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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  a^#(l(x1)) -> c_0(a^#(x1))
    , r^#(a(x1)) -> c_1(a^#(r(x1)))
    , b^#(l(x1)) -> c_2(b^#(a(r(x1))))
    , r^#(b(x1)) -> c_3(b^#(x1))}
  
  The usable rules are:
   {  a(l(x1)) -> l(a(x1))
    , r(a(x1)) -> a(r(x1))
    , r(b(x1)) -> l(b(x1))
    , b(l(x1)) -> b(a(r(x1)))}
  
  The estimated dependency graph contains the following edges:
   {a^#(l(x1)) -> c_0(a^#(x1))}
     ==> {a^#(l(x1)) -> c_0(a^#(x1))}
   {r^#(a(x1)) -> c_1(a^#(r(x1)))}
     ==> {a^#(l(x1)) -> c_0(a^#(x1))}
   {b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
     ==> {b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
   {r^#(b(x1)) -> c_3(b^#(x1))}
     ==> {b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
  
  We consider the following path(s):
   1) {  r^#(a(x1)) -> c_1(a^#(r(x1)))
       , a^#(l(x1)) -> c_0(a^#(x1))}
      
      The usable rules for this path are the following:
      {  r(a(x1)) -> a(r(x1))
       , r(b(x1)) -> l(b(x1))
       , a(l(x1)) -> l(a(x1))
       , b(l(x1)) -> b(a(r(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:
              {  r(a(x1)) -> a(r(x1))
               , r(b(x1)) -> l(b(x1))
               , a(l(x1)) -> l(a(x1))
               , b(l(x1)) -> b(a(r(x1)))
               , r^#(a(x1)) -> c_1(a^#(r(x1)))
               , a^#(l(x1)) -> c_0(a^#(x1))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a^#(l(x1)) -> c_0(a^#(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a^#(l(x1)) -> c_0(a^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  l(x1) = [1] x1 + [1]
                  r(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  r^#(x1) = [1] x1 + [1]
                  c_1(x1) = [1] x1 + [1]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {r^#(a(x1)) -> c_1(a^#(r(x1)))}
            and weakly orienting the rules
            {a^#(l(x1)) -> c_0(a^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {r^#(a(x1)) -> c_1(a^#(r(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  l(x1) = [1] x1 + [1]
                  r(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [0]
                  r^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {r(b(x1)) -> l(b(x1))}
            and weakly orienting the rules
            {  r^#(a(x1)) -> c_1(a^#(r(x1)))
             , a^#(l(x1)) -> c_0(a^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {r(b(x1)) -> l(b(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  l(x1) = [1] x1 + [0]
                  r(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [8]
                  a^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [0]
                  r^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [1]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] 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:
                {  r(a(x1)) -> a(r(x1))
                 , a(l(x1)) -> l(a(x1))
                 , b(l(x1)) -> b(a(r(x1)))}
              Weak Rules:
                {  r(b(x1)) -> l(b(x1))
                 , r^#(a(x1)) -> c_1(a^#(r(x1)))
                 , a^#(l(x1)) -> c_0(a^#(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:
                  {  r(a(x1)) -> a(r(x1))
                   , a(l(x1)) -> l(a(x1))
                   , b(l(x1)) -> b(a(r(x1)))}
                Weak Rules:
                  {  r(b(x1)) -> l(b(x1))
                   , r^#(a(x1)) -> c_1(a^#(r(x1)))
                   , a^#(l(x1)) -> c_0(a^#(x1))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  l_0(2) -> 2
                 , a^#_0(2) -> 5
                 , c_0_0(5) -> 5
                 , r^#_0(2) -> 7}
      
   2) {  r^#(b(x1)) -> c_3(b^#(x1))
       , b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
      
      The usable rules for this path are the following:
      {  a(l(x1)) -> l(a(x1))
       , r(a(x1)) -> a(r(x1))
       , r(b(x1)) -> l(b(x1))
       , b(l(x1)) -> b(a(r(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:
              {  a(l(x1)) -> l(a(x1))
               , r(a(x1)) -> a(r(x1))
               , r(b(x1)) -> l(b(x1))
               , b(l(x1)) -> b(a(r(x1)))
               , r^#(b(x1)) -> c_3(b^#(x1))
               , b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {r^#(b(x1)) -> c_3(b^#(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {r^#(b(x1)) -> c_3(b^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  l(x1) = [1] x1 + [0]
                  r(x1) = [1] x1 + [0]
                  b(x1) = [1] x1 + [8]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  r^#(x1) = [1] x1 + [1]
                  c_1(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [1]
                  c_2(x1) = [1] x1 + [1]
                  c_3(x1) = [1] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {r(b(x1)) -> l(b(x1))}
            and weakly orienting the rules
            {r^#(b(x1)) -> c_3(b^#(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {r(b(x1)) -> l(b(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  l(x1) = [1] x1 + [0]
                  r(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [0]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  r^#(x1) = [1] x1 + [10]
                  c_1(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [8]
                  c_2(x1) = [1] x1 + [2]
                  c_3(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:
                {  a(l(x1)) -> l(a(x1))
                 , r(a(x1)) -> a(r(x1))
                 , b(l(x1)) -> b(a(r(x1)))
                 , b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
              Weak Rules:
                {  r(b(x1)) -> l(b(x1))
                 , r^#(b(x1)) -> c_3(b^#(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:
                  {  a(l(x1)) -> l(a(x1))
                   , r(a(x1)) -> a(r(x1))
                   , b(l(x1)) -> b(a(r(x1)))
                   , b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
                Weak Rules:
                  {  r(b(x1)) -> l(b(x1))
                   , r^#(b(x1)) -> c_3(b^#(x1))}
              
