#(x1) -> c_0(g^#(d(x1)))}
      
      The usable rules for this path are the following:
      {  b(b(b(x1))) -> c(d(c(x1)))
       , b(b(x1)) -> a(g(g(x1)))
       , c(d(x1)) -> g(g(x1))
       , g(g(g(x1))) -> b(b(x1))
       , a(x1) -> g(d(x1))}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  b(b(b(x1))) -> c(d(c(x1)))
               , b(b(x1)) -> a(g(g(x1)))
               , c(d(x1)) -> g(g(x1))
               , g(g(g(x1))) -> b(b(x1))
               , a(x1) -> g(d(x1))
               , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
               , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
               , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
               , c^#(d(x1)) -> c_3(g^#(g(x1)))
               , a^#(x1) -> c_0(g^#(d(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a(x1) -> g(d(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(x1) -> g(d(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [15]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(d(x1)) -> c_3(g^#(g(x1)))}
            and weakly orienting the rules
            {a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(d(x1)) -> c_3(g^#(g(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [2]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
            and weakly orienting the rules
            {  c^#(d(x1)) -> c_3(g^#(g(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  g^#(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {a^#(x1) -> c_0(g^#(d(x1)))}
            and weakly orienting the rules
            {  b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(x1) -> c_0(g^#(d(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  g^#(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
            and weakly orienting the rules
            {  a^#(x1) -> c_0(g^#(d(x1)))
             , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  g^#(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  b(b(b(x1))) -> c(d(c(x1)))
             , b(b(x1)) -> a(g(g(x1)))}
            and weakly orienting the rules
            {  b^#(b(x1)) -> c_2(a^#(g(g(x1))))
             , a^#(x1) -> c_0(g^#(d(x1)))
             , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  b(b(b(x1))) -> c(d(c(x1)))
               , b(b(x1)) -> a(g(g(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  g(x1) = [1] x1 + [2]
                  d(x1) = [1] x1 + [2]
                  b(x1) = [1] x1 + [10]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [12]
                  c_0(x1) = [1] x1 + [0]
                  g^#(x1) = [1] x1 + [6]
                  b^#(x1) = [1] x1 + [10]
                  c_1(x1) = [1] x1 + [9]
                  c^#(x1) = [1] x1 + [14]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [8]
                  c_4(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(d(x1)) -> g(g(x1))}
            and weakly orienting the rules
            {  b(b(b(x1))) -> c(d(c(x1)))
             , b(b(x1)) -> a(g(g(x1)))
             , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
             , a^#(x1) -> c_0(g^#(d(x1)))
             , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(d(x1)) -> g(g(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [8]
                  c(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [8]
                  c_0(x1) = [1] x1 + [7]
                  g^#(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [8]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [3]
                  c_4(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules:
                {  g(g(g(x1))) -> b(b(x1))
                 , g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
              Weak Rules:
                {  c(d(x1)) -> g(g(x1))
                 , b(b(b(x1))) -> c(d(c(x1)))
                 , b(b(x1)) -> a(g(g(x1)))
                 , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
                 , a^#(x1) -> c_0(g^#(d(x1)))
                 , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
                 , c^#(d(x1)) -> c_3(g^#(g(x1)))
                 , a(x1) -> g(d(x1))}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules:
                  {  g(g(g(x1))) -> b(b(x1))
                   , g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
                Weak Rules:
                  {  c(d(x1)) -> g(g(x1))
                   , b(b(b(x1))) -> c(d(c(x1)))
                   , b(b(x1)) -> a(g(g(x1)))
                   , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
                   , a^#(x1) -> c_0(g^#(d(x1)))
                   , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
                   , c^#(d(x1)) -> c_3(g^#(g(x1)))
                   , a(x1) -> g(d(x1))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  g_0(3) -> 13
                 , d_0(3) -> 3
                 , a^#_0(3) -> 6
                 , c_0_0(8) -> 6
                 , g^#_0(3) -> 8
                 , g^#_0(13) -> 12
                 , b^#_0(3) -> 9
                 , c^#_0(3) -> 11
                 , c_3_0(12) -> 11}
      
