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

Details:         
  We have computed the following set of weak (innermost) dependency pairs:
   {  a^#(b(x1)) -> c_0(r^#(x1))
    , r^#(a(x1)) -> c_1(d^#(r(x1)))
    , r^#(x1) -> c_2(d^#(x1))
    , d^#(a(x1)) -> c_3(a^#(a(d(x1))))
    , d^#(x1) -> c_4(a^#(x1))}
  
  The usable rules are:
   {  a(b(x1)) -> b(r(x1))
    , r(a(x1)) -> d(r(x1))
    , r(x1) -> d(x1)
    , d(a(x1)) -> a(a(d(x1)))
    , d(x1) -> a(x1)}
  
  The estimated dependency graph contains the following edges:
   {a^#(b(x1)) -> c_0(r^#(x1))}
     ==> {r^#(x1) -> c_2(d^#(x1))}
   {a^#(b(x1)) -> c_0(r^#(x1))}
     ==> {r^#(a(x1)) -> c_1(d^#(r(x1)))}
   {r^#(a(x1)) -> c_1(d^#(r(x1)))}
     ==> {d^#(x1) -> c_4(a^#(x1))}
   {r^#(a(x1)) -> c_1(d^#(r(x1)))}
     ==> {d^#(a(x1)) -> c_3(a^#(a(d(x1))))}
   {r^#(x1) -> c_2(d^#(x1))}
     ==> {d^#(x1) -> c_4(a^#(x1))}
   {r^#(x1) -> c_2(d^#(x1))}
     ==> {d^#(a(x1)) -> c_3(a^#(a(d(x1))))}
   {d^#(a(x1)) -> c_3(a^#(a(d(x1))))}
     ==> {a^#(b(x1)) -> c_0(r^#(x1))}
   {d^#(x1) -> c_4(a^#(x1))}
     ==> {a^#(b(x1)) -> c_0(r^#(x1))}
  
  We consider the following path(s):
   1) {  a^#(b(x1)) -> c_0(r^#(x1))
       , d^#(x1) -> c_4(a^#(x1))
       , r^#(x1) -> c_2(d^#(x1))
       , r^#(a(x1)) -> c_1(d^#(r(x1)))
       , d^#(a(x1)) -> c_3(a^#(a(d(x1))))}
      
      The usable rules for this path are the following:
      {  a(b(x1)) -> b(r(x1))
       , r(a(x1)) -> d(r(x1))
       , r(x1) -> d(x1)
       , d(a(x1)) -> a(a(d(x1)))
       , d(x1) -> a(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(b(x1)) -> b(r(x1))
               , r(a(x1)) -> d(r(x1))
               , r(x1) -> d(x1)
               , d(a(x1)) -> a(a(d(x1)))
               , d(x1) -> a(x1)
               , a^#(b(x1)) -> c_0(r^#(x1))
               , d^#(x1) -> c_4(a^#(x1))
               , r^#(x1) -> c_2(d^#(x1))
               , r^#(a(x1)) -> c_1(d^#(r(x1)))
               , d^#(a(x1)) -> c_3(a^#(a(d(x1))))}
          