              Details:         
                The problem is Match-bounded by 1.
                The enriched problem is compatible with the following automaton:
                {  a_1(5) -> 4
                 , l_0(2) -> 2
                 , r_1(2) -> 5
                 , r^#_0(2) -> 1
                 , b^#_0(2) -> 1
                 , b^#_1(4) -> 3
                 , c_2_1(3) -> 1}
      
   3) {r^#(a(x1)) -> c_1(a^#(r(x1)))}
      
      The usable rules for this path are the following:
      {  r(a(x1)) -> a(r(x1))
       , r(b(x1)) -> l(b(x1))
       , a(l(x1)) -> l(a(x1))
       , b(l(x1)) -> b(a(r(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:
              {  r(a(x1)) -> a(r(x1))
               , r(b(x1)) -> l(b(x1))
               , a(l(x1)) -> l(a(x1))
               , b(l(x1)) -> b(a(r(x1)))
               , r^#(a(x1)) -> c_1(a^#(r(x1)))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {r^#(a(x1)) -> c_1(a^#(r(x1)))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {r^#(a(x1)) -> c_1(a^#(r(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  l(x1) = [1] x1 + [1]
                  r(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  r^#(x1) = [1] x1 + [5]
                  c_1(x1) = [1] x1 + [0]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] x1 + [0]
              
            Finally we apply the subprocessor
            We apply the weight gap principle, strictly orienting the rules
            {b(l(x1)) -> b(a(r(x1)))}
            and weakly orienting the rules
            {r^#(a(x1)) -> c_1(a^#(r(x1)))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {b(l(x1)) -> b(a(r(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [0]
                  l(x1) = [1] x1 + [9]
                  r(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [0]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  r^#(x1) = [1] x1 + [9]
                  c_1(x1) = [1] x1 + [8]
                  b^#(x1) = [0] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [0] 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:
                {  r(a(x1)) -> a(r(x1))
                 , r(b(x1)) -> l(b(x1))
                 , a(l(x1)) -> l(a(x1))}
              Weak Rules:
                {  b(l(x1)) -> b(a(r(x1)))
                 , r^#(a(x1)) -> c_1(a^#(r(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:
                  {  r(a(x1)) -> a(r(x1))
                   , r(b(x1)) -> l(b(x1))
                   , a(l(x1)) -> l(a(x1))}
                Weak Rules:
                  {  b(l(x1)) -> b(a(r(x1)))
                   , r^#(a(x1)) -> c_1(a^#(r(x1)))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  l_0(2) -> 2
                 , a^#_0(2) -> 5
                 , r^#_0(2) -> 7}
      
   4) {r^#(b(x1)) -> c_3(b^#(x1))}
      
      The usable rules for this path are empty.
      
        We have oriented the usable rules with the following strongly linear interpretation:
          Interpretation Functions:
           a(x1) = [0] x1 + [0]
           l(x1) = [0] x1 + [0]
           r(x1) = [0] x1 + [0]
           b(x1) = [0] x1 + [0]
           a^#(x1) = [0] x1 + [0]
           c_0(x1) = [0] x1 + [0]
           r^#(x1) = [0] x1 + [0]
           c_1(x1) = [0] x1 + [0]
           b^#(x1) = [0] x1 + [0]
           c_2(x1) = [0] x1 + [0]
           c_3(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: {r^#(b(x1)) -> c_3(b^#(x1))}
            Weak Rules: {}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {r^#(b(x1)) -> c_3(b^#(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {r^#(b(x1)) -> c_3(b^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [0] x1 + [0]
                  l(x1) = [0] x1 + [0]
                  r(x1) = [0] x1 + [0]
                  b(x1) = [1] x1 + [0]
                  a^#(x1) = [0] x1 + [0]
                  c_0(x1) = [0] x1 + [0]
                  r^#(x1) = [1] x1 + [1]
                  c_1(x1) = [0] x1 + [0]
                  b^#(x1) = [1] x1 + [0]
                  c_2(x1) = [0] x1 + [0]
                  c_3(x1) = [1] 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: {r^#(b(x1)) -> c_3(b^#(x1))}
            
            Details:         
              The given problem does not contain any strict rules