   2) {  b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
       , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
       , c^#(d(x1)) -> c_3(g^#(g(x1)))}
      
      The usable rules for this path are the following:
      {  b(b(b(x1))) -> c(d(c(x1)))
       , b(b(x1)) -> a(g(g(x1)))
       , c(d(x1)) -> g(g(x1))
       , g(g(g(x1))) -> b(b(x1))
       , a(x1) -> g(d(x1))}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  b(b(b(x1))) -> c(d(c(x1)))
               , b(b(x1)) -> a(g(g(x1)))
               , c(d(x1)) -> g(g(x1))
               , g(g(g(x1))) -> b(b(x1))
               , a(x1) -> g(d(x1))
               , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
               , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
               , c^#(d(x1)) -> c_3(g^#(g(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a(x1) -> g(d(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(x1) -> g(d(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [1] x1 + [7]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
            and weakly orienting the rules
            {a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [5]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [3]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(d(x1)) -> c_3(g^#(g(x1)))}
            and weakly orienting the rules
            {  g^#(g(g(x1))) -> c_4(b^#(b(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(d(x1)) -> c_3(g^#(g(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [2]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {g(g(g(x1))) -> b(b(x1))}
            and weakly orienting the rules
            {  c^#(d(x1)) -> c_3(g^#(g(x1)))
             , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {g(g(g(x1))) -> b(b(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [12]
                  g(x1) = [1] x1 + [8]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [8]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
            and weakly orienting the rules
            {  g(g(g(x1))) -> b(b(x1))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  g(x1) = [1] x1 + [5]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [3]
                  b^#(x1) = [1] x1 + [12]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [8]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(d(x1)) -> g(g(x1))}
            and weakly orienting the rules
            {  b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , g(g(g(x1))) -> b(b(x1))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(d(x1)) -> g(g(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [9]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [8]
                  b^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b(b(b(x1))) -> c(d(c(x1)))}
            and weakly orienting the rules
            {  c(d(x1)) -> g(g(x1))
             , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , g(g(g(x1))) -> b(b(x1))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b(b(b(x1))) -> c(d(c(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [15]
                  g(x1) = [1] x1 + [6]
                  d(x1) = [1] x1 + [9]
                  b(x1) = [1] x1 + [8]
                  c(x1) = [1] x1 + [3]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [8]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [5]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules: {b(b(x1)) -> a(g(g(x1)))}
              Weak Rules:
                {  b(b(b(x1))) -> c(d(c(x1)))
                 , c(d(x1)) -> g(g(x1))
                 , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
                 , g(g(g(x1))) -> b(b(x1))
                 , c^#(d(x1)) -> c_3(g^#(g(x1)))
                 , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
                 , a(x1) -> g(d(x1))}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules: {b(b(x1)) -> a(g(g(x1)))}
                Weak Rules:
                  {  b(b(b(x1))) -> c(d(c(x1)))
                   , c(d(x1)) -> g(g(x1))
                   , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
                   , g(g(g(x1))) -> b(b(x1))
                   , c^#(d(x1)) -> c_3(g^#(g(x1)))
                   , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
                   , a(x1) -> g(d(x1))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  g_0(3) -> 13
                 , d_0(3) -> 3
                 , g^#_0(3) -> 8
                 , g^#_0(13) -> 12
                 , b^#_0(3) -> 9
                 , c^#_0(3) -> 11
                 , c_3_0(12) -> 11}
      
   3) {  b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
       , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
       , c^#(d(x1)) -> c_3(g^#(g(x1)))
       , b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
      
      The usable rules for this path are the following:
      {  b(b(b(x1))) -> c(d(c(x1)))
       , b(b(x1)) -> a(g(g(x1)))
       , c(d(x1)) -> g(g(x1))
       , g(g(g(x1))) -> b(b(x1))
       , a(x1) -> g(d(x1))}
      