          Details:         
            We apply the weight gap principle, strictly orienting the rules
            {a(b(x1)) -> b(r(x1))}
            and weakly orienting the rules
            {}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {a(b(x1)) -> b(r(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [1]
                  b(x1) = [1] x1 + [0]
                  r(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [1]
                  c_0(x1) = [1] x1 + [1]
                  r^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [1]
                  d^#(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
            {  r(a(x1)) -> d(r(x1))
             , a^#(b(x1)) -> c_0(r^#(x1))
             , r^#(a(x1)) -> c_1(d^#(r(x1)))}
            and weakly orienting the rules
            {a(b(x1)) -> b(r(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  r(a(x1)) -> d(r(x1))
               , a^#(b(x1)) -> c_0(r^#(x1))
               , r^#(a(x1)) -> c_1(d^#(r(x1)))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [9]
                  r(x1) = [1] x1 + [1]
                  d(x1) = [1] x1 + [7]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  r^#(x1) = [1] x1 + [0]
                  c_1(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  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
            {r^#(x1) -> c_2(d^#(x1))}
            and weakly orienting the rules
            {  r(a(x1)) -> d(r(x1))
             , a^#(b(x1)) -> c_0(r^#(x1))
             , r^#(a(x1)) -> c_1(d^#(r(x1)))
             , a(b(x1)) -> b(r(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {r^#(x1) -> c_2(d^#(x1))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [13]
                  r(x1) = [1] x1 + [0]
                  d(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  r^#(x1) = [1] x1 + [8]
                  c_1(x1) = [1] x1 + [1]
                  d^#(x1) = [1] x1 + [0]
                  c_2(x1) = [1] x1 + [1]
                  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
            {r(x1) -> d(x1)}
            and weakly orienting the rules
            {  r^#(x1) -> c_2(d^#(x1))
             , r(a(x1)) -> d(r(x1))
             , a^#(b(x1)) -> c_0(r^#(x1))
             , r^#(a(x1)) -> c_1(d^#(r(x1)))
             , a(b(x1)) -> b(r(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {r(x1) -> d(x1)}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [8]
                  r(x1) = [1] x1 + [8]
                  d(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [8]
                  c_0(x1) = [1] x1 + [12]
                  r^#(x1) = [1] x1 + [4]
                  c_1(x1) = [1] x1 + [3]
                  d^#(x1) = [1] x1 + [1]
                  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
            {  d^#(x1) -> c_4(a^#(x1))
             , d^#(a(x1)) -> c_3(a^#(a(d(x1))))}
            and weakly orienting the rules
            {  r(x1) -> d(x1)
             , r^#(x1) -> c_2(d^#(x1))
             , r(a(x1)) -> d(r(x1))
             , a^#(b(x1)) -> c_0(r^#(x1))
             , r^#(a(x1)) -> c_1(d^#(r(x1)))
             , a(b(x1)) -> b(r(x1))}
            using the following strongly linear interpretation:
              Processor 'Matrix Interpretation' oriented the following rules strictly:
              
              {  d^#(x1) -> c_4(a^#(x1))
               , d^#(a(x1)) -> c_3(a^#(a(d(x1))))}
              
              Details:
                 Interpretation Functions:
                  a(x1) = [1] x1 + [8]
                  b(x1) = [1] x1 + [15]
                  r(x1) = [1] x1 + [8]
                  d(x1) = [1] x1 + [1]
                  a^#(x1) = [1] x1 + [0]
                  c_0(x1) = [1] x1 + [1]
                  r^#(x1) = [1] x1 + [12]
                  c_1(x1) = [1] x1 + [0]
                  d^#(x1) = [1] x1 + [9]
                  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:
                {  d(a(x1)) -> a(a(d(x1)))
                 , d(x1) -> a(x1)}
              Weak Rules:
                {  d^#(x1) -> c_4(a^#(x1))
                 , d^#(a(x1)) -> c_3(a^#(a(d(x1))))
                 , r(x1) -> d(x1)
                 , r^#(x1) -> c_2(d^#(x1))
                 , r(a(x1)) -> d(r(x1))
                 , a^#(b(x1)) -> c_0(r^#(x1))
                 , r^#(a(x1)) -> c_1(d^#(r(x1)))
                 , a(b(x1)) -> b(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:
                  {  d(a(x1)) -> a(a(d(x1)))
                   , d(x1) -> a(x1)}
                Weak Rules:
                  {  d^#(x1) -> c_4(a^#(x1))
                   , d^#(a(x1)) -> c_3(a^#(a(d(x1))))
                   , r(x1) -> d(x1)
                   , r^#(x1) -> c_2(d^#(x1))
                   , r(a(x1)) -> d(r(x1))
                   , a^#(b(x1)) -> c_0(r^#(x1))
                   , r^#(a(x1)) -> c_1(d^#(r(x1)))
                   , a(b(x1)) -> b(r(x1))}
              
              Details:         
                The problem is Match-bounded by 0.
                The enriched problem is compatible with the following automaton:
                {  b_0(2) -> 2
                 , a^#_0(2) -> 5
                 , c_0_0(7) -> 5
                 , r^#_0(2) -> 7
                 , d^#_0(2) -> 9
                 , c_2_0(9) -> 7
                 , c_4_0(5) -> 9}