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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  w^#(r(x)) -> c_0(w^#(x))
    , b^#(r(x)) -> c_1(b^#(x))
    , b^#(w(x)) -> c_2(w^#(b(x)))}
  
  The usable rules are:
   {  b(r(x)) -> r(b(x))
    , b(w(x)) -> w(b(x))
    , w(r(x)) -> r(w(x))}
  
  The estimated dependency graph contains the following edges:
   {w^#(r(x)) -> c_0(w^#(x))}
     ==> {w^#(r(x)) -> c_0(w^#(x))}
   {b^#(r(x)) -> c_1(b^#(x))}
     ==> {b^#(w(x)) -> c_2(w^#(b(x)))}
   {b^#(r(x)) -> c_1(b^#(x))}
     ==> {b^#(r(x)) -> c_1(b^#(x))}
   {b^#(w(x)) -> c_2(w^#(b(x)))}
     ==> {w^#(r(x)) -> c_0(w^#(x))}
  
  We consider the following path(s):
   1) {  b^#(r(x)) -> c_1(b^#(x))
       , b^#(w(x)) -> c_2(w^#(b(x)))
       , w^#(r(x)) -> c_0(w^#(x))}
      
      The usable rules for this path are the following:
      {  b(r(x)) -> r(b(x))
       , b(w(x)) -> w(b(x))
       , w(r(x)) -> r(w(x))}
      
        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(r(x)) -> r(b(x))
               , b(w(x)) -> w(b(x))
               , w(r(x)) -> r(w(x))
               , b^#(w(x)) -> c_2(w^#(b(x)))
               , b^#(r(x)) -> c_1(b^#(x))
               , w^#(r(x)) -> c_0(w^#(x))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {b^#(w(x)) -> c_2(w^#(b(x)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(w(x)) -> c_2(w^#(b(x)))}
              
              Details:
                 Interpretation Functions:
                  w(x1) = [1] x1 + [0]
                  r(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  w^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  b^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [8]
                  c_2(x1) = [1] x1 + [1]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {  b^#(r(x)) -> c_1(b^#(x))
             , w^#(r(x)) -> c_0(w^#(x))}
            and weakly orienting the rules
            {b^#(w(x)) -> c_2(w^#(b(x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  b^#(r(x)) -> c_1(b^#(x))
               , w^#(r(x)) -> c_0(w^#(x))}
              
              Details:
                 Interpretation Functions:
                  w(x1) = [1] x1 + [0]
                  r(x1) = [1] x1 + [6]
                  b(x1) = [1] x1 + [0]
                  w^#(x1) = [1] x1 + [4]
                  c_0(x1) = [1] x1 + [1]
                  b^#(x1) = [1] x1 + [4]
                  c_1(x1) = [1] x1 + [0]
                  c_2(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(r(x)) -> r(b(x))
                 , b(w(x)) -> w(b(x))
                 , w(r(x)) -> r(w(x))}
              Weak Rules:
                {  b^#(r(x)) -> c_1(b^#(x))
                 , w^#(r(x)) -> c_0(w^#(x))
                 , b^#(w(x)) -> c_2(w^#(b(x)))}
            
            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(r(x)) -> r(b(x))
                   , b(w(x)) -> w(b(x))
                   , w(r(x)) -> r(w(x))}
                Weak Rules:
                  {  b^#(r(x)) -> c_1(b^#(x))
                   , w^#(r(x)) -> c_0(w^#(x))
                   , b^#(w(x)) -> c_2(w^#(b(x)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  r_0(2) -> 2
                 , w^#_0(2) -> 4
                 , c_0_0(4) -> 4
                 , b^#_0(2) -> 6
                 , c_1_0(6) -> 6}
      
   2) {  b^#(r(x)) -> c_1(b^#(x))
       , b^#(w(x)) -> c_2(w^#(b(x)))}
      
      The usable rules for this path are the following:
      {  b(r(x)) -> r(b(x))
       , b(w(x)) -> w(b(x))
       , w(r(x)) -> r(w(x))}
      
        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(r(x)) -> r(b(x))
               , b(w(x)) -> w(b(x))
               , w(r(x)) -> r(w(x))
               , b^#(r(x)) -> c_1(b^#(x))
               , b^#(w(x)) -> c_2(w^#(b(x)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {b^#(w(x)) -> c_2(w^#(b(x)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(w(x)) -> c_2(w^#(b(x)))}
              
              Details:
                 Interpretation Functions:
                  w(x1) = [1] x1 + [0]
                  r(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [1]
                  w^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [5]
                  c_1(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b^#(r(x)) -> c_1(b^#(x))}
            and weakly orienting the rules
            {b^#(w(x)) -> c_2(w^#(b(x)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(r(x)) -> c_1(b^#(x))}
              
              Details:
                 Interpretation Functions:
                  w(x1) = [1] x1 + [8]
                  r(x1) = [1] x1 + [4]
                  b(x1) = [1] x1 + [1]
                  w^#(x1) = [1] x1 + [3]
                  c_0(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  c_2(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(r(x)) -> r(b(x))
                 , b(w(x)) -> w(b(x))
                 , w(r(x)) -> r(w(x))}
              Weak Rules:
                {  b^#(r(x)) -> c_1(b^#(x))
                 , b^#(w(x)) -> c_2(w^#(b(x)))}
            
            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(r(x)) -> r(b(x))
                   , b(w(x)) -> w(b(x))
                   , w(r(x)) -> r(w(x))}
                Weak Rules:
                  {  b^#(r(x)) -> c_1(b^#(x))
                   , b^#(w(x)) -> c_2(w^#(b(x)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  r_0(2) -> 2
                 , w^#_0(2) -> 4
                 , b^#_0(2) -> 6
                 , c_1_0(6) -> 6}
      
   3) {b^#(r(x)) -> c_1(b^#(x))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           w(x1) = [0] x1 + [0]
           r(x1) = [0] x1 + [0]
           b(x1) = [0] x1 + [0]
           w^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           b^#(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           c_2(x1) = [0] x1 + [0]
        
        We have applied the subprocessor on the resulting DP-problem:
        
          'Weight Gap Principle'
          ----------------------
          Answer:           YES(?,O(n^1))
          Input Problem:    innermost DP runtime-complexity with respect to
            Strict Rules: {b^#(r(x)) -> c_1(b^#(x))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {b^#(r(x)) -> c_1(b^#(x))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b^#(r(x)) -> c_1(b^#(x))}
              
              Details:
                 Interpretation Functions:
                  w(x1) = [0] x1 + [0]
                  r(x1) = [1] x1 + [8]
                  b(x1) = [0] x1 + [0]
                  w^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [3]
                  c_2(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            'Empty TRS'
            -----------
            Answer:           YES(?,O(1))
            Input Problem:    innermost DP runtime-complexity with respect to
              Strict Rules: {}
              Weak Rules: {b^#(r(x)) -> c_1(b^#(x))}
            
            Details:         
              The given problem does not contain any strict rules