        We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost runtime-complexity with respect to
            Rules:
              {  b(b(b(x1))) -> c(d(c(x1)))
               , b(b(x1)) -> a(g(g(x1)))
               , c(d(x1)) -> g(g(x1))
               , g(g(g(x1))) -> b(b(x1))
               , a(x1) -> g(d(x1))
               , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
               , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
               , c^#(d(x1)) -> c_3(g^#(g(x1)))
               , b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a(x1) -> g(d(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(x1) -> g(d(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
            and weakly orienting the rules
            {a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c^#(d(x1)) -> c_3(g^#(g(x1)))}
            and weakly orienting the rules
            {  b^#(b(x1)) -> c_2(a^#(g(g(x1))))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c^#(d(x1)) -> c_3(g^#(g(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  c^#(x1) = [1] x1 + [1]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  g(g(g(x1))) -> b(b(x1))
             , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
            and weakly orienting the rules
            {  c^#(d(x1)) -> c_3(g^#(g(x1)))
             , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  g(g(g(x1))) -> b(b(x1))
               , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  g(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [13]
                  c_1(x1) = [1] x1 + [4]
                  c^#(x1) = [1] x1 + [8]
                  c_2(x1) = [1] x1 + [8]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
            and weakly orienting the rules
            {  g(g(g(x1))) -> b(b(x1))
             , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [9]
                  g(x1) = [1] x1 + [3]
                  d(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [2]
                  b^#(x1) = [1] x1 + [6]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [5]
                  c_2(x1) = [1] x1 + [0]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {c(d(x1)) -> g(g(x1))}
            and weakly orienting the rules
            {  g^#(g(g(x1))) -> c_4(b^#(b(x1)))
             , g(g(g(x1))) -> b(b(x1))
             , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {c(d(x1)) -> g(g(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [9]
                  g(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [0]
                  c(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [4]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [9]
                  b^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [1]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b(b(b(x1))) -> c(d(c(x1)))}
            and weakly orienting the rules
            {  c(d(x1)) -> g(g(x1))
             , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
             , g(g(g(x1))) -> b(b(x1))
             , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
             , c^#(d(x1)) -> c_3(g^#(g(x1)))
             , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
             , a(x1) -> g(d(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b(b(b(x1))) -> c(d(c(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [7]
                  g(x1) = [1] x1 + [6]
                  d(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [8]
                  c(x1) = [1] x1 + [11]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  g^#(x1) = [1] x1 + [5]
                  b^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [0]
                  c^#(x1) = [1] x1 + [13]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
                  c_4(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
            ------------------------------------------------------------------------------------------
            Answer:           YES(?,O(n^1))
            Input Problem:    innermost relative runtime-complexity with respect to
              Strict Rules: {b(b(x1)) -> a(g(g(x1)))}
              Weak Rules:
                {  b(b(b(x1))) -> c(d(c(x1)))
                 , c(d(x1)) -> g(g(x1))
                 , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
                 , g(g(g(x1))) -> b(b(x1))
                 , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
                 , c^#(d(x1)) -> c_3(g^#(g(x1)))
                 , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
                 , a(x1) -> g(d(x1))}
            
            Details:         
              The problem was solved by processor 'Bounds with default enrichment':
              'Bounds with default enrichment'
              --------------------------------
              Answer:           YES(?,O(n^1))
              Input Problem:    innermost relative runtime-complexity with respect to
                Strict Rules: {b(b(x1)) -> a(g(g(x1)))}
                Weak Rules:
                  {  b(b(b(x1))) -> c(d(c(x1)))
                   , c(d(x1)) -> g(g(x1))
                   , g^#(g(g(x1))) -> c_4(b^#(b(x1)))
                   , g(g(g(x1))) -> b(b(x1))
                   , b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
                   , c^#(d(x1)) -> c_3(g^#(g(x1)))
                   , b^#(b(x1)) -> c_2(a^#(g(g(x1))))
                   , a(x1) -> g(d(x1))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  g_0(3) -> 13
                 , d_0(3) -> 3
                 , a^#_0(3) -> 6
                 , g^#_0(3) -> 8
                 , g^#_0(13) -> 12
                 , b^#_0(3) -> 9
                 , c^#_0(3) -> 11
                 , c_3_0(12) -> 11}